%! /TeXDict 200 dict def TeXDict begin /Mtrx matrix def /@start { /StartTime usertime def vmstatus pop /@VMused exch def pop 72 720 translate } def /@letter { /letter where { pop letter 72 720 translate } if } def /@legal { /legal where { pop legal 72 936 translate } if } def /@note { /note where { pop note 72 720 translate } if } def /@a4 { /a4 where { pop a4 72 765 translate } if } def /@landscape { [ 0 1 -1 0 0 0 ] concat -648 0 translate } def /@manualfeed { /manualfeed where { /manualfeed true put } if } def /@duplex { /setduplexmode where { /setduplexmode true put } if } def /@copies { /#copies exch def } def /@startdoc { dup 72 exch div dup neg scale 0 0 transform round exch round exch itransform translate Mtrx currentmatrix pop currentdict /isoddpage known { /SecondMtrx matrix def 8.5 exch mul 0 translate SecondMtrx currentmatrix pop }{ pop } ifelse /@TeXSave save def } def /@restore /restore load def /restore { vmstatus pop dup @VMused lt {pop @VMused} if exch pop exch @restore /@VMused exch def } bind def /@pri { ( ) print ( ) cvs print }def /@newfont { /newname exch def newname 7 dict def newname load begin /FontType 3 def dup neg matrix scale /FontMatrix exch def /FontBBox [0 0 1 1] def /BitMaps 256 array def /BuildChar {CharBuilder} def /Encoding 256 array def 0 1 255 {Encoding exch /.undef put} for end newname newname load definefont pop } bind def /ch-image {ch-data 0 get} bind def /ch-width {ch-data 1 get} bind def /ch-height {ch-data 2 get} bind def /ch-xoff {ch-data 3 get} bind def /ch-yoff {ch-data 4 get} bind def /ch-tfmw {ch-data 5 get} bind def /CharBuilder { exch /BitMaps get exch get /ch-data exch def ch-data null ne { ch-tfmw 0 ch-xoff neg ch-yoff neg ch-width ch-xoff sub ch-height ch-yoff sub setcachedevice ch-width ch-height true [1 0 0 1 ch-xoff ch-yoff] {ch-image} imagemask } if } bind def /@dc{ /ch-code exch def dup 0 get length 2 lt { pop [ <00> 1 1 0 0 8.00 ] } if /ch-data exch def currentfont /BitMaps get ch-code ch-data put currentfont /Encoding get ch-code dup ( ) cvs cvn put } bind def /@pc { pop /ch-data exch def currentpoint translate ch-width ch-height true [1 0 0 -1 ch-xoff ch-yoff ] {ch-image} imagemask } bind def /@sf { setfont } def /@flushfonts { @TeXSave restore /@TeXSave save def} def /@bop0 { pop } bind def /@bop1 { pop Mtrx setmatrix /SaveImage save def } def /@eop { showpage SaveImage restore } def /p { moveto } bind def /r { 0 rmoveto } bind def /s { show pop } bind def /ks { 2 1 counttomark 2 sub { -1 roll } for counttomark 2 idiv { show 0 rmoveto } repeat show pop } bind def /m { mark } bind def /ru { /dy exch 1 sub neg def /dx exch 1 sub def gsave currentpoint newpath moveto dx 0 rlineto 0 dy rlineto dx neg 0 rlineto closepath fill grestore } bind def /@end { (Time used: ) print usertime StartTime sub @pri (VM used: ) print @VMused @pri (. Unused: ) print vmstatus @VMused sub @pri pop pop (\n) print flush end } def /DraftFirst { gsave initmatrix newpath /Helvetica-Bold findfont setfont (Draft) dup stringwidth pop 8.875 exch div dup 72 mul dup scale 52.3 rotate 2.5 exch div -.35 translate 0.95 setgray 0 0 moveto show grestore } bind def /DraftMore { gsave initmatrix newpath /Helvetica-Bold findfont 72 scalefont setfont 0.95 setgray 576 702 moveto (Draft) stringwidth exch neg exch neg rmoveto (Draft) show grestore } bind def /@draft { /@bop1 { 1 le {DraftFirst} {DraftMore} ifelse Mtrx setmatrix /SaveImage save def } bind def } bind def /@twoup { initmatrix [ 0 1 -1 0 0 0 ] concat 36 -87 translate 5.5 8.5 div dup scale /isoddpage true def /@eop { isoddpage not {showpage} if SaveImage restore /isoddpage isoddpage not def } def /@bop1 { pop isoddpage { SecondMtrx setmatrix } { Mtrx setmatrix } ifelse /SaveImage save def } def /@end { isoddpage not { showpage } if end } def } def /@beginspecial { SpecialDict begin gsave /SpecialSave save def currentpoint transform initgraphics itransform translate 0 0 moveto SpecialDefaults IfMacSetUp } def /@setspecial { hof vof translate setclipper hsc vsc scale angl rotate } def /@endspecial { SpecialSave restore grestore end } def /@beginfullpage { gsave /SpecialSave save def SpecialDict begin /showpage {} def initgraphics } def /@endfullpage { clear end SpecialSave restore grestore showpage } def /SpecialDict 50 dict def SpecialDict begin /Inch { 72 mul} bind def /SpecialDefaults { /hsi 8.5 Inch def /vsi 11 Inch def /hof 0 def /vof 0 def /hsc 1 def /vsc 1 def /angl 0 def /CLIP false def } def /@hsize {/hsi exch def /CLIP true def} def /@vsize {/vsi exch def /CLIP true def} def /@hoffset {/hof exch def} def /@voffset {/vof exch def} def /@hscale {/hsc exch def} def /@vscale {/vsc exch def} def /@rotate {/angl exch def} def /setclipper { CLIP { newpath 0 0 moveto hsi 0 rlineto 0 vsi rlineto hsi neg 0 rlineto closepath clip } if } def end /IfMacSetUp { userdict /md known { userdict /md get type /dicttype eq { md begin /txpose{ 1 -1 scale } bind def /cp {pop pop} bind def end } if } if } def end /TeXDict where {/TeXDict get begin} { (Need to load basic TeXDict first\n) print currentfile flushfile} ifelse /@texmakefont { /TeXsize exch def [ TeXsize 0 0 TeXsize neg 0 0 ] makefont def } bind def /@ps2texfont { dup FontDirectory exch known { exch pop findfont } { ReEncodeForTeX } ifelse exch @texmakefont } bind def /ReEncodeForTeX { /newfontname exch def /basefontname exch def /TeXstr 30 string def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall basefontdict /MathFontName known not { /TeXvec basefontname TeXstr cvs (Courier) search { pop pop pop TeXcourvec } { pop TeXnormalvec } ifelse def TeXvec aload pop TeXvec length 2 idiv { newfont /Encoding get 3 1 roll put } repeat } if newfontname newfont definefont } bind def /TeXnormalvec [ 8#014 /fi 8#015 /fl 8#020 /dotlessi 8#022 /grave 8#023 /acute 8#024 /caron 8#025 /breve 8#026 /macron 8#027 /ring 8#030 /cedilla 8#031 /germandbls 8#032 /ae 8#033 /oe 8#034 /oslash 8#035 /AE 8#036 /OE 8#037 /Oslash 8#042 /quotedblright 8#074 /exclamdown 8#076 /questiondown 8#134 /quotedblleft 8#136 /circumflex 8#137 /dotaccent 8#173 /endash 8#174 /emdash 8#175 /hungarumlat 8#176 /tilde 8#177 /dieresis ] def /TeXcourvec [ 8#016 /exclamdown 8#017 /questiondown 8#020 /dotlessi 8#022 /grave 8#023 /acute 8#024 /caron 8#025 /breve 8#026 /macron 8#027 /ring 8#030 /cedilla 8#031 /germandbls 8#032 /ae 8#033 /oe 8#034 /oslash 8#035 /AE 8#036 /OE 8#037 /Oslash 8#074 /less 8#076 /greater 8#134 /backslash 8#136 /circumflex 8#137 /underscore 8#173 /braceleft 8#174 /bar 8#175 /braceright 8#177 /dieresis ] def /ObliqueFont { /ObliqueAngle exch def /ObliqueBaseName exch def /ObliqueFontName exch def /ObliqueTransform [1 0 ObliqueAngle sin ObliqueAngle cos div 1 0 0] def /basefontdict ObliqueBaseName findfont ObliqueTransform makefont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName ObliqueFontName put ObliqueFontName newfont definefont pop } bind def /SmallCapsFont { /SmallCapsBaseName exch def /SmallCapsFontName exch def /basefontdict SmallCapsBaseName findfont def /newfont basefontdict maxlength 10 add dict def newfont begin /FontName SmallCapsFontName def /FontBaseDict basefontdict 1000 scalefont def /FontSCDict FontBaseDict .8 scalefont def /FontType 3 def /BuildChar {SmallCapChar} def /FontMatrix dup basefontdict exch get def /FontBBox dup basefontdict exch get def /Encoding dup basefontdict exch get def end SmallCapsFontName newfont definefont pop } bind def /SmallCapChar { /achar (A) def achar exch 0 exch put begin achar 0 get dup 97 ge exch 122 le and { achar 0 achar 0 get 32 sub put FontSCDict } { FontBaseDict } ifelse setfont achar stringwidth FontBBox setcachedevice newpath 0 0 moveto achar show end } bind def /MathEncodingVec [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon1 /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon /zeta /nu /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /omega1 /rho /sigma /phi1 /arrowleft /arrowleft /arrowright /arrowright /suchthat /congruent /proportional /angle /zero /one /two /three /four /five /six /seven /eight /nine /period /comma /less /slash /greater /asteriskmath /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /registerserif /copyrightserif /trademarkserif /logicalor /logicaland /section /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /j /weierstrass /macron /breve ] def /MathCharVec [ 71 68 81 76 88 80 83 161 70 89 87 97 98 103 100 101 122 110 113 105 107 108 109 110 120 112 114 115 116 117 102 99 121 119 101 74 118 114 115 106 172 172 174 174 39 64 181 208 48 49 50 51 52 53 54 55 56 57 46 44 60 47 62 42 182 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 210 211 212 218 217 167 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 245 106 195 197 198 ] def /MathDictVec [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 ] def /MathOblique { 3 copy ObliqueFont pop pop MathFont } bind def /MathFont { /MathSymName exch def /MathBaseName exch def /MathFontName exch def /basefontdict MathBaseName findfont def /newfont basefontdict maxlength 10 add dict def newfont begin /FontName MathFontName def /FontDictArray [ basefontdict 1000 scalefont MathSymName findfont 950 scalefont /Symbol findfont 950 scalefont ] def /FontType 3 def /BuildChar {MathFontChar} def /FontMatrix dup basefontdict exch get def /FontBBox dup basefontdict exch get def /Encoding dup basefontdict exch get dup length array copy def /WhichFont MathDictVec def /WhichChar MathCharVec def MathEncodingVec Encoding copy pop /achar (A) def end MathFontName newfont definefont pop } bind def /MathFontChar { exch begin achar exch 0 exch put FontDictArray WhichFont achar 0 get get get setfont achar dup 0 get WhichChar exch get 0 exch put achar stringwidth FontBBox setcachedevice newpath 0 0 moveto achar show end } bind def end /psf$TeXscale { 65536 div } def /DocumentInitState [ matrix currentmatrix currentlinewidth currentlinecap currentlinejoin currentdash currentgray currentmiterlimit ] cvx def /startTexFig { /psf$SavedState save def userdict maxlength dict begin currentpoint transform DocumentInitState setmiterlimit setgray setdash setlinejoin setlinecap setlinewidth setmatrix itransform moveto /psf$ury exch psf$TeXscale def /psf$urx exch psf$TeXscale def /psf$lly exch psf$TeXscale def /psf$llx exch psf$TeXscale def /psf$y exch psf$TeXscale def /psf$x exch psf$TeXscale def currentpoint /psf$cy exch def /psf$cx exch def /psf$sx psf$x psf$urx psf$llx sub div def /psf$sy psf$y psf$ury psf$lly sub div def psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub translate /DefFigCTM matrix currentmatrix def /initmatrix { DefFigCTM setmatrix } def /defaultmatrix { DefFigCTM exch copy } def /initgraphics { DocumentInitState setmiterlimit setgray setdash setlinejoin setlinecap setlinewidth setmatrix DefFigCTM setmatrix } def /showpage { initgraphics } def /erasepage { initgraphics } def /copypage {} def } bind def /clipFig { currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll exch lineto exch lineto exch lineto closepath clip newpath moveto } bind def /doclip { psf$llx 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ks 168 648 p m (S.) 22 (Hammarling,) 23 (A.) 22 (McKenney) -3 (,) 24 (and) 23 (D.) 21 (Sorensen.) 41 (LAP) -3 (A) 0 (CK:) 21 (A) 22 (p) 1 (ortable) 23 (linear) 23 (algebra) ks 168 731 p m (library) 15 (for) 14 (high-p) 1 (erformance) 16 (computers.) 18 (In) ks cmti10.329 @sf 16 r m (Sup) -1 (er) -2 (c) -2 (omputer) 15 (90) ks cmr10.329 @sf m (,) 15 (New) 14 (Y) -3 (ork,) 13 (NY,) 15 (1990.) 13 (IEEE) ks 168 814 p m (Press.) s 97 934 p m ([2]) s 168 934 p m (Z.) 12 (Bai,) 13 (J.) 12 (Demmel,) 14 (and) 12 (A.) 13 (McKenney) -3 (.) 16 (On) 13 (the) 13 (conditioning) 14 (of) 12 (the) 13 (nonsymmetric) 13 (eigenprob-) ks 168 1017 p m (lem:) 19 (Theory) 14 (and) 14 (soft) 0 (w) -2 (are.) 15 (Computer) 13 (Science) 16 (Dept.) 13 (T) -3 (ec) 0 (hnical) 14 (Rep) 1 (ort) 14 (CS-89-86,) 12 (Univ) 0 (ersit) 0 (y) ks 168 1100 p m (of) 14 (T) -3 (ennessee,) 15 (Kno) 0 (xville,) 16 (TN,) 15 (Octob) 1 (er) 16 (1990.) 18 (\(LAP) -3 (A) 0 (CK) 14 (W) -3 (orking) 14 (Note) 15 (#13\).) ks 97 1221 p m ([3]) s 168 1221 p m (C.) 18 (Ba) 0 (v) -1 (ely) 18 (and) 20 (G.W.) 17 (Stew) 0 (art.) 30 (An) 19 (algorithm) 19 (for) 19 (computing) 19 (reducing) 21 (subspaces) 19 (b) 0 (y) 18 (blo) 1 (c) 0 (k) ks 168 1304 p m (diagonalization.) s cmti10.329 @sf 21 r m (SIAM) 15 (J.) 16 (Numer.) 16 (A) 0 (nal.) ks cmr10.329 @sf m (,) 13 (16:359{367) -1 (,) 13 (1979.) ks 97 1424 p m ([4]) s 168 1424 p m (James) 16 (Demmel.) 24 (Three) 17 (metho) 1 (ds) 16 (for) 16 (re\014ning) 18 (estimates) 16 (of) 16 (in) 0 (v) -2 (arian) 0 (t) 14 (subspaces.) ks cmti10.329 @sf 24 r m (Computing) s cmr10.329 @sf m (,) s 168 1507 p m (38:43{57) -1 (,) 13 (1987.) ks 97 1627 p m ([5]) s 168 1627 p m (B.) 13 (Kagstrom) 12 (and) 14 (A.) 14 (Ruhe.) 18 (Algorithm) 14 (560:) 18 (An) 14 (algorithm) 14 (for) 13 (n) 0 (umerical) 14 (computation) 14 (of) 13 (the) ks 168 1710 p m (jordan) 17 (normal) 19 (form) 17 (of) 18 (a) 18 (complex) 19 (matrix.) ks cmti10.329 @sf 28 r m (A) 0 (CM) 17 (T) -2 (r) -2 (an) -1 (sactions) 17 (on) 18 (Mathematic) -1 (al) 18 (Softwar) -1 (e) ks cmr10.329 @sf m (,) s 168 1793 p m (6:398{41) -1 (9,) 13 (1980.) ks 97 1914 p m ([6]) s 168 1914 p m (B.) 15 (Kagstrom) 14 (and) 15 (A.) 15 (Ruhe.) 22 (An) 16 (algorithm) 15 (for) 15 (n) 0 (umerical) 15 (computation) 16 (of) 15 (the) 15 (jordan) 15 (normal) ks 168 1997 p m (form) 13 (of) 15 (a) 14 (complex) 15 (matrix.) ks cmti10.329 @sf 19 r m (A) 0 (CM) 14 (T) -2 (r) -3 (ansactions) 13 (on) 16 (Mathematic) -1 (al) 14 (Softwar) -1 (e) ks cmr10.329 @sf m (,) 14 (6:437{443) -1 (,) 13 (1980.) ks 97 2117 p m ([7]) s 168 2117 p m (K.C.) 12 (Ng) 13 (and) 13 (B.N.) 13 (P) 0 (arlett.) 15 (Programs) 11 (to) 13 (sw) 0 (ap) 11 (diagonal) 14 (blo) 1 (c) 0 (ks.) 16 (Cen) 0 (ter) 12 (for) 12 (Pure) 14 (and) 13 (Applied) ks 168 2200 p m (Math.) 14 (Tec) 0 (hnical) 15 (Rep) 1 (ort) 16 (381,) 14 (Univ) 0 (ersit) 0 (y) 14 (of) 15 (California,) 15 (Berk) 0 (eley) -3 (,) 14 (Berk) 0 (eley) -3 (,) 14 (CA,) 15 (1988.) ks 97 2321 p m ([8]) s 168 2321 p m 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001C00003C00003C0000380000780000700000700000E00000E00001C00001C000038000038000 0700000700000E00000C00001C0000180000300000700000600000C00000> 22 75 -1 74 36] 49 @dc [ 5 27 -18 26 36] 67 @dc [ 22 75 -1 72 36] 65 @dc cmmi10.329 @sf [<7FFFC00001F0000000F0000000F0000000F0000000F00000007800000078000000780000007800 00003C0000003C0000003C0000003C0000001E0000001E0000001E0000001E0000000F0000000F 0000000F0000800F0010400780084007800860078008200780082003C0081003C0081803C0181E 03C0380FFFFFFC> 30 31 -1 0 27] 84 @dc [<0F00188038403820382038201C101C001C001C000E000E000E000E0007000700070007000380FF F80380038001C001C001C001C000E000C0> 13 28 -1 0 16] 116 @dc [<0F0780308C40705C40703C20F01C20F01C20F00E10F00E00F00E00F00E00780700780700780700 3807003C03801C03800E03800707800389C000F1C00001C00001C00000E00000E00000E00000E0 0000700000700000700000700003F8000078> 21 32 -2 0 24] 100 @dc [<70F8F8F870> 5 5 -4 0 13] 58 @dc cmr10.329 @sf [<40202010101008080878F8F8F0700000000000000000000070F8F8F870> 5 29 -4 9 13] 59 @dc [<7FE3FE3FF007007007000700700700070070070007007007000700700700070070070007007007 000700700700070070070007007007000700700700070070070007007007000700700700070070 0700070070070007007007000700700700FFFFFFFF000700700000070070000007007000000700 700000070070000007007000000700F00F000300F00F000380F80F0001C07C060000F04F040000 1F81F800> 36 32 0 0 38] 14 @dc 1.000000 /cmmi8.300 @newfont cmmi8.300 @sf [<1C0032003200310031001800180018000C008C004600460024001C000000000000000000000000 00030003800300> 9 23 -1 0 12] 105 @dc cmbx10.432 @sf [<00FF800003FFF0000FFFF8001F01FE003C007F0078003F8078003F80FC001FC0FE001FC0FE001F E0FE001FE0FE001FE07C001FE018001FE000001FE000001FE000001FC000001FC000001F800C00 3F000E003E000F80FC000FFFF8000E7FC0000E0000000E0000000E0000000E0000000E0000000E 0000000FFE00000FFFC0000FFFE0000FFFF0000FFFF8000FFFFC000FFFFE000F803F000C000300> 27 39 -3 0 34] 53 @dc [<00003FF800000003FFFF0000000FFFFFC000003FF007E00000FF8000F80001FE00003C0003FC00 001E0007F000000E000FF0000007001FE0000007001FE0000003803FC0000003803FC000000380 7FC0000003807F80000000007F8000000000FF8000000000FF8000000000FF8000000000FF8000 000000FF8000000000FF8000000000FF8000000000FF8000000000FF80000000007F8000000000 7F80000003807FC0000003803FC0000003803FC0000007801FE0000007801FE0000007800FF000 000F8007F000001F8003FC00003F8001FE00007F8000FF8001FF80003FF007DF80000FFFFF8780 0003FFFE038000003FF00180> 41 41 -4 0 50] 67 @dc [<003FE00001FFFC0007F07F000FC01F801F800FC03F800FE03F800FE07F0007F07F0007F0FF0007 F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F87F0007F07F0007F07F00 07F03F0007E03F800FE01F800FC00FC01F8003F07E0001FFFC00003FE000> 29 27 -2 0 34] 111 @dc [ 15 42 -3 0 20] 108 @dc [<003FC3FF8001FFF3FF8003F03BFF8007E00FF80007E007F8000FE007F8000FE003F8000FE003F8 000FE003F8000FE003F8000FE003F8000FE003F8000FE003F8000FE003F8000FE003F8000FE003 F8000FE003F8000FE003F8000FE003F8000FE003F8000FE003F8000FE003F8000FE003F8000FE0 03F800FFE03FF800FFE03FF800FFE03FF800> 33 27 -3 0 38] 117 @dc [ 15 43 -3 0 20] 105 @dc 22 @bop1 cmr10.329 @sf 124 67 p m (F) -3 (or) 10 (example,) 13 (supp) 1 (ose) 13 (w) 0 (e) 11 (group) 11 (\() ks cmmi10.329 @sf 765 67 p m (\025) s cmr8.300 @sf 792 74 p m (9) s cmmi10.329 @sf 811 67 p m (;) 15 (\025) ks cmr8.300 @sf 866 74 p m (8) s cmr10.329 @sf 885 67 p m (\)) s 903 67 p m (,) s cmmi10.329 @sf 12 r m (\025) s cmr8.300 @sf 955 74 p m (6) s cmr10.329 @sf 974 67 p m (,) 13 (\() ks cmmi10.329 @sf 1017 67 p m (\025) s cmr8.300 @sf 1044 74 p m (4) s cmmi10.329 @sf 1063 67 p m (;) 15 (\025) ks cmr8.300 @sf 1118 74 p m (3) s cmr10.329 @sf 1137 67 p m (\)) s 1155 67 p m (,) 12 (where) 12 (\() ks cmmi10.329 @sf 1326 67 p m (\025) s cmr8.300 @sf 1353 74 p m (9) s cmmi10.329 @sf 1372 67 p m (;) 15 (\025) ks cmr8.300 @sf 1427 74 p m (8) s cmr10.329 @sf 1446 67 p m (\)) s 1476 67 p m (and) 12 (\() ks cmmi10.329 @sf 1579 67 p m (\025) s cmr8.300 @sf 1606 74 p m (4) s cmmi10.329 @sf 1625 67 p m (;) 15 (\025) ks cmr8.300 @sf 1680 74 p m (3) s cmr10.329 @sf 1699 67 p m (\)) s 1729 67 p m (are) 11 (complex) ks 74 150 p m (pairs.) 20 (W) -3 (e) 15 (ha) 0 (v) -1 (e) ks 575 525 p m (\() s cmmi10.329 @sf m (x) s cmr8.300 @sf 619 532 p m (3) s cmmi10.329 @sf 639 525 p m (;) 8 (x) ks cmr8.300 @sf 686 532 p m (4) s cmmi10.329 @sf 705 525 p m (;) 8 (x) ks cmr8.300 @sf 752 532 p m (6) s cmmi10.329 @sf 771 525 p m (;) 8 (x) ks cmr8.300 @sf 818 532 p m (8) s cmmi10.329 @sf 836 525 p m (;) 8 (x) ks cmr8.300 @sf 883 532 p m (9) s cmr10.329 @sf 902 525 p m (\)) 13 (=) ks cmex10.300 @sf 981 167 p m (0) s 981 240 p m (B) s 981 265 p m (B) s 981 290 p m (B) s 981 315 p m (B) s 981 340 p m (B) s 981 365 p m (B) s 981 390 p m (B) s 981 415 p m (B) s 981 440 p m (B) s 981 465 p m (B) s 981 490 p m (B) s 981 515 p m (B) s 981 540 p m (B) s 981 565 p m (B) s 981 590 p m (B) s 981 615 p m (B) s 981 640 p m (B) s 981 665 p m (B) s 981 690 p m (B) s 981 715 p m (B) s 981 740 p m (B) s 981 765 p m (B) s 981 791 p m (@) s cmsy10.329 @sf 1065 203 p m (\003) s 1138 203 p m (\003) s 1185 220 p 2 57 ru 1210 203 p m (\003) s 1257 220 p 2 57 ru 1283 203 p m (\003) s 1355 203 p m (\003) s 1065 259 p m (\003) s 1138 259 p m (\003) s 1185 276 p 2 57 ru 1210 259 p m (\003) s 1257 276 p 2 57 ru 1283 259 p m (\003) s 1355 259 p m (\003) s cmr10.329 @sf 1065 316 p m (1) s 1138 316 p m (0) s 1185 333 p 2 57 ru 1210 316 p m (0) s 1257 333 p 2 57 ru 1283 316 p m (0) s 1355 316 p m (0) s 1065 372 p m (0) s 1138 372 p m (1) s 1185 389 p 2 57 ru 1210 372 p m (0) s 1257 389 p 2 57 ru 1283 372 p m (0) s 1355 372 p m (0) s 1040 391 p 363 2 ru 1065 429 p m (0) s 1138 429 p m (0) s 1185 446 p 2 57 ru cmsy10.329 @sf 1210 429 p m (\003) s 1257 446 p 2 57 ru 1283 429 p m (\003) s 1355 429 p m (\003) s cmr10.329 @sf 1065 485 p m (0) s 1138 485 p m (0) s 1185 502 p 2 57 ru 1210 485 p m (1) s 1257 502 p 2 57 ru 1283 485 p m (0) s 1355 485 p m (0) s 1040 504 p 363 2 ru 1065 542 p m (0) s 1138 542 p m (0) s 1185 559 p 2 57 ru 1210 542 p m (0) s 1257 559 p 2 57 ru cmsy10.329 @sf 1283 542 p m (\003) s 1355 542 p m (\003) s cmr10.329 @sf 1065 598 p m (0) s 1138 598 p m (0) s 1185 615 p 2 57 ru 1210 598 p m (0) s 1257 615 p 2 57 ru 1283 598 p m (1) s 1355 598 p m (0) s 1065 654 p m (0) s 1138 654 p m (0) s 1185 671 p 2 57 ru 1210 654 p m (0) s 1257 671 p 2 57 ru 1283 654 p m (0) s 1355 654 p m (1) s 1040 673 p 363 2 ru 1065 711 p m (0) s 1138 711 p m (0) s 1185 728 p 2 57 ru 1210 711 p m (0) s 1257 728 p 2 57 ru 1283 711 p m (0) s 1355 711 p m (0) s 1070 759 p m (.) s 1070 776 p m (.) s 1070 793 p m (.) s 1143 759 p m (.) s 1143 776 p m (.) s 1143 793 p m (.) s 1185 808 p 2 80 ru 1215 759 p m (.) s 1215 776 p m (.) s 1215 793 p m (.) s 1257 808 p 2 80 ru 1288 759 p m (.) s 1288 776 p m (.) s 1288 793 p m (.) s 1361 759 p m (.) s 1361 776 p m (.) s 1361 793 p m (.) s 1065 847 p m (0) s 1138 847 p m (0) s 1185 864 p 2 57 ru 1210 847 p m (0) s 1257 864 p 2 57 ru 1283 847 p m (0) s 1355 847 p m (0) s cmex10.300 @sf 1427 167 p m (1) s 1427 240 p m (C) s 1427 265 p m (C) s 1427 290 p m (C) s 1427 315 p m (C) s 1427 340 p m (C) s 1427 365 p m (C) s 1427 390 p m (C) s 1427 415 p m (C) s 1427 440 p m (C) s 1427 465 p m (C) s 1427 490 p m (C) s 1427 515 p m (C) s 1427 540 p m (C) s 1427 565 p m (C) s 1427 590 p m (C) s 1427 615 p m (C) s 1427 640 p m (C) s 1427 665 p m (C) s 1427 690 p m (C) s 1427 715 p m (C) s 1427 740 p m (C) s 1427 765 p m (C) s 1427 791 p m (A) s cmr10.329 @sf 74 916 p m (and) 16 (\014nally) ks cmmi10.329 @sf 328 1106 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x) s cmr8.300 @sf 405 1113 p m (3) s cmmi10.329 @sf 425 1106 p m (;) 8 (x) ks cmr8.300 @sf 472 1113 p m (4) s cmmi10.329 @sf 491 1106 p m (;) 8 (x) ks cmr8.300 @sf 538 1113 p m (6) s cmmi10.329 @sf 557 1106 p m (;) 8 (x) ks cmr8.300 @sf 604 1113 p m (8) s cmmi10.329 @sf 622 1106 p m (;) 8 (x) ks cmr8.300 @sf 669 1113 p m (9) s cmr10.329 @sf 688 1106 p m (\)) 13 (=) 13 (\() ks cmmi10.329 @sf m (x) s cmr8.300 @sf 811 1113 p m (3) s cmmi10.329 @sf 830 1106 p m (;) 8 (x) ks cmr8.300 @sf 877 1113 p m (4) s cmmi10.329 @sf 896 1106 p m (;) 8 (x) ks cmr8.300 @sf 943 1113 p m (6) s cmmi10.329 @sf 962 1106 p m (;) 8 (x) ks cmr8.300 @sf 1009 1113 p m (8) s cmmi10.329 @sf 1028 1106 p m (;) 8 (x) ks cmr8.300 @sf 1075 1113 p m (9) s cmr10.329 @sf 1094 1106 p m (\)) s cmex10.300 @sf 1120 959 p m (0) s 1119 1032 p m (B) s 1119 1057 p m (B) s 1119 1082 p m (B) s 1119 1107 p m (B) s 1119 1132 p m (B) s 1119 1159 p m (@) s cmmi10.329 @sf 1176 993 p m (t) s cmr8.300 @sf 1192 1000 p m (33) s cmmi10.329 @sf 1271 993 p m (t) s cmr8.300 @sf 1287 1000 p m (34) s cmmi10.329 @sf 1366 993 p m (d) s cmr8.300 @sf 1390 1000 p m (36) s cmmi10.329 @sf 1469 993 p m (d) s cmr8.300 @sf 1493 1000 p m (38) s cmmi10.329 @sf 1572 993 p m (d) s cmr8.300 @sf 1596 1000 p m (39) s cmmi10.329 @sf 1176 1049 p m (t) s cmr8.300 @sf 1192 1056 p m (43) s cmmi10.329 @sf 1271 1049 p m (t) s cmr8.300 @sf 1287 1056 p m (44) s cmmi10.329 @sf 1366 1049 p m (d) s cmr8.300 @sf 1390 1056 p m (46) s cmmi10.329 @sf 1469 1049 p m (d) s cmr8.300 @sf 1493 1056 p m (48) s cmmi10.329 @sf 1572 1049 p m (d) s cmr8.300 @sf 1596 1056 p m (49) s cmmi10.329 @sf 1370 1106 p m (t) s cmr8.300 @sf 1386 1113 p m (66) s cmmi10.329 @sf 1469 1106 p m (d) s cmr8.300 @sf 1493 1113 p m (68) s cmmi10.329 @sf 1572 1106 p m (d) s cmr8.300 @sf 1596 1113 p m (69) s cmmi10.329 @sf 1473 1162 p m (t) s cmr8.300 @sf 1489 1169 p m (88) s cmmi10.329 @sf 1575 1162 p m (t) s cmr8.300 @sf 1591 1169 p m (89) s cmmi10.329 @sf 1473 1219 p m (t) s cmr8.300 @sf 1489 1226 p m (98) s cmmi10.329 @sf 1575 1219 p m (t) s cmr8.300 @sf 1591 1226 p m (99) s cmex10.300 @sf 1653 959 p m (1) s 1653 1032 p m (C) s 1653 1057 p m (C) s 1653 1082 p m (C) s 1653 1107 p m (C) s 1653 1132 p m (C) s 1653 1159 p m (A) s cmmi10.329 @sf 1697 1106 p m (:) s cmr10.329 @sf 74 1295 p m (The) 20 (elemen) 0 (ts) 20 (named) ks cmmi10.329 @sf 20 r m (d) s cmr8.300 @sf 538 1302 p m (36) s cmr10.329 @sf 595 1295 p m (and) s cmmi10.329 @sf 20 r m (d) s cmr8.300 @sf 712 1302 p m (46) s cmr10.329 @sf 769 1295 p m (w) 0 (ould) 19 (ha) 0 (v) 0 (e) 18 (b) 1 (een) 21 (determined) 21 (when) 21 (computing) ks cmmi10.329 @sf 20 r m (x) s cmr8.300 @sf 1749 1302 p m (6) s cmr10.329 @sf 1789 1295 p m (when) 20 (w) 0 (e) ks 74 1378 p m (reac) 0 (hed) 16 (ro) 0 (ws) 15 (3) 17 (and) 17 (4;) 17 (the) 17 (elemen) 0 (ts) ks cmmi10.329 @sf 17 r m (d) s cmr8.300 @sf 823 1385 p m (68) s cmr10.329 @sf 877 1378 p m (and) s cmmi10.329 @sf 17 r m (d) s cmr8.300 @sf 991 1385 p m (69) s cmr10.329 @sf 1045 1378 p m (w) 0 (ould) 16 (ha) 0 (v) 0 (e) 15 (b) 1 (een) 18 (determined) 18 (when) 18 (computing) ks cmmi10.329 @sf 74 1461 p m (x) s cmr8.300 @sf 100 1468 p m (8) s cmr10.329 @sf 140 1461 p m (and) s cmmi10.329 @sf 20 r m (x) s cmr8.300 @sf 259 1468 p m (9) s cmr10.329 @sf 298 1461 p m (when) 21 (w) 0 (e) 18 (reac) 0 (hed) 20 (elemen) 0 (t) 19 (6;) 22 (and) 20 (the) 20 (elemen) 0 (ts) ks cmmi10.329 @sf 19 r m (d) s cmr8.300 @sf 1284 1468 p m (48) s cmr10.329 @sf 1321 1461 p m (,) s cmmi10.329 @sf 21 r m (d) s cmr8.300 @sf 1379 1468 p m (49) s cmr10.329 @sf 1416 1461 p m (,) s cmmi10.329 @sf 21 r m (d) s cmr8.300 @sf 1474 1468 p m (38) s cmr10.329 @sf 1511 1461 p m (,) s cmmi10.329 @sf 20 r m (d) s cmr8.300 @sf 1568 1468 p m (39) s cmr10.329 @sf 1625 1461 p m (w) 0 (ould) 19 (ha) 0 (v) 0 (e) 18 (b) 1 (een) ks 74 1545 p m (determined) 17 (when) 15 (w) 0 (e) 14 (reac) 0 (hed) 15 (elemen) 0 (ts) 15 (4) 15 (and) 15 (3.) ks 124 1669 p m (If) 18 (w) 0 (e) 16 (ha) 0 (v) 0 (e) 15 (made) 18 (a) 17 (go) 1 (o) 1 (d) 18 (decision) 19 (ab) 1 (out) 17 (our) 18 (grouping,) 18 (ro) 0 (ws) 15 (of) 17 (the) 18 (v) 0 (ectors) 16 (will) 19 (not) 17 (b) 1 (e) 18 (large,) ks 74 1752 p m (though) 14 (this) 14 (w) 0 (ould) 13 (not) 14 (b) 1 (e) 15 (su\016cien) 0 (t) 13 (to) 13 (decide) 16 (that) 13 (the) 14 (grouping) 14 (is) 15 (complete.) 20 (First,) 13 (there) 14 (ma) 0 (y) 12 (b) 1 (e) ks 74 1835 p m (some) s cmmi10.329 @sf 14 r m (\025) s cmmi8.300 @sf 214 1842 p m (i) s cmr10.329 @sf 242 1835 p m (whic) 0 (h) 15 (should) 15 (also) 15 (b) 1 (e) 15 (asso) 1 (ciated) 15 (with) 14 (these) 15 (\014v) 0 (e.) 19 (Second,) 15 (the) 15 (v) 0 (ectors) ks cmmi10.329 @sf 12 r m (x) s cmr8.300 @sf 1625 1842 p m (3) s cmr10.329 @sf 1645 1835 p m (,) s cmmi10.329 @sf 14 r m (x) s cmr8.300 @sf 1698 1842 p m (4) s cmr10.329 @sf 1718 1835 p m (,) s cmmi10.329 @sf 14 r m (x) s cmr8.300 @sf 1771 1842 p m (6) s cmr10.329 @sf 1791 1835 p m (,) s cmmi10.329 @sf 14 r m (x) s cmr8.300 @sf 1844 1842 p m (8) s cmr10.329 @sf 1864 1835 p m (,) 14 (and) ks cmmi10.329 @sf 74 1918 p m (x) s cmr8.300 @sf 100 1925 p m (9) s cmr10.329 @sf 135 1918 p m (migh) 0 (t) 14 (not) 15 (b) 1 (e) 16 (as) 15 (linearly) 17 (indep) 1 (enden) 1 (t) 15 (as) 15 (w) 0 (e) 14 (w) 0 (ould) 14 (lik) 0 (e.) ks 124 2043 p m (Other) 22 (approac) 0 (hes) 20 (ha) 0 (v) 0 (e) 19 (b) 1 (een) 23 (suggested) 21 (for) 21 (computing) 22 (the) 22 (in) 0 (v) -2 (arian) 0 (t) 19 (subspace) 22 (directly) -3 (,) 24 (see) ks 74 2126 p m ([) 0 (6) ks 110 2126 p m (,) 14 (5) ks 160 2126 p m (,) 15 (4) ks 211 2126 p m (].) 19 (These) 16 (are) 14 (most) 15 (lik) 0 (ely) 16 (more) 14 (stable) 16 (but) 15 (more) 15 (exp) 1 (ensiv) 0 (e) 16 (to) 14 (compute.) ks cmbx10.432 @sf 74 2307 p m (5) s 177 2307 p m (Conclusions) s cmr10.329 @sf 74 2477 p m (The) 17 (metho) 1 (ds) 18 (describ) 1 (ed) 18 (in) 18 (Section) 18 (2) 17 (has) 17 (b) 1 (een) 18 (impro) 0 (v) 0 (ed) 15 (and) 17 (generalized) 19 (b) 0 (y) 16 (Ng) 17 (and) 17 (P) 0 (arlett) 15 ([) 0 (7) ks 1951 2477 p m (]) s 74 2560 p m (and) 14 (implemen) 0 (ted) 15 (in) 15 (LAP) -3 (A) 0 (CK) 12 ([) 0 (1) ks 720 2560 p m (].) 19 (The) 15 (LAP) -3 (A) 0 (CK) 12 (implemen) 0 (tation) 14 (includ) 1 (es) 16 (tolerance) 14 (c) 0 (hec) 0 (ks) 12 (and) ks 74 2643 p m (scaling) 14 (to) 11 (ensure) 13 (n) 0 (umerical) 13 (stabilit) 0 (y) 12 ([) 0 (2) ks 828 2643 p m (].) 18 (This) 13 (is) 13 (essen) 0 (tially) 13 (ac) 0 (hiev) 0 (ed) 12 (b) 0 (y) 11 (not) 12 (sw) 0 (apping) 12 (blo) 1 (c) 0 (ks) 12 (that) ks 74 2726 p m (are) 15 (regarded) 15 (as) 15 (b) 1 (eing) 16 (to) 1 (o) 15 (close.) ks 996 2850 p m (22) s @eop 21 @bop0 cmsy10.329 @sf [ 26 2 -4 -10 35] 0 @dc 1.000000 /cmsy8.300 @newfont cmsy8.300 @sf [ 20 2 -3 -7 27] 0 @dc cmr8.300 @sf [<7FF007000700070007000700070007000700070007000700070007000700070007000700FF0007 000300> 12 21 -2 0 18] 49 @dc [<07C01C703018701C600C600CE00EE00EE00EE00EE00EE00EE00EE00EE00E600C600C701C301818 3007C0> 15 21 -1 0 18] 48 @dc cmmi10.329 @sf [<1F000021C000606000707000783800781800001C00000E00000E0003EE00061E000E0F000C0700 0C07000E07000E03800E03800E03800E03800701C00701C00701C08381C04380E041C0E021C0E0 2180E01180700F0060> 20 29 -2 9 22] 121 @dc cmr8.300 @sf [ 13 21 -2 0 18] 50 @dc cmex10.300 @sf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dc [ 21 124 -1 122 33] 33 @dc cmsy10.329 @sf [ 28 32 -3 1 35] 6 @dc cmmi10.329 @sf [<0E00190038803840384038401C201C001C000E000E00070007008700438043804380238013000E 00000000000000000000000000000000C001E001E000E0> 11 31 -2 0 16] 105 @dc cmr8.300 @sf [<1F0020C040608030E038E0384038003800380030207030E02F80200020002000200024003FC03F E02030> 13 21 -2 0 18] 53 @dc cmr10.329 @sf [<000600000006000000060000000600000006000000060000000600000006000000060000000600 000006000000060000000600000006000000060000FFFFFFF0FFFFFFF000060000000600000006 000000060000000600000006000000060000000600000006000000060000000600000006000000 0600000006000000060000> 28 32 -3 5 35] 43 @dc cmmi10.329 @sf [ 16 45 -3 11 23] 61 @dc [<01FC0000078380000E00E0001C00700038003C0038001E0078000F007000070070000380F00003 C0F00001C0F00001E0F00000E0F00000F0F00000F0780000787800007878000078780000783C00 00383C0000381C0000381E0000380E00003807000038078000380380007001C0007000E0006000 3000E0001C01C0000707000001FC00> 29 33 -2 1 35] 79 @dc cmsy10.329 @sf [<400004C0000C6000183000301800600C00C006018003030001860000CC00007800003000003000 00780000CC000186000303000601800C00C0180060300030600018C0000C400004> 22 24 -6 1 35] 2 @dc cmr10.329 @sf [<003FF800038000038000038000038000038000038000038000038003E3800E13801C0B80380780 380380780380700380F00380F00380F00380F00380F00380F003807003807803803803803C0780 1C058006198003E080> 21 29 -2 9 24] 113 @dc cmr8.300 @sf [<003000003000003000003000003000003000003000003000003000003000003000FFFFFCFFFFFC 003000003000003000003000003000003000003000003000003000003000003000> 22 24 -2 4 27] 43 @dc cmmi8.300 @sf [ 16 20 0 6 18] 112 @dc [<40402020101070F0F060> 4 10 -3 6 10] 59 @dc cmr8.300 @sf [<0F001C8038403840384038403800380038003800380038003800FF803800180018000800080008 00> 10 20 -1 0 14] 116 @dc [ 19 23 0 0 20] 104 @dc cmr10.329 @sf [<40201010080804040474FCFCF870> 6 14 -4 -18 13] 39 @dc 21 @bop1 cmr10.329 @sf 74 67 p m (t) 0 (w) -1 (o) 12 (eigen) 0 (v) 0 (ectors) 12 (are) 14 (close,) 14 (pro) 0 (vided) 14 (the) 14 (earlier) 15 (eigen) 0 (v) -2 (alues) 14 (are) 14 (w) 0 (ell) 14 (separated) 14 (from) 13 (them.) 19 (Th) 0 (us,) ks 74 150 p m (for) s cmex10.300 @sf 375 156 p m (0) s 375 229 p m (B) s 375 256 p m (@) s cmr10.329 @sf 432 196 p m (3) s 558 196 p m (1) s 684 196 p m (2) s 432 253 p m (0) s 558 253 p m (1) s 684 253 p m (1) s 432 309 p m (0) s cmsy10.329 @sf 496 309 p m (\000) s cmr10.329 @sf m (10) s cmsy8.300 @sf 577 294 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 684 309 p m (1) s cmex10.300 @sf 727 156 p m (1) s 727 229 p m (C) s 727 256 p m (A) s 771 156 p m (0) s 771 229 p m (B) s 771 256 p m (@) s cmmi10.329 @sf 828 196 p m (x) s cmr8.300 @sf 854 203 p m (1) s cmmi10.329 @sf 915 196 p m (y) s cmr8.300 @sf 937 203 p m (2) s cmr10.329 @sf 840 253 p m (1) s 925 253 p m (0) s 840 309 p m (0) s 925 309 p m (1) s cmex10.300 @sf 978 156 p m (1) s 978 229 p m (C) s 978 256 p m (A) s cmr10.329 @sf 1027 253 p m (=) s cmex10.300 @sf 1075 156 p m (0) s 1075 229 p m (B) s 1075 256 p m (@) s cmmi10.329 @sf 1132 196 p m (x) s cmr8.300 @sf 1158 203 p m (1) s cmmi10.329 @sf 1219 196 p m (y) s cmr8.300 @sf 1241 203 p m (1) s cmr10.329 @sf 1143 253 p m (1) s 1229 253 p m (0) s 1143 309 p m (0) s 1229 309 p m (1) s cmex10.300 @sf 1282 156 p m (1) s 1282 229 p m (C) s 1282 256 p m (A) s 1326 181 p m ( ) s cmr10.329 @sf 1441 226 p m (1) s 1567 226 p m (1) s cmsy10.329 @sf 1379 281 p m (\000) s cmr10.329 @sf m (10) s cmsy8.300 @sf 1460 266 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 1567 281 p m (1) s cmex10.300 @sf 1610 181 p m (!) s cmmi10.329 @sf 1651 253 p m (;) s cmr10.329 @sf 74 398 p m (the) 12 (eigen) 0 (v) -2 (alues) 12 (are) 12 (1) ks cmsy10.329 @sf 4 r m (\006) s cmmi10.329 @sf 4 r m (i) s cmr10.329 @sf m (10) s cmsy8.300 @sf 582 383 p m (\000) s cmr8.300 @sf m (5) s cmr10.329 @sf 627 398 p m (;) 12 (they) 12 (are) 12 (close,) 13 (but) 12 (w) 0 (ell) 12 (separated) 11 (from) 12 (the) 12 (other) 11 (eigen) 0 (v) -2 (alue) ks cmmi10.329 @sf 12 r m (\025) s cmr8.300 @sf 1849 405 p m (1) s cmr10.329 @sf 1881 398 p m (=) 13 (3.) ks 74 481 p m (The) 16 (comp) 1 (onen) 0 (ts) ks cmmi10.329 @sf 14 r m (x) s cmr8.300 @sf 444 488 p m (1) s cmr10.329 @sf 479 481 p m (and) s cmmi10.329 @sf 15 r m (y) s cmr8.300 @sf 589 488 p m (1) s cmr10.329 @sf 624 481 p m (satisfy) s 760 582 p m (3) s cmmi10.329 @sf m (x) s cmr8.300 @sf 809 589 p m (1) s cmr10.329 @sf 838 582 p m (+) 11 (1) ks 948 582 p m (=) s cmmi10.329 @sf 1025 582 p m (x) s cmr8.300 @sf 1051 589 p m (1) s cmsy10.329 @sf 1081 582 p m (\000) s cmr10.329 @sf 10 r m (10) s cmsy8.300 @sf 1172 567 p m (\000) s cmr8.300 @sf m (10) s cmmi10.329 @sf 1236 582 p m (y) s cmr8.300 @sf 1258 589 p m (1) s cmr10.329 @sf 764 638 p m (3) s cmmi10.329 @sf m (y) s cmr8.300 @sf 809 645 p m (1) s cmr10.329 @sf 838 638 p m (+) 11 (2) ks 948 638 p m (=) s cmmi10.329 @sf 1025 638 p m (x) s cmr8.300 @sf 1051 645 p m (1) s cmr10.329 @sf 1081 638 p m (+) s cmmi10.329 @sf 10 r m (y) s cmr8.300 @sf 1148 645 p m (1) s cmmi10.329 @sf 1168 638 p m (:) s cmr10.329 @sf 74 738 p m (T) -3 (o) 14 (eigh) 0 (t) 14 (decimals,) ks cmmi10.329 @sf 17 r m (x) s cmr8.300 @sf 477 745 p m (1) s cmr10.329 @sf 509 738 p m (=) s cmsy10.329 @sf 13 r m (\000) s cmr10.329 @sf m (1) s cmmi10.329 @sf m (=) s cmr10.329 @sf m (2) 15 (and) ks cmmi10.329 @sf 15 r m (y) s cmr8.300 @sf 786 745 p m (1) s cmr10.329 @sf 819 738 p m (=) s cmsy10.329 @sf 13 r m (\000) s cmr10.329 @sf m (5) s cmmi10.329 @sf m (=) s cmr10.329 @sf m (4.) 19 (The) 15 (v) 0 (ectors) 14 (are) 15 (extremely) 15 (w) 0 (ell) 16 (separated) 15 (and) ks cmmi10.329 @sf 572 873 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x;) 14 (y) ks cmr10.329 @sf 2 r m (\)) s cmsy10.329 @sf 10 r m (\000) s cmr10.329 @sf 11 r m (\() s cmmi10.329 @sf m (x;) 14 (y) ks cmr10.329 @sf 2 r m (\)) s cmex10.300 @sf 895 801 p m ( ) s cmr10.329 @sf 1009 846 p m (1) s 1135 846 p m (1) s cmsy10.329 @sf 948 901 p m (\000) s cmr10.329 @sf m (10) s cmsy8.300 @sf 1029 886 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 1135 901 p m (1) s cmex10.300 @sf 1178 801 p m (!) s cmr10.329 @sf 1224 873 p m (=) s cmmi10.329 @sf 13 r m (O) s cmr10.329 @sf 1 r m (\(10) s cmsy8.300 @sf 1372 855 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 1436 873 p m (\)) s cmmi10.329 @sf m (:) s cmr10.329 @sf 124 1043 p m (If,) 17 (when) 18 (computing) 17 (the) 17 (t) 0 (w) -1 (o) 15 (v) 0 (ectors) 15 (corresp) 1 (onding) 18 (to) 16 (a) 17 (complex) 17 (pair,) 18 (w) 0 (e) 15 (encoun) 0 (ter) 16 (another) ks 74 1126 p m (2) s cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 11 r m (2) 16 (blo) 1 (c) 0 (k,) 15 (sa) 0 (y) 15 (in) 16 (p) 1 (osition) ks cmmi10.329 @sf 17 r m (i;) 8 (i) ks cmr10.329 @sf 9 r m (+) 11 (1,) 16 (then) 16 (comp) 1 (onen) 0 (ts) ks cmmi10.329 @sf 15 r m (i) s cmr10.329 @sf 15 r m (and) s cmmi10.329 @sf 16 r m (i) s cmr10.329 @sf 11 r m (+) 11 (1) 15 (of) ks cmmi10.329 @sf 16 r m (x) s cmr10.329 @sf 16 r m (and) s cmmi10.329 @sf 16 r m (y) s cmr10.329 @sf 17 r m (are) 16 (determined) 17 (b) 0 (y) ks 74 1209 p m (solving) 17 (a) 15 (set) 15 (of) 15 (four) 16 (linear) 16 (equations) 16 (deriv) 0 (ed) 16 (b) 0 (y) 14 (equating) 16 (ro) 0 (ws) ks cmmi10.329 @sf 14 r m (i) s cmr10.329 @sf 15 r m (and) s cmmi10.329 @sf 16 r m (i) s cmr10.329 @sf 10 r m (+) 11 (1) 15 (of) 15 (\(12\).) 20 (This) 16 (will) 17 (b) 1 (e) ks 74 1292 p m (a) 15 (w) 0 (ell-conditioned) 17 (4) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (4) 15 (system) 15 (if) ks cmmi10.329 @sf 16 r m (\025) s cmmi8.300 @sf 785 1299 p m (i) s cmr10.329 @sf 798 1292 p m (,) s cmmi10.329 @sf 15 r m (\025) s cmmi8.300 @sf 853 1299 p m (i) s cmr8.300 @sf m (+1) s cmr10.329 @sf 927 1292 p m (are) 15 (w) 0 (ell) 15 (separated) 15 (from) ks cmmi10.329 @sf 14 r m (\025) s cmmi8.300 @sf 1434 1299 p m (p) s cmr10.329 @sf 1454 1292 p m (,) s cmmi10.329 @sf 14 r m (\025) s cmmi8.300 @sf 1508 1299 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1573 1292 p m (.) s 124 1417 p m (When) 15 (w) 0 (e) 13 (wish) 14 (to) 14 (asso) 1 (ciate) 14 (\() ks cmmi10.329 @sf 687 1417 p m (\025) s cmmi8.300 @sf 714 1424 p m (p) s cmmi10.329 @sf 733 1417 p m (;) 8 (\025) ks cmmi8.300 @sf 781 1424 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 845 1417 p m (\)) s 877 1417 p m (with) 15 (some) 13 (of) 14 (the) 15 (earlier) 15 (eigen) 0 (v) -2 (alues) 14 (\(for) 13 (whic) 0 (h) 14 (w) 0 (e) 13 (ha) 0 (v) 0 (e) ks 74 1500 p m (already) 22 (done) 21 (the) 21 (bac) 0 (k) 20 (substitution\),) 23 (the) 21 (solution) 22 (is) 22 (quite) 22 (clear.) 38 (When) 22 (w) 0 (e) 20 (encoun) 0 (ter) 20 (a) 21 (real) ks 74 1583 p m (eigen) 0 (v) -2 (alue) ks cmmi10.329 @sf 15 r m (\025) s cmmi8.300 @sf 318 1590 p m (i) s cmr10.329 @sf 347 1583 p m (that) 15 (is) 15 (to) 15 (b) 1 (e) 16 (asso) 1 (ciated) 15 (with) 16 (them,) 14 (w) 0 (e) 14 (solv) 0 (e) 15 (from) 14 (that) 15 (p) 1 (oin) 0 (t) 14 (on:) ks cmmi10.329 @sf 423 1711 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x) s cmmi8.300 @sf 500 1718 p m (p) s cmmi10.329 @sf 519 1711 p m (;) 15 (x) ks cmmi8.300 @sf 573 1718 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 638 1711 p m (\)) 12 (=) 13 (\() ks cmmi10.329 @sf m (x) s cmmi8.300 @sf 760 1718 p m (p) s cmmi10.329 @sf 779 1711 p m (;) 15 (x) ks cmmi8.300 @sf 833 1718 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 898 1711 p m (\)) s cmex10.300 @sf 924 1639 p m ( ) s cmmi10.329 @sf 999 1684 p m (t) s cmmi8.300 @sf 1015 1691 p m (p;p) s cmmi10.329 @sf 1150 1684 p m (t) s cmmi8.300 @sf 1166 1691 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 977 1740 p m (t) s cmmi8.300 @sf 993 1747 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmmi10.329 @sf 1127 1740 p m (t) s cmmi8.300 @sf 1143 1747 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmr8.300 @sf m (+1) s cmex10.300 @sf 1302 1639 p m (!) s cmr10.329 @sf 1345 1711 p m (+) 10 (\() ks cmmi10.329 @sf m (x) s cmmi8.300 @sf 1434 1718 p m (i) s cmr10.329 @sf 1448 1711 p m (\)\() s cmmi10.329 @sf m (d) s cmr8.300 @sf 1508 1718 p m (1) s cmmi10.329 @sf 1527 1711 p m (;) 15 (d) ks cmr8.300 @sf 1579 1718 p m (2) s cmr10.329 @sf 1598 1711 p m (\)) s 74 1852 p m (and) 15 (w) 0 (e) 14 (c) 0 (hose) ks cmmi10.329 @sf 13 r m (d) s cmr8.300 @sf 372 1859 p m (1) s cmr10.329 @sf 406 1852 p m (and) s cmmi10.329 @sf 15 r m (d) s cmr8.300 @sf 518 1859 p m (2) s cmr10.329 @sf 553 1852 p m (so) 14 (that) 14 (the) ks cmmi10.329 @sf 15 r m (i) s cmr8.300 @sf 800 1837 p m (th) s cmr10.329 @sf 850 1852 p m (comp) 1 (onen) 0 (t) 14 (of) ks cmmi10.329 @sf 14 r m (x) s cmmi8.300 @sf 1159 1859 p m (p) s cmr10.329 @sf 1193 1852 p m (and) s cmmi10.329 @sf 15 r m (x) s cmmi8.300 @sf 1307 1859 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1387 1852 p m (are) 14 (zero.) 20 (This) 15 (giv) 0 (es) 14 (us) 15 (a) 15 (pair) ks 74 1935 p m (of) 16 (equations) 17 (for) ks cmmi10.329 @sf 16 r m (d) s cmr8.300 @sf 427 1942 p m (1) s cmr10.329 @sf 463 1935 p m (and) s cmmi10.329 @sf 17 r m (d) s cmr8.300 @sf 577 1942 p m (2) s cmr10.329 @sf 596 1935 p m (.) 24 (If) ks cmmi10.329 @sf 17 r m (\025) s cmmi8.300 @sf 707 1942 p m (p) s cmr10.329 @sf 726 1935 p m (,) s cmmi10.329 @sf 16 r m (\025) s cmmi8.300 @sf 782 1942 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 847 1935 p m (,) 16 (and) ks cmmi10.329 @sf 17 r m (\025) s cmmi8.300 @sf 993 1942 p m (i) s cmr10.329 @sf 1023 1935 p m (w) 0 (ere) 15 (the) 17 (only) 17 (three) 17 (to) 16 (b) 1 (e) 17 (asso) 1 (ciated,) 16 (w) 0 (e) 16 (w) 0 (ould) ks 74 2018 p m (ha) 0 (v) 0 (e) 13 (for) 15 (the) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (3-space) ks cmmi10.329 @sf 576 2175 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x) s cmmi8.300 @sf 653 2182 p m (i) s cmmi10.329 @sf 667 2175 p m (;) 8 (x) ks cmmi8.300 @sf 714 2182 p m (p) s cmmi10.329 @sf 733 2175 p m (;) 14 (x) ks cmmi8.300 @sf 786 2182 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 851 2175 p m (\)) 13 (=) ks cmex10.300 @sf 930 2078 p m (0) s 930 2151 p m (B) s 930 2178 p m (@) s cmmi10.329 @sf 987 2118 p m (t) s cmmi8.300 @sf 1003 2125 p m (i;i) s cmmi10.329 @sf 1113 2118 p m (d) s cmr8.300 @sf 1137 2125 p m (1) s cmmi10.329 @sf 1286 2118 p m (d) s cmr8.300 @sf 1310 2125 p m (2) s cmr10.329 @sf 1002 2175 p m (0) s cmmi10.329 @sf 1103 2175 p m (t) s cmmi8.300 @sf 1119 2182 p m (p;p) s cmmi10.329 @sf 1253 2175 p m (t) s cmmi8.300 @sf 1269 2182 p m (p;p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1002 2231 p m (0) s cmmi10.329 @sf 1081 2231 p m (t) s cmmi8.300 @sf 1097 2238 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmmi10.329 @sf 1231 2231 p m (t) s cmmi8.300 @sf 1247 2238 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmr8.300 @sf m (+1) s cmex10.300 @sf 1406 2078 p m (1) s 1406 2151 p m (C) s 1406 2178 p m (A) s cmmi10.329 @sf 1450 2175 p m (:) s cmr10.329 @sf 74 2332 p m (If) 22 (during) 23 (the) 21 (bac) 0 (k) 21 (substitution) 22 (for) ks cmmi10.329 @sf 21 r m (x) s cmmi8.300 @sf 835 2339 p m (p) s cmmi10.329 @sf 855 2332 p m (;) 15 (x) ks cmmi8.300 @sf 909 2339 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 995 2332 p m (w) 0 (e) 21 (encoun) 0 (ter) 21 (a) 21 (pair) ks cmmi10.329 @sf 22 r m (\025) s cmmi8.300 @sf 1455 2339 p m (i) s cmmi10.329 @sf 1469 2332 p m (;) 14 (\025) ks cmmi8.300 @sf 1523 2339 p m (i) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1604 2332 p m (whic) 0 (h) 21 (w) 0 (e) 20 (wish) 23 (to) ks 74 2415 p m (asso) 1 (ciate) 15 (with) 16 (them,) 15 (w) 0 (e) 14 (solv) 0 (e) 14 (from) 14 (that) 15 (p) 1 (oin) 0 (t) 15 (on) ks cmmi10.329 @sf 229 2544 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x) s cmmi8.300 @sf 306 2551 p m (p) s cmmi10.329 @sf 326 2544 p m (;) 14 (x) ks cmmi8.300 @sf 379 2551 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 444 2544 p m (\)) 13 (=) 13 (\() ks cmmi10.329 @sf m (x) s cmmi8.300 @sf 567 2551 p m (p) s cmmi10.329 @sf 586 2544 p m (;) 15 (x) ks cmmi8.300 @sf 640 2551 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 705 2544 p m (\)) s cmex10.300 @sf 731 2472 p m ( ) s cmmi10.329 @sf 806 2517 p m (t) s cmmi8.300 @sf 822 2524 p m (p;p) s cmmi10.329 @sf 957 2517 p m (t) s cmmi8.300 @sf 973 2524 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 784 2572 p m (t) s cmmi8.300 @sf 800 2579 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmmi10.329 @sf 934 2572 p m (t) s cmmi8.300 @sf 950 2579 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmr8.300 @sf m (+1) s cmex10.300 @sf 1109 2472 p m (!) s cmr10.329 @sf 1152 2544 p m (+) 10 (\() ks cmmi10.329 @sf m (x) s cmmi8.300 @sf 1241 2551 p m (i) s cmmi10.329 @sf 1255 2544 p m (;) 8 (x) ks cmmi8.300 @sf 1302 2551 p m (i) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1360 2544 p m (\)) s cmex10.300 @sf 1386 2472 p m ( ) s cmmi10.329 @sf 1462 2517 p m (d) s cmmi8.300 @sf 1486 2524 p m (i;i) s cmmi10.329 @sf 1608 2517 p m (d) s cmmi8.300 @sf 1632 2524 p m (i;i) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 1439 2572 p m (d) s cmmi8.300 @sf 1463 2579 p m (i) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;i) s cmmi10.329 @sf 1585 2572 p m (d) s cmmi8.300 @sf 1609 2579 p m (i) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;i) s cmr8.300 @sf m (+1) s cmex10.300 @sf 1756 2472 p m (!) s cmmi10.329 @sf 1796 2544 p m (;) s cmr10.329 @sf 74 2672 p m (where) 14 (the) 13 (four) ks cmmi10.329 @sf 13 r m (d) s cmr10.329 @sf m ('s) 12 (are) 13 (c) 0 (hosen) 13 (so) 13 (as) 13 (to) 12 (mak) 0 (e) 12 (comp) 1 (onen) 0 (ts) ks cmmi10.329 @sf 12 r m (i) s cmr10.329 @sf 13 r m (and) s cmmi10.329 @sf 14 r m (i) s cmr10.329 @sf 6 r m (+) 6 (1) 13 (of) ks cmmi10.329 @sf 13 r m (x) s cmmi8.300 @sf 1475 2679 p m (p) s cmr10.329 @sf 1508 2672 p m (and) s cmmi10.329 @sf 14 r m (x) s cmmi8.300 @sf 1621 2679 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1699 2672 p m (equal) 14 (to) 13 (zero.) ks 996 2850 p m (21) s @eop 20 @bop0 cmmi10.329 @sf [<0000F0000001F8000003FC000003FE000003860000038300000101000001018001FD0080078380 800E81E0001C81700038813C0038811E0070600F00701C070070000380F00003C0F00001C0F000 01E0F00000E0F00000F0F00000F0780000787800007878000078780000783C0000783C0000381E 0000381E0000380F00003807000038078000780380007001C0007000E000E0003000E0001C01C0 000707000001FC00> 29 41 -2 9 36] 81 @dc [<000003E0FFFC071807C00E0403C01E0403C01E0203C01E0203C01E0001E00F0001E00F0001E00F 0001E00F0000F0078000F0070000F00F0000F00E0000781C00007FF80000780F0000780380003C 01E0003C00F0003C0078003C0078001E003C001E003C001E003C001E003C000F003C000F003800 0F0070000F01E000FFFF80> 31 32 -2 1 35] 82 @dc [ 44 31 -2 0 44] 77 @dc cmr8.300 @sf [<0102040C1818303070606060E0E0E0E0E0E0E0E0E0E060606070303018180C040201> 8 34 -3 9 14] 40 @dc [<8040203018180C0C0E060606070707070707070707070606060E0C0C181830204080> 8 34 -2 9 14] 41 @dc 20 @bop1 cmr10.329 @sf 74 67 p m (diagonal) 18 (elemen) 0 (ts) 16 (to) 17 (asso) 1 (ciate) 17 (together.) 24 (W) -3 (e) 16 (ma) 0 (y) 15 (need) 18 (to) 17 (asso) 1 (ciate) 16 (eigen) 0 (v) -2 (alues) 17 (that) 17 (are) 16 (b) 0 (y) 16 (no) ks 74 150 p m (means) 15 (pathologically) 17 (close.) 21 (If) 15 (w) 0 (e) 14 (ha) 0 (v) 0 (e) 13 (decided) 17 (whic) 0 (h) 15 (eigen) 0 (v) -2 (alues) 15 (w) 0 (e) 14 (wish) 16 (to) 15 (asso) 1 (ciate,) 15 (then) 15 (w) 0 (e) ks 74 233 p m (pro) 1 (ceed) 16 (exactly) 16 (as) 15 (describ) 1 (ed.) ks 124 358 p m (So) 22 (far) 21 (in) 23 (this) 23 (section) 22 (w) 0 (e) 21 (ha) 0 (v) -1 (e) 21 (tacitly) 22 (assumed) 22 (that) ks cmmi10.329 @sf 22 r m (T) s cmr10.329 @sf 28 r m (is) 22 (exactly) 22 (triangular,) 24 (but) 22 (the) ks cmmi10.329 @sf 22 r m (QR) s cmr10.329 @sf 74 441 p m (algorithm) 15 (ma) 0 (y) 13 (giv) 0 (e) 15 (2) ks cmsy10.329 @sf 9 r m (\002) s cmr10.329 @sf 10 r m (2's) 14 (on) 15 (the) 15 (diagonal.) 20 (If) 16 (a) 14 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2) 14 (corresp) 1 (onds) 15 (to) 15 (a) 14 (pair) 16 (of) 14 (real) 16 (eigen) 0 (v) -2 (alues,) ks 74 524 p m (w) 0 (e) 15 (can) 16 (get) 15 (rid) 17 (of) 15 (it) 16 (b) 0 (y) 15 (an) 16 (orthogonal) 15 (transformation.) 21 (If) 16 (it) 16 (corresp) 1 (onds) 16 (to) 15 (a) 16 (complex) 16 (conjugate) ks 74 607 p m (pair,) 15 (w) 0 (e) 14 (cannot.) 20 (W) -3 (e) 14 (assume) 15 (then) 16 (that) 14 (all) 16 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2's) 15 (corresp) 1 (ond) 15 (to) 15 (complex) 16 (conjugate) 15 (eigen) 0 (v) -2 (alues.) ks 124 731 p m (W) -3 (e) 14 (turn) 15 (no) 0 (w) 13 (to) 14 (the) 15 (case) 15 (of) 15 (2) ks cmsy10.329 @sf 9 r m (\002) s cmr10.329 @sf 10 r m (2) 14 (blo) 1 (c) 0 (ks.) 20 (If) 15 (w) 0 (e) 14 (asso) 1 (ciate) 14 (only) 16 (real) 15 (eigen) 0 (v) -2 (alues) 15 (in) 16 (an) 14 (in) 0 (v) -2 (arian) 0 (t) ks 74 814 p m (subspace,) 14 (there) 13 (are) 13 (no) 13 (real) 13 (new) 13 (p) 1 (oin) 0 (ts.) 19 (W) -3 (e) 12 (merely) 13 (need) 14 (to) 12 (kno) 0 (w) 12 (ho) 0 (w) 11 (to) 13 (get) 12 (the) 13 (t) 0 (w) -1 (o) 11 (comp) 1 (onen) 0 (ts) ks 74 897 p m (of) 18 (an) 0 (y) 16 (of) 18 (our) 17 (v) 0 (ectors) 16 (in) 19 (the) 18 (p) 1 (osition) 19 (of) 17 (a) 18 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 13 r m (2) 17 (blo) 1 (c) 0 (k) 18 (in) 18 (the) 18 (matrix.) 28 (Clearly) 18 (w) 0 (e) 17 (solv) 0 (e) 17 (a) 18 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 12 r m (2) s 74 980 p m (system) 16 (of) 15 (equations) 16 (for) 15 (the) 16 (t) 0 (w) -1 (o) 14 (comp) 1 (onen) 0 (ts.) 21 (The) 16 (tec) 0 (hnique) 16 (for) 15 (getting) 16 (the) 16 (generators) 15 (and) 16 (the) ks cmmi10.329 @sf 74 1063 p m (M) s cmr10.329 @sf 20 r m (is) 16 (unc) 0 (hanged.) ks 124 1188 p m (No) 0 (w,) 18 (consider) 20 (obtaining) 20 (a) 18 (pair) 20 (of) 18 (v) 0 (ectors) 18 (spanning) 19 (the) 20 (t) 0 (w) -1 (o) -1 (-space) 18 (asso) 1 (ciated) 19 (with) 19 (complex) ks 74 1271 p m (conjugate) 14 (pairs) 14 (of) 13 (eigen) 0 (v) -2 (alues,) 14 (assuming) 14 (for) 13 (the) 14 (momen) 0 (t) 12 (that) 13 (w) 0 (e) 12 (are) 14 (not) 13 (asso) 1 (ciating) 14 (it) 14 (with) 14 (an) 0 (y) ks 74 1354 p m (other) 15 (eigen) 0 (v) -2 (alues.) 20 (F) -3 (or) ks cmmi10.329 @sf 13 r m (T) s cmr10.329 @sf 6 r m (,) 15 (illustrated) 17 (b) 0 (y) ks cmmi10.329 @sf 733 1590 p m (T) s cmr10.329 @sf 19 r m (=) s cmex10.300 @sf 827 1418 p m (0) s 827 1491 p m (B) s 827 1516 p m (B) s 827 1541 p m (B) s 827 1566 p m (B) s 827 1591 p m (B) s 827 1616 p m (B) s 827 1641 p m (B) s 827 1668 p m (@) s cmsy10.329 @sf 884 1449 p m (\003) s 948 1449 p m (\003) s 1012 1449 p m (\003) s 1077 1449 p m (\003) s 1141 1449 p m (\003) s 1205 1449 p m (\003) s 948 1505 p m (\003) s 1012 1505 p m (\003) s 1077 1505 p m (\003) s 1141 1505 p m (\003) s 1205 1505 p m (\003) s 1012 1562 p m (\003) s 1077 1562 p m (\003) s 1141 1562 p m (\003) s 1205 1562 p m (\003) s 1077 1618 p m (\003) s 1141 1618 p m (\003) s 1205 1618 p m (\003) s 1077 1674 p m (\003) s 1141 1674 p m (\003) s 1205 1674 p m (\003) s 1205 1731 p m (\003) s cmex10.300 @sf 1248 1418 p m (1) s 1248 1491 p m (C) s 1248 1516 p m (C) s 1248 1541 p m (C) s 1248 1566 p m (C) s 1248 1591 p m (C) s 1248 1616 p m (C) s 1248 1641 p m (C) s 1248 1668 p m (A) s cmmi10.329 @sf 1292 1590 p m (;) s cmr10.329 @sf 74 1826 p m (w) 0 (e) 14 (merely) 16 (solv) 0 (e) 14 (the) 16 (equations) ks cmmi10.329 @sf 529 1951 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x) s cmmi8.300 @sf 606 1958 p m (p) s cmmi10.329 @sf 625 1951 p m (;) 8 (x) ks cmmi8.300 @sf 672 1958 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 736 1951 p m (\)) 13 (=) 13 (\() ks cmmi10.329 @sf m (x) s cmmi8.300 @sf 859 1958 p m (p) s cmmi10.329 @sf 878 1951 p m (;) 8 (x) ks cmmi8.300 @sf 925 1958 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 989 1951 p m (\)) s cmex10.300 @sf 1015 1879 p m ( ) s cmmi10.329 @sf 1091 1924 p m (t) s cmmi8.300 @sf 1107 1931 p m (p;p) s cmmi10.329 @sf 1241 1924 p m (t) s cmmi8.300 @sf 1257 1931 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 1068 1979 p m (t) s cmmi8.300 @sf 1084 1986 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmmi10.329 @sf 1219 1979 p m (t) s cmmi8.300 @sf 1235 1986 p m (p) s cmr8.300 @sf m (+1) s cmmi8.300 @sf m (;p) s cmr8.300 @sf m (+1) s cmex10.300 @sf 1393 1879 p m (!) s cmr10.329 @sf 1883 1951 p m (\(12\)) s 74 2076 p m (and) 16 (tak) 0 (e) ks 788 2301 p m (\() s cmmi10.329 @sf 806 2301 p m (x) s cmmi8.300 @sf 832 2308 p m (p) s cmmi10.329 @sf 852 2301 p m (;) 15 (x) ks cmmi8.300 @sf 906 2308 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 971 2301 p m (\)) s 1001 2301 p m (=) s cmex10.300 @sf 1049 2092 p m (0) s 1049 2165 p m (B) s 1049 2190 p m (B) s 1049 2215 p m (B) s 1049 2240 p m (B) s 1049 2265 p m (B) s 1049 2290 p m (B) s 1049 2315 p m (B) s 1049 2340 p m (B) s 1049 2365 p m (B) s 1049 2390 p m (B) s 1049 2417 p m (@) s cmsy10.329 @sf 1106 2120 p m (\003) s 1170 2120 p m (\003) s 1106 2176 p m (\003) s 1170 2176 p m (\003) s 1106 2232 p m (\003) s 1170 2232 p m (\003) s cmr10.329 @sf 1106 2289 p m (1) s 1170 2289 p m (0) s 1106 2345 p m (0) s 1170 2345 p m (1) s 1106 2402 p m (0) s 1170 2402 p m (0) s 1111 2450 p m (.) s 1111 2467 p m (.) s 1111 2484 p m (.) s 1175 2450 p m (.) s 1175 2467 p m (.) s 1175 2484 p m (.) s cmex10.300 @sf 1214 2092 p m (1) s 1214 2165 p m (C) s 1214 2190 p m (C) s 1214 2215 p m (C) s 1214 2240 p m (C) s 1214 2265 p m (C) s 1214 2290 p m (C) s 1214 2315 p m (C) s 1214 2340 p m (C) s 1214 2365 p m (C) s 1214 2390 p m (C) s 1214 2417 p m (A) s cmr10.329 @sf 74 2560 p m (so) 20 (that) 20 (they) 21 (are) 20 (certainly) 22 (indep) 1 (end) 1 (en) 0 (t.) 36 (The) 21 (t) 0 (w) -1 (o) 19 (bac) 0 (k) 19 (substitutions) 21 (for) 20 (determining) ks cmmi10.329 @sf 22 r m (x) s cmmi8.300 @sf 1850 2567 p m (p) s cmr10.329 @sf 1891 2560 p m (and) s cmmi10.329 @sf 74 2643 p m (x) s cmmi8.300 @sf 100 2650 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 181 2643 p m (are) 16 (done) 17 (as) 16 (b) 1 (efore.) 23 (W) -3 (e) 15 (determine) ks cmmi10.329 @sf 18 r m (x) s cmr8.300 @sf 901 2620 p m (\() s cmmi8.300 @sf m (p) s cmr8.300 @sf m (\)) s cmmi8.300 @sf 901 2656 p m (i) s cmr10.329 @sf 964 2643 p m (and) s cmmi10.329 @sf 17 r m (x) s cmr8.300 @sf 1080 2620 p m (\() s cmmi8.300 @sf m (p) s cmr8.300 @sf m (+1\)) s cmmi8.300 @sf 1080 2656 p m (i) s cmr10.329 @sf 1188 2643 p m (from) 16 (the) 16 (pair) 17 (of) 16 (equations) 16 (obtained) 17 (b) 0 (y) ks 74 2726 p m (equating) 16 (ro) 0 (w) ks cmmi10.329 @sf 14 r m (i) s cmr10.329 @sf 15 r m (on) 15 (b) 1 (oth) 16 (sides) 16 (of) 15 (\(12\).) 20 (This) 16 (giv) 0 (es) 15 (a) 15 (w) 0 (ell-separated) 15 (pair) 16 (of) 15 (v) 0 (ectors) 14 (ev) 0 (en) 15 (when) 16 (the) ks 996 2850 p m (20) s @eop 19 @bop0 cmmi10.329 @sf [<07C01C1838323004617003E07000E0F000E0F000E0F000F0F000F0F000E87800E47800E47800E2 3C00E21C00E21E00C10E01C107018101C300007C00> 24 20 -2 0 29] 11 @dc [ 22 31 -1 0 20] 73 @dc [<80700040F80061FC003F060010030008010004008002000001000000C000006000001000000800 0004000002000801000C1F8007F8C003F04001E020> 19 20 -2 0 21] 122 @dc [<3C0000620000F3000079800031800001C00000C00000E00000E00000E00000E000007000007000 007000007000007000003800003800003800003800003800001C00001C00001C00001C00001C00 000E00000E0000FFF0000E00000E00000E0000070000070000070000070000030C00039E00019E 0000CE00007C> 23 41 -2 9 22] 102 @dc [<07C018383004700270017000F000F000F000F0007C007BF0780C38023C011C010E01070101C200 7C> 16 20 -2 0 21] 101 @dc cmmi8.300 @sf [<3F030060C480C02C00C01C00800C00800E00C00E00C00D00C00C806018802018801010400C3040 07C000> 18 14 -2 0 23] 11 @dc cmsy10.329 @sf [<70F8F8F870> 5 5 -4 -9 13] 1 @dc cmmi8.300 @sf [<7FFC0003C00001C00001C00001C00001C00000E00000E00000E00000E000007000007000007000 007000003800003800803804403804401C04601C04201C04381C0E1FFFFE> 23 23 -1 0 21] 84 @dc [<1FC00300030003000180018039804780C1C080C080C080C0C060C060C06060606070307018D007 10> 12 20 -2 6 16] 113 @dc [<3000300018001800180018000C000C000C008C0046304638451838F0> 13 14 -1 0 16] 114 @dc cmmi10.329 @sf [<0FFE0000E00000E0000070000070000070000070000038000038000F380030B800705C00703C00 F01C00F01C00F00E00F00E00F00E00F00E007807007807007807003807003C03801C03800E0380 0705800388C000F040> 18 29 -2 9 20] 113 @dc [<1800003800001C00001C00001C00001C00000E00000E00000E00000E0000070000070000070000 87000043830043878043C78023C3802321001E1E00> 17 20 -2 0 21] 114 @dc cmsy10.329 @sf [ 28 20 -3 -1 35] 17 @dc [<400000C000006000006000003000003000001800001800000C00000C0000060000060000060000 03000003000001800001800000C00000C000006000006000003000003000001800001800000C00 000C000006000006000003000003000001800001800001800000C00000C0000060000060000030 00003000001800001800000C00000C> 22 44 -6 11 0] 54 @dc cmmi10.329 @sf [ 30 32 -2 0 34] 65 @dc 19 @bop1 cmr10.329 @sf 74 67 p m (or) s 427 216 p m (\() s cmmi10.329 @sf m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)\() s cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) 10 (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) s cmex10.300 @sf 972 119 p m (0) s 970 192 p m (B) s 970 219 p m (@) s cmmi10.329 @sf 1027 159 p m (\013) s 1100 159 p m (d) s 1169 159 p m (f) s 1098 215 p m (\013) s 1172 215 p m (e) s 1168 272 p m (\013) s cmex10.300 @sf 1218 119 p m (1) s 1218 192 p m (C) s 1218 219 p m (A) s cmr10.329 @sf 1267 216 p m (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) s cmmi10.329 @sf m (T) s cmmi8.300 @sf 1493 223 p m (\013) s cmmi10.329 @sf 1515 216 p m (:) s cmr10.329 @sf 1883 216 p m (\(11\)) s cmmi10.329 @sf 244 385 p m (x) s cmr10.329 @sf 311 385 p m (=) s 388 385 p m (\() s cmmi10.329 @sf 406 385 p m (x) s cmr8.300 @sf 432 392 p m (1) s cmmi10.329 @sf 452 385 p m (;) 15 (x) ks cmr8.300 @sf 506 392 p m (2) s cmmi10.329 @sf 525 385 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 621 385 p m (;) 8 (x) ks cmmi8.300 @sf 668 392 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 732 385 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 14 r m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 8 r m (\001) 7 (\001) 7 (\001) ks cmr10.329 @sf 957 385 p m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 7 r m (\001) 7 (\001) 7 (\001) ks cmmi10.329 @sf 1146 385 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1207 365 p m (T) s cmmi10.329 @sf 246 445 p m (y) s cmr10.329 @sf 311 445 p m (=) s 388 445 p m (\() s cmmi10.329 @sf 406 445 p m (y) s cmr8.300 @sf 428 452 p m (1) s cmmi10.329 @sf 448 445 p m (;) 15 (y) ks cmr8.300 @sf 498 452 p m (2) s cmmi10.329 @sf 518 445 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 614 445 p m (;) 8 (y) ks cmmi8.300 @sf 657 452 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 721 445 p m (;) s cmr10.329 @sf 15 r m (0) s cmmi10.329 @sf m (;) 14 (y) ks cmmi8.300 @sf 821 452 p m (p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 887 445 p m (;) 14 (y) ks cmmi8.300 @sf 936 452 p m (p) s cmr8.300 @sf m (+2) s cmmi10.329 @sf 1002 445 p m (;) s cmsy10.329 @sf 8 r m (\001) 8 (\001) 7 (\001) ks cmmi10.329 @sf 1090 445 p m (y) s cmmi8.300 @sf 1112 452 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1176 445 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 7 r m (\001) 8 (\001) 7 (\001) ks cmmi10.329 @sf 1343 445 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1404 425 p m (T) s cmmi10.329 @sf 247 506 p m (z) s cmr10.329 @sf 311 506 p m (=) s 388 506 p m (\() s cmmi10.329 @sf 406 506 p m (z) s cmr8.300 @sf 427 513 p m (1) s cmmi10.329 @sf 447 506 p m (;) 15 (z) ks cmr8.300 @sf 496 513 p m (2) s cmmi10.329 @sf 516 506 p m (;) s cmsy10.329 @sf 14 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 611 506 p m (;) 8 (z) ks cmmi8.300 @sf 653 513 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 718 506 p m (;) s cmr10.329 @sf 15 r m (0) s cmmi10.329 @sf m (;) 14 (z) ks cmmi8.300 @sf 817 513 p m (p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 882 506 p m (;) 15 (z) ks cmmi8.300 @sf 931 513 p m (p) s cmr8.300 @sf m (+2) s cmmi10.329 @sf 996 506 p m (;) s cmsy10.329 @sf 8 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 1084 506 p m (z) s cmmi8.300 @sf 1105 513 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1170 506 p m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) 8 (z) ks cmmi8.300 @sf 1256 513 p m (q) s cmr8.300 @sf 1 r m (+1) s cmmi10.329 @sf 1318 506 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 1414 506 p m (;) 8 (z) ks cmmi8.300 @sf 1456 513 p m (r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1519 506 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 7 r m (\001) 8 (\001) 7 (\001) ks cmmi10.329 @sf 1686 506 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1747 486 p m (T) s cmmi10.329 @sf 1782 506 p m (:) s cmr10.329 @sf 124 627 p m (Clearly) -3 (,) ks cmmi10.329 @sf 16 r m (x;) 15 (y) 2 (;) 8 (z) ks cmr10.329 @sf 17 r m (are) 16 (linearly) 18 (indep) 1 (en) 1 (den) 0 (t,) 17 (and) 16 (they) 16 (span) 17 (the) 16 (three-dimensional) 19 (in) 0 (v) -2 (arian) 0 (t) 14 (sub-) ks 74 710 p m (space) 11 (asso) 1 (ciated) 11 (with) ks cmmi10.329 @sf 11 r m (\013) s cmr10.329 @sf m (.) 19 (They) 11 (are) 11 (not) 10 (orthogonal,) 11 (in) 11 (general,) 12 (but) 11 (w) 0 (e) 10 (could) 11 (dev) 0 (elop) 11 (an) 11 (orthogonal) ks 74 793 p m (basis) 16 (from) 14 (this.) 21 (Sp) 1 (eci\014call) 1 (y) -3 (,) 16 (if) ks 542 950 p m (\() s cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) 11 (=) 13 (\() ks cmmi10.329 @sf m (q) s cmr8.300 @sf 790 957 p m (1) s cmmi10.329 @sf 809 950 p m (;) 8 (q) ks cmr8.300 @sf 850 957 p m (2) s cmmi10.329 @sf 869 950 p m (;) 8 (q) ks cmr8.300 @sf 910 957 p m (3) s cmr10.329 @sf 930 950 p m (\)) s cmex10.300 @sf 956 853 p m (0) s 955 926 p m (B) s 955 953 p m (@) s cmmi10.329 @sf 1012 894 p m (r) s cmr8.300 @sf 1033 901 p m (11) s cmmi10.329 @sf 1111 894 p m (r) s cmr8.300 @sf 1132 901 p m (12) s cmmi10.329 @sf 1211 894 p m (r) s cmr8.300 @sf 1232 901 p m (13) s cmmi10.329 @sf 1111 950 p m (r) s cmr8.300 @sf 1132 957 p m (22) s cmmi10.329 @sf 1211 950 p m (r) s cmr8.300 @sf 1232 957 p m (23) s cmmi10.329 @sf 1211 1007 p m (r) s cmr8.300 @sf 1232 1014 p m (33) s cmex10.300 @sf 1289 853 p m (1) s 1289 926 p m (C) s 1289 953 p m (A) s cmsy10.329 @sf 1338 950 p m (\021) s cmmi10.329 @sf 13 r m (Q) s cmr8.300 @sf 1422 957 p m (3) s cmmi10.329 @sf 1442 950 p m (R) s cmr8.300 @sf 1477 957 p m (3) s cmr10.329 @sf 761 1110 p m (\() s cmmi10.329 @sf 779 1110 p m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 945 1110 p m (Q) s cmr8.300 @sf 981 1117 p m (3) s cmmi10.329 @sf 1001 1110 p m (R) s cmr8.300 @sf 1036 1117 p m (3) s cmr10.329 @sf 1068 1110 p m (=) s cmmi10.329 @sf 13 r m (Q) s cmr8.300 @sf 1152 1117 p m (3) s cmmi10.329 @sf 1172 1110 p m (R) s cmr8.300 @sf 1207 1117 p m (3) s cmmi10.329 @sf 1226 1110 p m (T) s cmmi8.300 @sf 1253 1117 p m (\013) s cmr10.329 @sf 74 1220 p m (or) s 647 1303 p m (\() s cmmi10.329 @sf 665 1303 p m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 831 1303 p m (Q) s cmr8.300 @sf 867 1310 p m (3) s cmr10.329 @sf 899 1303 p m (=) s cmmi10.329 @sf 13 r m (Q) s cmr8.300 @sf 983 1310 p m (3) s cmr10.329 @sf 1003 1303 p m ([) s cmmi10.329 @sf m (R) s cmr8.300 @sf 1051 1310 p m (3) s cmmi10.329 @sf 1070 1303 p m (T) s cmmi8.300 @sf 1097 1310 p m (\013) s cmmi10.329 @sf 1121 1303 p m (R) s cmsy8.300 @sf 1156 1285 p m (\000) s cmr8.300 @sf m (1) s 1156 1315 p m (3) s cmr10.329 @sf 1203 1303 p m (]) 12 (=) ks cmmi10.329 @sf 13 r m (Q) s cmr8.300 @sf 1312 1310 p m (3) s cmmi10.329 @sf 1332 1303 p m (M) 2 (:) ks 74 1413 p m (Q) s cmr8.300 @sf 110 1420 p m (3) s cmr10.329 @sf 145 1413 p m (is) 16 (no) 0 (w) 13 (an) 15 (orthogonal) 15 (basis,) 15 (and) ks cmmi10.329 @sf 16 r m (M) s cmr10.329 @sf 20 r m (has) s cmmi10.329 @sf 15 r m (\013) s cmr10.329 @sf 15 r m (as) 15 (a) 15 (triple) 16 (eigen) 0 (v) -2 (alue.) ks 124 1538 p m (A) 14 (derogatory) 14 (matrix) 14 (will) 16 (b) 1 (e) 15 (rev) 0 (ealed) 14 (b) 0 (y) 14 (zero) 14 (v) -2 (alues) 15 (among) ks cmmi10.329 @sf 13 r m (d;) 8 (e;) 8 (f) ks cmr10.329 @sf 5 r m (.) 18 (Th) 0 (us) 14 (if) ks cmmi10.329 @sf 15 r m (d) s cmr10.329 @sf 12 r m (=) s cmmi10.329 @sf 13 r m (e) s cmr10.329 @sf 13 r m (=) s cmmi10.329 @sf 13 r m (f) s cmr10.329 @sf 17 r m (=) 13 (0,) 14 (w) 0 (e) ks 74 1621 p m (get) 15 (three) 15 (indep) 1 (end) 1 (en) 0 (t) 16 (eigen) 0 (v) 0 (ectors,) 13 (and) ks cmmi10.329 @sf 667 1778 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) 11 (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) s cmex10.300 @sf 1068 1681 p m (0) s 1066 1754 p m (B) s 1066 1781 p m (@) s cmmi10.329 @sf 1123 1721 p m (\013) s 1194 1778 p m (\013) s 1264 1834 p m (\013) s cmex10.300 @sf 1314 1681 p m (1) s 1314 1754 p m (C) s 1314 1781 p m (A) s cmmi10.329 @sf 1358 1778 p m (:) s cmr10.329 @sf 74 1935 p m (If) s cmmi10.329 @sf 16 r m (d) s cmr10.329 @sf 12 r m (=) s cmmi10.329 @sf 13 r m (f) s cmr10.329 @sf 18 r m (=) 13 (0) 15 (and) ks cmmi10.329 @sf 15 r m (e) s cmsy10.329 @sf 13 r m (6) s cmr10.329 @sf m (=) 13 (0,) 14 (w) 0 (e) 14 (ha) 0 (v) 0 (e) ks cmmi10.329 @sf 667 2092 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) 11 (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) 2 (;) 8 (z) ks cmr10.329 @sf 2 r m (\)) s cmex10.300 @sf 1068 1995 p m (0) s 1066 2068 p m (B) s 1066 2095 p m (@) s cmmi10.329 @sf 1123 2036 p m (\013) s 1194 2092 p m (\013) s 1269 2092 p m (e) s 1264 2149 p m (\013) s cmex10.300 @sf 1314 1995 p m (1) s 1314 2068 p m (C) s 1314 2095 p m (A) s cmmi10.329 @sf 1358 2092 p m (:) s cmr10.329 @sf 74 2260 p m (Then) 16 (w) 0 (e) 14 (ha) 0 (v) -1 (e) 14 (a) 14 (linear) 17 (divisor) 16 (\() ks cmmi10.329 @sf 695 2260 p m (\025) s cmsy10.329 @sf 9 r m (\000) s cmmi10.329 @sf 11 r m (\013) s cmr10.329 @sf m (\)) s 839 2260 p m (and) 15 (one) 15 (quadratic,) 15 (\() ks cmmi10.329 @sf 1244 2260 p m (\025) s cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 10 r m (\013) s cmr10.329 @sf m (\)) s cmr8.300 @sf 1373 2240 p m (2) s cmr10.329 @sf 1393 2260 p m (.) s 124 2385 p m (If) 18 (all) 18 (computations) 17 (are) 17 (exact) 17 (and) ks cmmi10.329 @sf 18 r m (T) s cmr10.329 @sf 23 r m (comes) 17 (from) 17 (exact) 17 (computation,) 17 (then) 18 (w) 0 (e) 16 (asso) 1 (ciate) 18 (only) ks 74 2468 p m (the) 15 (eigen) 0 (v) -2 (alues) 15 (that) 14 (are) 14 (truly) 15 (equal,) 15 (and) 15 (the) 15 (v) 0 (ectors) 13 (obtained) 15 (in) 16 (the) 14 (w) 0 (a) -1 (y) 13 (w) 0 (e) 13 (ha) 0 (v) 0 (e) 13 (describ) 1 (ed) 16 (are) ks 74 2551 p m (truly) 16 (indep) 1 (en) 1 (den) 0 (t.) 23 (In) 16 (practice,) 17 (ho) 0 (w) -1 (ev) -1 (er,) ks cmmi10.329 @sf 14 r m (T) s cmr10.329 @sf 21 r m (will) 18 (rarely) 16 (b) 1 (e) 16 (an) 16 (exact) 16 (matrix.) 22 (Usually) 17 (it) 16 (will) 17 (ha) 0 (v) 0 (e) ks 74 2634 p m (b) 1 (een) 16 (obtained) 15 (from) 14 (a) 14 (matrix) ks cmmi10.329 @sf 14 r m (A) s cmr10.329 @sf 15 r m (b) 0 (y) -3 (,) 12 (sa) 0 (y) -3 (,) 12 (the) ks cmmi10.329 @sf 15 r m (QR) s cmr10.329 @sf 14 r m (algorithm.) 20 (Ev) 0 (en) 14 (if) ks cmmi10.329 @sf 15 r m (A) s cmr10.329 @sf 14 r m (had) 15 (defectiv) 0 (e) 15 (eigen) 0 (v) -2 (alues,) ks cmmi10.329 @sf 74 2717 p m (T) s cmr10.329 @sf 26 r m (will) 22 (usually) 22 (not) 20 (ha) 0 (v) 0 (e) 18 (an) 0 (y) 19 (rep) 1 (eated) 21 (diagonal) 21 (elemen) 0 (ts.) 35 (A) 21 (real) 20 (problem) 21 (is) 21 (to) 20 (decide) 22 (whic) 0 (h) ks 996 2850 p m (19) s @eop 18 @bop0 cmr10.329 @sf [<7FE3FF000700700007007000070070000700700007007000070070000700700007007000070070 00070070000700700007007000070070000700700007007000070070000700700007007000FFFF FF80070070000700700007007000070070000700700007007000070070000300F0300380F87801 C0787800F06E30001F83E0> 29 32 0 0 27] 11 @dc cmmi10.329 @sf [ 26 26 -4 2 35] 62 @dc [ 23 29 1 9 23] 112 @dc 18 @bop1 cmr10.329 @sf 74 67 p m (giving) s 939 150 p m (0) s cmmi10.329 @sf m (y) s cmmi8.300 @sf 984 157 p m (p) s cmr10.329 @sf 1016 150 p m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 253 p m (Again) s cmmi10.329 @sf 18 r m (y) s cmmi8.300 @sf 231 260 p m (p) s cmr10.329 @sf 268 253 p m (is) 17 (arbitrary) -3 (,) 16 (and) 17 (it) 17 (is) 18 (simplest) 18 (to) 16 (tak) 0 (e) ks cmmi10.329 @sf 16 r m (y) s cmmi8.300 @sf 1064 260 p m (p) s cmr10.329 @sf 1101 253 p m (to) 16 (b) 1 (e) 18 (zero.) 25 (There) 17 (are) 17 (no) 17 (further) 17 (problems,) ks 74 336 p m (and) 16 (w) 0 (e) 13 (ha) 0 (v) 0 (e) ks cmmi10.329 @sf 479 392 p m (x) s cmr10.329 @sf 547 392 p m (=) s 624 392 p m (\() s cmmi10.329 @sf 641 392 p m (x) s cmr8.300 @sf 667 399 p m (1) s cmmi10.329 @sf 687 392 p m (;) 15 (x) ks cmr8.300 @sf 741 399 p m (2) s cmmi10.329 @sf 760 392 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 856 392 p m (;) 8 (x) ks cmmi8.300 @sf 903 399 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 967 392 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 14 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 1157 392 p m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf -1 r m (;) s cmsy10.329 @sf 8 r m (\001) 7 (\001) 7 (\001) ks cmmi10.329 @sf 1346 392 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1407 372 p m (T) s cmmi10.329 @sf 481 451 p m (y) s cmr10.329 @sf 547 451 p m (=) s 624 451 p m (\() s cmmi10.329 @sf 641 451 p m (y) s cmr8.300 @sf 663 458 p m (1) s cmmi10.329 @sf 683 451 p m (;) 15 (y) ks cmr8.300 @sf 733 458 p m (2) s cmmi10.329 @sf 753 451 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 849 451 p m (;) 8 (y) ks cmmi8.300 @sf 892 458 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 956 451 p m (;) s cmr10.329 @sf 15 r m (0) s cmmi10.329 @sf m (;) 15 (y) ks cmmi8.300 @sf 1057 458 p m (p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 1122 451 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 1218 451 p m (y) s cmmi8.300 @sf 1240 458 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1304 451 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 7 r m (\001) 8 (\001) 7 (\001) ks cmmi10.329 @sf 1471 451 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1532 431 p m (T) s cmr10.329 @sf 74 526 p m (with) 16 (\() ks cmmi10.329 @sf 195 526 p m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 11 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 362 526 p m (x) s cmr10.329 @sf 12 r m (=) 13 (0,) 15 (\() ks cmmi10.329 @sf 517 526 p m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 683 526 p m (y) s cmr10.329 @sf 15 r m (=) s cmmi10.329 @sf 13 r m (dx) s cmr10.329 @sf m (,) 14 (or) ks cmmi10.329 @sf 749 647 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) 11 (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) s cmex10.300 @sf 1063 575 p m ( ) s cmmi10.329 @sf 1115 620 p m (\013) s 1189 620 p m (d) s 1186 675 p m (\013) s cmex10.300 @sf 1236 575 p m (!) s cmmi10.329 @sf 1276 647 p m (:) s cmr10.329 @sf 124 803 p m (No) 0 (w) 9 (for) 11 (the) 11 (third) 11 (v) 0 (ector,) 10 (w) 0 (e) 10 (shall) 12 (ignore) 11 (the) 12 (p) 1 (ossibilit) 0 (y) 12 (of) 10 (its) 12 (b) 1 (eing) 12 (derogatory) 10 (for) 10 (the) 11 (momen) 0 (t.) ks 74 886 p m (W) -3 (e) 14 (attempt) 15 (to) 14 (solv) 0 (e) ks 859 969 p m (\() s cmmi10.329 @sf 877 969 p m (T) s cmmi8.300 @sf 904 976 p m (r) 1 (r) ks cmsy10.329 @sf 949 969 p m (\000) s cmmi10.329 @sf 11 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 1073 969 p m (z) s cmr10.329 @sf 15 r m (=) 13 (0) ks 74 1072 p m (starting) 15 (with) ks cmmi10.329 @sf 15 r m (z) s cmmi8.300 @sf 368 1079 p m (r) s cmr10.329 @sf 400 1072 p m (=) 13 (1.) 20 (W) -3 (e) 14 (pro) 1 (ceed) 16 (as) 15 (usual) 16 (un) 0 (til) 15 (w) 0 (e) 14 (reac) 0 (h) ks cmmi10.329 @sf 14 r m (z) s cmmi8.300 @sf 1240 1079 p m (q) s cmr10.329 @sf 1259 1072 p m (.) 20 (A) 0 (t) 14 (this) 15 (stage) 15 (w) 0 (e) 14 (ha) 0 (v) -1 (e) ks 466 1159 p m (0) s cmmi10.329 @sf m (z) s cmmi8.300 @sf 510 1166 p m (q) s cmr10.329 @sf 539 1159 p m (+) s cmmi10.329 @sf 11 r m (t) s cmmi8.300 @sf 601 1166 p m (q) 1 (;q) ks cmr8.300 @sf 1 r m (+1) s cmmi10.329 @sf 692 1159 p m (z) s cmmi8.300 @sf 713 1166 p m (q) s cmr8.300 @sf 1 r m (+1) s cmr10.329 @sf 787 1159 p m (+) s cmsy10.329 @sf 11 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 896 1159 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 957 1166 p m (q) 1 (;r) ks cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1049 1159 p m (z) s cmmi8.300 @sf 1070 1166 p m (r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1144 1159 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1205 1166 p m (q) 1 (;r) ks cmr10.329 @sf 1264 1159 p m (=) s 1341 1159 p m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 1433 1159 p m (so) 15 (that) ks cmmi10.329 @sf 585 1215 p m (t) s cmmi8.300 @sf 601 1222 p m (q) 1 (;q) ks cmr8.300 @sf 1 r m (+1) s cmmi10.329 @sf 692 1215 p m (z) s cmmi8.300 @sf 713 1222 p m (q) s cmr8.300 @sf 1 r m (+1) s cmr10.329 @sf 787 1215 p m (+) s cmsy10.329 @sf 11 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 896 1215 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 957 1222 p m (q) 1 (;r) ks cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1049 1215 p m (z) s cmmi8.300 @sf 1070 1222 p m (r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1144 1215 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1205 1222 p m (q) 1 (;r) ks cmr10.329 @sf 1264 1215 p m (=) s cmmi10.329 @sf 1342 1215 p m (e:) s cmr10.329 @sf 74 1301 p m (Hence,) 16 (w) 0 (e) 14 (solv) 0 (e) ks 845 1384 p m (\() s cmmi10.329 @sf m (T) s cmmi8.300 @sf 890 1391 p m (r) 1 (r) ks cmsy10.329 @sf 936 1384 p m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (z) s cmr10.329 @sf 14 r m (=) s cmmi10.329 @sf 13 r m (ey) 2 (:) ks cmr10.329 @sf 74 1487 p m (This) 16 (do) 1 (es) 15 (not) 15 (a\013ect) 15 (the) 15 (comp) 1 (onen) 0 (ts) 14 (already) 16 (computed) 15 (since) ks cmmi10.329 @sf 17 r m (y) s cmmi8.300 @sf 1316 1494 p m (i) s cmr10.329 @sf 1343 1487 p m (=) 13 (0,) 14 (\() ks cmmi10.329 @sf 1459 1487 p m (i) 12 (>) 13 (q) ks cmr10.329 @sf 2 r m (\)) s 1575 1487 p m (.) s 124 1611 p m (F) -3 (or) 14 (con) 0 (v) -1 (enience) 15 (w) 0 (e) 14 (then) 16 (tak) 0 (e) ks cmmi10.329 @sf 13 r m (z) s cmmi8.300 @sf 744 1618 p m (q) s cmr10.329 @sf 776 1611 p m (=) 13 (0.) 20 (W) -3 (e) 14 (con) 0 (tin) 0 (ue) 14 (un) 0 (til) 15 (reac) 0 (hing) ks cmmi10.329 @sf 15 r m (z) s cmmi8.300 @sf 1449 1618 p m (p) s cmr10.329 @sf 1469 1611 p m (.) 19 (W) -3 (e) 15 (no) 0 (w) 13 (ha) 0 (v) 0 (e) ks 562 1726 p m (0) s cmmi10.329 @sf m (z) s cmmi8.300 @sf 606 1733 p m (p) s cmr10.329 @sf 636 1726 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 697 1733 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 790 1726 p m (z) s cmmi8.300 @sf 811 1733 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 886 1726 p m (+) s cmsy10.329 @sf 11 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 995 1726 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1056 1733 p m (p;r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1149 1726 p m (z) s cmmi8.300 @sf 1170 1733 p m (r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1244 1726 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1305 1733 p m (p;r) s cmr10.329 @sf 1365 1726 p m (=) s cmmi10.329 @sf 13 r m (ey) s cmmi8.300 @sf 1456 1733 p m (p) s cmr10.329 @sf 74 1840 p m (i.e.,) s cmmi10.329 @sf 635 1923 p m (t) s cmmi8.300 @sf 651 1930 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 744 1923 p m (z) s cmmi8.300 @sf 765 1930 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 840 1923 p m (+) s cmsy10.329 @sf 10 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 948 1923 p m (+) s cmmi10.329 @sf 11 r m (t) s cmmi8.300 @sf 1010 1930 p m (p;r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1102 1923 p m (z) s cmmi8.300 @sf 1123 1930 p m (r) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1197 1923 p m (+) s cmmi10.329 @sf 11 r m (t) s cmmi8.300 @sf 1259 1930 p m (p;r) s cmr10.329 @sf 1318 1923 p m (=) s cmmi10.329 @sf 13 r m (f) 2 (:) ks cmr10.329 @sf 74 2026 p m (If) s cmmi10.329 @sf 16 r m (f) s cmsy10.329 @sf 17 r m (6) s cmr10.329 @sf m (=) 13 (0,) 15 (w) 0 (e) 14 (w) 0 (ould) 14 (get) ks cmmi10.329 @sf 15 r m (z) s cmmi8.300 @sf 554 2033 p m (p) s cmr10.329 @sf 587 2026 p m (=) 13 (0.) 19 (T) -3 (o) 14 (a) 0 (v) -1 (oid) 14 (this) 16 (situation,) 15 (w) 0 (e) 14 (solv) 0 (e) ks 787 2141 p m (\() s cmmi10.329 @sf 805 2141 p m (T) s cmmi8.300 @sf 832 2148 p m (r) 1 (r) ks cmsy10.329 @sf 877 2141 p m (\000) s cmmi10.329 @sf 11 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 1001 2141 p m (z) s cmr10.329 @sf 15 r m (=) s cmmi10.329 @sf 13 r m (ey) s cmr10.329 @sf 12 r m (+) s cmmi10.329 @sf 10 r m (f) 5 (x:) ks cmr10.329 @sf 74 2255 p m (This) 21 (do) 1 (es) 20 (not) 19 (a\013ect) 19 (previous) 21 (comp) 1 (onen) 0 (ts) 19 (since) ks cmmi10.329 @sf 21 r m (x) s cmmi8.300 @sf 1085 2262 p m (i) s cmr10.329 @sf 1119 2255 p m (=) 21 (0) 19 (for) ks cmmi10.329 @sf 19 r m (i) 20 (>) 21 (p) ks cmr10.329 @sf m (.) 34 (The) 20 (equation) 20 (for) ks cmmi10.329 @sf 19 r m (z) s cmmi8.300 @sf 1836 2262 p m (p) s cmr10.329 @sf 1876 2255 p m (then) s 74 2338 p m (b) 1 (ecomes) ks 939 2421 p m (0) s cmmi10.329 @sf m (z) s cmmi8.300 @sf 983 2428 p m (p) s cmr10.329 @sf 1016 2421 p m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 2524 p m (If) 16 (w) 0 (e) 14 (tak) 0 (e) ks cmmi10.329 @sf 13 r m (z) s cmmi8.300 @sf 306 2531 p m (p) s cmr10.329 @sf 339 2524 p m (=) 13 (0) 14 (and) 16 (then) 15 (determine) ks cmmi10.329 @sf 17 r m (z) s cmmi8.300 @sf 850 2531 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 915 2524 p m (;) 8 (z) ks cmmi8.300 @sf 957 2531 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (2) s cmmi10.329 @sf 1021 2524 p m (;) s cmsy10.329 @sf 8 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 1102 2524 p m (;) 15 (z) ks cmr8.300 @sf 1151 2531 p m (1) s cmr10.329 @sf 1170 2524 p m (,) 15 (w) 0 (e) 14 (then) 16 (ha) 0 (v) -1 (e) ks 782 2612 p m (\() s cmmi10.329 @sf m (T) s cmsy10.329 @sf 15 r m (\000) s cmmi10.329 @sf 11 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (x) s cmr10.329 @sf 1026 2612 p m (=) s 1103 2612 p m (0) s 784 2669 p m (\() s cmmi10.329 @sf m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (y) s cmr10.329 @sf 1026 2669 p m (=) s cmmi10.329 @sf 1103 2669 p m (dx) s cmr10.329 @sf 785 2725 p m (\() s cmmi10.329 @sf m (T) s cmsy10.329 @sf 15 r m (\000) s cmmi10.329 @sf 11 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (z) s cmr10.329 @sf 1026 2725 p m (=) s cmmi10.329 @sf 1103 2725 p m (ey) s cmr10.329 @sf 12 r m (+) s cmmi10.329 @sf 10 r m (f) 5 (x) ks cmr10.329 @sf 996 2850 p m (18) s @eop 17 @bop0 cmmi10.329 @sf [<1801800F0038038018801C01C018401C01C038401C01C038201C01C01C200E00E01C100E00E00E 000E00E00E000E00E00E0007007007000700700700070070070087007007004380380380438038 038043C034030023A032030023186186001E07C07C00> 36 20 -2 0 40] 109 @dc cmmi8.300 @sf [ 29 23 -1 0 29] 72 @dc cmmi10.329 @sf [<07E00C1818003000700070007000F000F000F0007FF07800780038003C001C000E00070001C000 7E> 15 20 -2 0 18] 15 @dc cmmi8.300 @sf [<387046C8C1C480C480C480C0C060C060C06060606070307018F007300018001800180018000C00 0C000C000C003E> 15 23 -2 0 18] 100 @dc [<1F0030C04000C000C000C000C000C000FF006000600030001C0007C0> 10 14 -2 0 14] 15 @dc cmsy10.329 @sf [<0000000400000000020000000002000000000100000000008000000000400000000020FFFFFFFF FCFFFFFFFFFC000000002000000000400000000080000000010000000002000000000200000000 0400> 38 16 -3 -3 45] 33 @dc 17 @bop1 cmr10.329 @sf 124 67 p m (It) 15 (is) 16 (simplest) 16 (to) 15 (tak) 0 (e) ks cmmi10.329 @sf 13 r m (y) s cmmi8.300 @sf 572 74 p m (p) s cmr10.329 @sf 605 67 p m (=) 13 (0.) 20 (Hence,) 15 (when) ks cmmi10.329 @sf 16 r m (d) s cmr10.329 @sf 12 r m (=) 13 (0,) 15 (w) 0 (e) 14 (obtain) ks cmmi10.329 @sf 501 171 p m (x) s cmr10.329 @sf 568 171 p m (=) s 645 171 p m (\() s cmmi10.329 @sf m (x) s cmr8.300 @sf 689 178 p m (1) s cmmi10.329 @sf 708 171 p m (;) 15 (x) ks cmr8.300 @sf 762 178 p m (2) s cmmi10.329 @sf 782 171 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 878 171 p m (;) 8 (x) ks cmmi8.300 @sf 925 178 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 989 171 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 14 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 14 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 1178 171 p m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 8 r m (\001) 7 (\001) 8 (\001) ks cmmi10.329 @sf 1325 171 p m (;) s cmr10.329 @sf 8 r m (0\)) s cmmi8.300 @sf 1387 155 p m (T) s cmmi10.329 @sf 503 230 p m (y) s cmr10.329 @sf 568 230 p m (=) s 645 230 p m (\() s cmmi10.329 @sf 663 230 p m (y) s cmr8.300 @sf 685 237 p m (1) s cmmi10.329 @sf 705 230 p m (;) 15 (y) ks cmr8.300 @sf 755 237 p m (2) s cmmi10.329 @sf 775 230 p m (;) s cmsy10.329 @sf 14 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 870 230 p m (;) 8 (y) ks cmmi8.300 @sf 913 237 p m (p) s cmsy8.300 @sf m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 978 230 p m (;) s cmr10.329 @sf 15 r m (0) s cmmi10.329 @sf m (;) 14 (y) ks cmmi8.300 @sf 1078 237 p m (p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 1143 230 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 1239 230 p m (y) s cmmi8.300 @sf 1261 237 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1326 230 p m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 8 r m (\001) 7 (\001) 7 (\001) ks cmmi10.329 @sf 1449 230 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1510 210 p m (T) s cmr10.329 @sf 74 329 p m (These) 19 (t) 0 (w) -1 (o) 16 (v) 0 (ectors) 16 (are) 18 (ob) 0 (viously) 18 (linearly) 20 (indep) 1 (end) 1 (en) 0 (t.) 29 (Hence) 19 (w) 0 (e) 17 (ha) 0 (v) 0 (e) 16 (t) 0 (w) -1 (o) 16 (eigen) 0 (v) 0 (ectors) 16 (corre-) ks 74 412 p m (sp) 1 (onding) 17 (to) ks cmmi10.329 @sf 14 r m (\013) s cmr10.329 @sf m (.) 20 (Both) 15 (satisfy) 15 (\() ks cmmi10.329 @sf m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (x) s cmr10.329 @sf 12 r m (=) 13 (0,) 15 (and) 15 (\() ks cmmi10.329 @sf m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (y) s cmr10.329 @sf 14 r m (=) 13 (0.) ks 124 536 p m (If) 14 (w) 0 (e) 13 (had) 14 (tak) 0 (en) ks cmmi10.329 @sf 13 r m (y) s cmmi8.300 @sf 466 543 p m (p) s cmr10.329 @sf 500 536 p m (to) 14 (b) 1 (e) ks cmmi10.329 @sf 14 r m (m) s cmr10.329 @sf 14 r m (instead) 15 (of) 14 (zero,) 13 (the) 15 (solution) 14 (w) 0 (ould) 14 (ha) 0 (v) -1 (e) 13 (b) 1 (een) ks cmmi10.329 @sf 15 r m (y) s cmr10.329 @sf 10 r m (+) s cmmi10.329 @sf 8 r m (mx) s cmr10.329 @sf m (.) 19 (This) 15 (is) 14 (\014ne) ks 74 619 p m (since) s cmmi10.329 @sf 17 r m (y) s cmr10.329 @sf 12 r m (+) s cmmi10.329 @sf 10 r m (mx) s cmr10.329 @sf 15 r m (is) 16 (also) 15 (an) 15 (eigen) 0 (v) 0 (ector.) 18 (W) -3 (e) 14 (could) 16 (ha) 0 (v) 0 (e) 13 (c) 0 (hosen) ks cmmi10.329 @sf 15 r m (y) s cmr10.329 @sf 12 r m (+) s cmmi10.329 @sf 10 r m (mx) s cmr10.329 @sf 15 r m (orthogonal) 15 (to) ks cmmi10.329 @sf 14 r m (x) s cmr10.329 @sf m (,) s cmmi10.329 @sf 654 748 p m (x) s cmmi8.300 @sf 680 730 p m (H) s cmr10.329 @sf 714 748 p m (\() s cmmi10.329 @sf m (y) s cmr10.329 @sf 12 r m (+) s cmmi10.329 @sf 10 r m (mx) s cmr10.329 @sf m (\)) 12 (=) 13 (0) ks cmmi10.329 @sf m (;) 17 (m) ks cmr10.329 @sf 13 r m (=) 13 (\() ks cmsy10.329 @sf m (\000) s cmmi10.329 @sf m (x) s cmmi8.300 @sf 1188 730 p m (H) s cmmi10.329 @sf 1221 748 p m (y) 2 (=x) ks cmmi8.300 @sf 1294 730 p m (H) s cmmi10.329 @sf 1328 748 p m (x) s cmr10.329 @sf m (\)) s cmmi10.329 @sf m (:) s cmr10.329 @sf 124 918 p m (That) 17 (the) 17 (matrix) 18 (will) 19 (b) 1 (e) 18 (derogatory) 16 (is) 18 (m) 0 (uc) 0 (h) 15 (less) 18 (probable) 19 (than) 17 (that) 17 (it) 18 (will) 19 (b) 1 (e) 18 (defectiv) 0 (e.) 26 (In) ks 74 1001 p m (fact,) 21 (ev) 0 (en) 20 (if) ks cmmi10.329 @sf 21 r m (A) s cmr10.329 @sf 21 r m (w) 0 (ere) 19 (exactly) 21 (derogatory) -3 (,) ks cmmi10.329 @sf 19 r m (T) s cmr10.329 @sf 26 r m (w) 0 (ould) 20 (probably) 22 (not) 20 (b) 1 (e,) 22 (ev) 0 (en) 20 (if) 21 (it) 21 (still) 22 (had) 20 (exact) ks 74 1084 p m (m) 0 (ultiple) 16 (eigen) 0 (v) -2 (alues.) ks 124 1209 p m (Supp) 1 (ose) 16 (no) 0 (w) ks cmmi10.329 @sf 14 r m (d) s cmsy10.329 @sf 12 r m (6) s cmr10.329 @sf m (=) 13 (0.) 20 (T) -3 (o) 14 (get) ks cmmi10.329 @sf 14 r m (y) s cmmi8.300 @sf 701 1216 p m (p) s cmr10.329 @sf 722 1209 p m (,) 14 (w) 0 (e) 14 (w) 0 (ould) 15 (need) 16 (to) 14 (solv) 0 (e) ks 921 1338 p m (0) s cmmi10.329 @sf m (y) s cmmi8.300 @sf 966 1345 p m (p) s cmr10.329 @sf 998 1338 p m (=) s cmsy10.329 @sf 13 r m (\000) s cmmi10.329 @sf m (d:) s cmr10.329 @sf 74 1466 p m (Hence) 15 (w) 0 (e) 13 (cannot) 13 (get) 14 (a) 13 (second) 15 (eigen) 0 (v) 0 (ector.) 17 (Notice) 15 (that) 13 (if) ks cmmi10.329 @sf 14 r m (\025) s cmmi8.300 @sf 1237 1473 p m (q) s cmr10.329 @sf 1269 1466 p m (w) 0 (ere) ks cmmi10.329 @sf 13 r m (\025) s cmmi8.300 @sf 1400 1473 p m (p) s cmr10.329 @sf 1427 1466 p m (+) s cmmi10.329 @sf 7 r m (\017) s cmr10.329 @sf 15 r m (instead) 15 (of) ks cmmi10.329 @sf 13 r m (\025) s cmmi8.300 @sf 1735 1473 p m (p) s cmr10.329 @sf 1754 1466 p m (,) 14 (w) 0 (e) 13 (w) 0 (ould) ks 74 1549 p m (b) 1 (e) 16 (solving) ks cmmi10.329 @sf 929 1632 p m (\017y) s cmmi8.300 @sf 969 1639 p m (p) s cmr10.329 @sf 1002 1632 p m (=) s cmsy10.329 @sf 13 r m (\000) s cmmi10.329 @sf m (d) s cmr10.329 @sf 74 1742 p m (at) 15 (this) 17 (stage,) 14 (giving) 17 (an) 16 (erroneous) 16 (v) -2 (alue) 16 (of) ks cmmi10.329 @sf 15 r m (y) s cmmi8.300 @sf 947 1749 p m (p) s cmr10.329 @sf 967 1742 p m (.) 22 (Ob) 0 (viously) -3 (,) 15 (in) 17 (this) 16 (case) 16 (the) 16 (\014rst) ks cmmi10.329 @sf 15 r m (p) s cmr10.329 @sf 16 r m (comp) 1 (onen) 0 (ts) 15 (of) ks cmmi10.329 @sf 74 1825 p m (y) s cmr10.329 @sf 19 r m (w) 0 (ould) 17 (b) 1 (e) 17 (essen) 0 (tially) ks cmsy8.300 @sf 538 1808 p m (\000) s cmmi8.300 @sf m (d) s 538 1815 p 46 2 ru 554 1841 p m (\017) s cmmi10.329 @sf 589 1825 p m (x) s cmr10.329 @sf 11 r m (+) 12 (\(v) 0 (ector) 13 (that) 14 (is) 16 (not) 15 (to) 1 (o) 14 (large\)) ks 1245 1825 p m (.) 26 (As) ks cmmi10.329 @sf 17 r m (\017) s cmsy10.329 @sf 16 r m (!) s cmr10.329 @sf 16 r m (0,) 17 (the) 17 (v) 0 (ector) ks cmmi10.329 @sf 16 r m (y) s cmr10.329 @sf 19 r m (tends) 17 (to) 16 (a) ks 74 1908 p m (m) 0 (ultiple) 15 (of) ks cmmi10.329 @sf 14 r m (x) s cmr10.329 @sf 14 r m (with) 15 (a) 14 (relativ) 0 (ely) 14 (negligibl) 1 (e) 16 (amoun) 0 (t) 13 (of) 14 (in) 0 (terference.) 19 (In) 15 (the) 15 (limit) 15 (w) 0 (e) 13 (\014nd) 15 (that) ks cmmi10.329 @sf 14 r m (y) s cmr10.329 @sf 16 r m (and) s cmmi10.329 @sf 74 1991 p m (x) s cmr10.329 @sf 14 r m (are) 14 (in) 14 (exactly) 15 (the) 14 (same) 13 (direction;) 15 (the) 14 (last) ks cmmi10.329 @sf 14 r m (q) s cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 8 r m (p) s cmr10.329 @sf 13 r m (comp) 1 (onen) 0 (ts) 13 (of) ks cmmi10.329 @sf 14 r m (y) s cmr10.329 @sf 16 r m (are) 13 (negligibl) 1 (e) 16 (compared) 14 (with) ks 74 2074 p m (the) 15 (rest) 15 (when) ks cmmi10.329 @sf 16 r m (q) s cmr10.329 @sf 17 r m (is) 16 (small,) 15 (and) 16 (arbitrarily) 15 (v) -2 (anish) 16 (altogether) 15 (in) 16 (the) 15 (normalized) ks cmmi10.329 @sf 16 r m (y) s cmr10.329 @sf 2 r m (.) s 124 2199 p m (W) -3 (e) 14 (cannot) 15 (\014nd) 16 (a) 15 (second) 16 (eigen) 0 (v) 0 (ector.) 18 (W) -3 (e) 14 (can,) 15 (ho) 0 (w) -1 (ev) -1 (er,) 13 (\014nd) 16 (a) 15 (v) 0 (ector) ks cmmi10.329 @sf 13 r m (y) s cmr10.329 @sf 17 r m (suc) 0 (h) 15 (that) ks 849 2328 p m (\() s cmmi10.329 @sf m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmmi8.300 @sf 982 2335 p m (q) s cmmi10.329 @sf 1001 2328 p m (I) s cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf m (y) s cmr10.329 @sf 14 r m (=) s cmmi10.329 @sf 13 r m (dx:) s cmr10.329 @sf 74 2456 p m (Hence) 23 (the) 22 (determination) 23 (of) ks cmmi10.329 @sf 22 r m (y) s cmr10.329 @sf 23 r m (pro) 1 (ceeds) 23 (as) 22 (b) 1 (efore,) 23 (from) ks cmmi10.329 @sf 22 r m (y) s cmmi8.300 @sf 1255 2463 p m (q) s cmr10.329 @sf 1296 2456 p m (to) s cmmi10.329 @sf 21 r m (y) s cmmi8.300 @sf 1380 2463 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 1445 2456 p m (,) 24 (since) ks cmmi10.329 @sf 23 r m (x) s cmr10.329 @sf 22 r m (is) 22 (zero) 22 (in) 23 (these) ks 74 2539 p m (comp) 1 (onen) 0 (ts.) 19 (W) -3 (e) 14 (no) 0 (w) 14 (ha) 0 (v) -1 (e) ks 382 2641 p m (0) s cmmi10.329 @sf m (y) s cmmi8.300 @sf 427 2648 p m (p) s cmr10.329 @sf 457 2641 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 518 2648 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 611 2641 p m (y) s cmmi8.300 @sf 633 2648 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 708 2641 p m (+) s cmsy10.329 @sf 11 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 817 2641 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 878 2648 p m (p;q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 970 2641 p m (y) s cmmi8.300 @sf 992 2648 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1067 2641 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1128 2648 p m (p;q) s cmr10.329 @sf 1188 2641 p m (=) s cmmi10.329 @sf 13 r m (dx) s cmmi8.300 @sf 1286 2648 p m (p) s cmr10.329 @sf 1347 2641 p m (=) s cmmi10.329 @sf 1425 2641 p m (d;) s cmr10.329 @sf 1518 2641 p m (so) 14 (that) ks cmmi10.329 @sf 584 2696 p m (t) s cmmi8.300 @sf 600 2703 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 693 2696 p m (y) s cmmi8.300 @sf 715 2703 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 790 2696 p m (+) s cmsy10.329 @sf 11 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 899 2696 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 960 2703 p m (p;q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1052 2696 p m (y) s cmmi8.300 @sf 1074 2703 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1149 2696 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1210 2703 p m (p;q) s cmr10.329 @sf 1270 2696 p m (=) s cmmi10.329 @sf 1347 2696 p m (d;) s cmr10.329 @sf 996 2850 p m (17) s @eop 16 @bop0 cmmi8.300 @sf [ 15 23 -2 0 18] 107 @dc cmmi10.329 @sf [<600F00E01880701880703840703840703840381C203838003838003870001FC0001E00001D0000 1C80000E40600E20F00E10F00E08700706100701E0070000070000038000038000038000038000 01C00001C00001C00001C0000FE00001E000> 20 32 -2 0 24] 107 @dc 1.000000 /cmr9.300 @newfont cmr9.300 @sf [<03800E401C201C201C201C201C201C001C001C001C001C001C001C001C00FFC03C001C000C000C 00040004000400> 11 23 -1 0 15] 116 @dc [ 18 26 -1 0 21] 104 @dc cmmi8.300 @sf [<300F00300880181880181840180C40180C000C0C000C06000C06008E060046020047020044C600 383C00> 18 14 -1 0 21] 110 @dc cmmi10.329 @sf [<000001C00000078000001E00000078000001E00000078000000E00000038000000F0000003C000 000F0000003C000000F0000000F00000003C0000000F00000003C0000000F0000000380000000E 0000000780000001E0000000780000001E0000000780000001C0> 26 26 -4 2 35] 60 @dc 16 @bop1 cmr10.329 @sf 74 67 p m (of) 16 (the) 16 (matrix) ks cmmi10.329 @sf 16 r m (T) s cmr10.329 @sf 6 r m (.) 22 (W) -3 (e) 16 (assume) 16 (that) 15 (the) 16 (matrix) ks cmmi10.329 @sf 16 r m (T) s cmr10.329 @sf 22 r m (is) 17 (deriv) 0 (ed) 16 (from) 15 (some) 16 (square) 16 (general) 17 (matrix) ks cmmi10.329 @sf 15 r m (A) s cmr10.329 @sf m (.) s 74 150 p m (Supp) 1 (ose) ks cmmi10.329 @sf 17 r m (\025) s cmmi8.300 @sf 280 157 p m (k) s cmr10.329 @sf 316 150 p m (is) 16 (the) ks cmmi10.329 @sf 16 r m (k) s cmr9.300 @sf 466 135 p m (th) s cmr10.329 @sf 520 150 p m (eigen) 0 (v) -2 (alue) 15 (along) 16 (the) 15 (diagonal) 16 (of) ks cmmi10.329 @sf 16 r m (T) s cmr10.329 @sf 21 r m (and) s cmmi10.329 @sf 16 r m (T) s cmmi8.300 @sf 1336 157 p m (k) 1 (k) ks cmr10.329 @sf 1391 150 p m (is) 16 (the) 16 (leading) ks cmmi10.329 @sf 17 r m (k) s cmsy10.329 @sf 11 r m (\002) s cmmi10.329 @sf 11 r m (k) s cmr10.329 @sf 16 r m (minor) 16 (in) ks 74 233 p m (the) 15 (matrix) ks cmmi10.329 @sf 15 r m (T) s cmr10.329 @sf 6 r m (.) s 124 358 p m (If) s cmmi10.329 @sf 15 r m (\025) s cmmi8.300 @sf 196 365 p m (k) s cmr10.329 @sf 232 358 p m (is) 16 (a) 15 (simple) 17 (eigen) 0 (v) -2 (alue,) 14 (w) 0 (e) 14 (just) 15 (solv) 0 (e) ks 857 486 p m (\() s cmmi10.329 @sf 874 486 p m (T) s cmsy10.329 @sf 16 r m (\000) s cmmi10.329 @sf 11 r m (\025) s cmmi8.300 @sf 990 493 p m (k) s cmmi10.329 @sf 1011 486 p m (I) s cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 1060 486 p m (x) s cmr10.329 @sf 12 r m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 615 p m (This) 16 (giv) 0 (es) ks cmmi10.329 @sf 14 r m (x) s cmmi8.300 @sf 315 622 p m (k) s cmr8.300 @sf 1 r m (+1) s cmmi10.329 @sf 382 615 p m (;) 8 (x) ks cmmi8.300 @sf 429 622 p m (k) s cmr8.300 @sf 1 r m (+2) s cmmi10.329 @sf 494 615 p m (;) s cmsy10.329 @sf 8 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 575 615 p m (;) 8 (x) ks cmmi8.300 @sf 622 622 p m (n) s cmr10.329 @sf 657 615 p m (=) 13 (0.) 20 (Next,) 15 (w) 0 (e) 14 (tak) 0 (e) ks cmmi10.329 @sf 13 r m (x) s cmmi8.300 @sf 1076 622 p m (k) s cmr10.329 @sf 1110 615 p m (=) 13 (1) 15 (and) 15 (solv) 0 (e) ks 846 744 p m (\() s cmmi10.329 @sf 864 744 p m (T) s cmmi8.300 @sf 891 751 p m (k) 1 (k) ks cmsy10.329 @sf 941 744 p m (\000) s cmmi10.329 @sf 10 r m (\025) s cmmi8.300 @sf 1013 751 p m (k) s cmmi10.329 @sf 1034 744 p m (I) s cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 1083 744 p m (x) s cmr10.329 @sf 13 r m (=) 13 (0) ks 74 872 p m (for) s cmmi10.329 @sf 15 r m (x) s cmmi8.300 @sf 170 879 p m (k) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 236 872 p m (;) 15 (x) ks cmmi8.300 @sf 290 879 p m (k) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (2) s cmmi10.329 @sf 356 872 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 452 872 p m (;) 8 (x) ks cmr8.300 @sf 499 879 p m (2) s cmmi10.329 @sf 518 872 p m (;) 8 (x) ks cmr8.300 @sf 565 879 p m (1) s cmr10.329 @sf 584 872 p m (,) 15 (so) 15 (the) 15 (v) 0 (ector) ks cmmi10.329 @sf 13 r m (x) s cmr10.329 @sf 16 r m (will) 16 (ha) 0 (v) 0 (e) 13 (the) 15 (form) ks cmmi10.329 @sf 647 1001 p m (x) s cmr10.329 @sf 12 r m (=) 13 (\() ks cmmi10.329 @sf 751 1001 p m (x) s cmr8.300 @sf 777 1008 p m (1) s cmmi10.329 @sf 797 1001 p m (;) 14 (x) ks cmr8.300 @sf 850 1008 p m (2) s cmmi10.329 @sf 870 1001 p m (;) s cmsy10.329 @sf 15 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 966 1001 p m (;) 15 (x) ks cmmi8.300 @sf 1020 1008 p m (k) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1086 1001 p m (;) s cmr10.329 @sf 15 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 15 r m (0) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 14 r m (\001) 8 (\001) 8 (\001) ks cmmi10.329 @sf 1283 1001 p m (;) s cmr10.329 @sf 8 r m (0) 0 (\)) ks cmmi8.300 @sf 1344 981 p m (T) s cmmi10.329 @sf 1379 1001 p m (:) s cmr10.329 @sf 124 1171 p m (No) 0 (w) 14 (supp) 1 (ose) ks cmmi10.329 @sf 15 r m (\013) s cmr10.329 @sf 16 r m (is) 16 (a) 14 (m) 0 (ultiple) 16 (eigen) 0 (v) -2 (alue,) 15 (sa) 0 (y) 14 (a) 15 (triple,) 16 (suc) 0 (h) 14 (that) ks cmmi10.329 @sf 697 1300 p m (\013) s cmr10.329 @sf 13 r m (=) s cmmi10.329 @sf 13 r m (\025) s cmmi8.300 @sf 814 1307 p m (p) s cmr10.329 @sf 846 1300 p m (=) s cmmi10.329 @sf 13 r m (\025) s cmmi8.300 @sf 921 1307 p m (q) s cmr10.329 @sf 952 1300 p m (=) s cmmi10.329 @sf 13 r m (\025) s cmmi8.300 @sf 1027 1307 p m (r) s cmmi10.329 @sf 1045 1300 p m (;) s cmr10.329 @sf 17 r m (\() s cmmi10.329 @sf 1093 1300 p m (p) 20 (<) 21 (q) 22 (<) 13 (r) ks cmr10.329 @sf 1 r m (\)) s cmmi10.329 @sf 1329 1300 p m (:) s cmr10.329 @sf 74 1429 p m (In) 14 (general,) 13 (there) 14 (will) 14 (b) 1 (e) 14 (only) 14 (one) 13 (eigen) 0 (v) 0 (ector) 11 (corresp) 1 (onding) 14 (to) ks cmmi10.329 @sf 13 r m (\013) s cmr10.329 @sf 13 r m (\(unless) s cmmi10.329 @sf 14 r m (T) s cmr10.329 @sf 19 r m (is) 13 (derogatory\).) 18 (First,) ks 74 1512 p m (w) 0 (e) 14 (\014nd) 16 (the) 15 (eigen) 0 (v) 0 (ector) ks cmmi10.329 @sf 13 r m (x) s cmr10.329 @sf 16 r m (corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 14 r m (\025) s cmmi8.300 @sf 962 1519 p m (p) s cmr10.329 @sf 997 1512 p m (b) 0 (y) 14 (solving) ks 850 1640 p m (\() s cmmi10.329 @sf 868 1640 p m (T) s cmmi8.300 @sf 895 1647 p m (pp) s cmsy10.329 @sf 942 1640 p m (\000) s cmmi10.329 @sf 11 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 1066 1640 p m (x) s cmr10.329 @sf 12 r m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 1769 p m (Next,) 15 (w) 0 (e) 14 (attempt) 14 (to) 15 (\014nd) ks cmmi10.329 @sf 16 r m (y) s cmr10.329 @sf 17 r m (corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 14 r m (\025) s cmmi8.300 @sf 999 1776 p m (q) s cmr10.329 @sf 1033 1769 p m (b) 0 (y) 14 (taking) ks cmmi10.329 @sf 15 r m (y) s cmmi8.300 @sf 1258 1776 p m (q) s cmr10.329 @sf 1290 1769 p m (=) 13 (1) 15 (and) 15 (attempting) 15 (to) 15 (solv) 0 (e) ks 619 1898 p m (\() s cmmi10.329 @sf 637 1898 p m (T) s cmmi8.300 @sf 664 1905 p m (q) 1 (q) ks cmsy10.329 @sf 710 1898 p m (\000) s cmmi10.329 @sf 10 r m (\025) s cmmi8.300 @sf 782 1905 p m (q) s cmmi10.329 @sf 801 1898 p m (I) s cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 849 1898 p m (y) s cmr10.329 @sf 15 r m (=) 13 (0) ks cmmi10.329 @sf m (;) s cmr10.329 @sf 22 r m (i.e.,) s 1085 1898 p m (\() s cmmi10.329 @sf 1103 1898 p m (T) s cmmi8.300 @sf 1130 1905 p m (q) 1 (q) ks cmsy10.329 @sf 1176 1898 p m (\000) s cmmi10.329 @sf 10 r m (\013) 0 (I) ks cmr10.329 @sf 4 r m (\)) s cmmi10.329 @sf 1299 1898 p m (y) s cmr10.329 @sf 15 r m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 2026 p m (All) 17 (is) 15 (\014ne) 16 (un) 0 (til) 16 (w) 0 (e) 13 (reac) 0 (h) 15 (the) 15 (determination) 16 (of) ks cmmi10.329 @sf 15 r m (y) s cmmi8.300 @sf 1020 2033 p m (p) s cmr10.329 @sf 1040 2026 p m (.) 20 (W) -3 (e) 14 (ha) 0 (v) -1 (e) ks 574 2155 p m (0) s cmmi10.329 @sf m (y) s cmmi8.300 @sf 619 2162 p m (p) s cmr10.329 @sf 649 2155 p m (+) s cmmi10.329 @sf 11 r m (t) s cmmi8.300 @sf 711 2162 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 804 2155 p m (y) s cmmi8.300 @sf 826 2162 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 901 2155 p m (+) s cmsy10.329 @sf 10 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 1010 2155 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1071 2162 p m (p;q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1163 2155 p m (y) s cmmi8.300 @sf 1185 2162 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1260 2155 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1321 2162 p m (p;q) s cmr10.329 @sf 1381 2155 p m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 2284 p m (If) 16 (w) 0 (e) 14 (let) ks cmmi10.329 @sf 634 2367 p m (t) s cmmi8.300 @sf 650 2374 p m (p;p) s cmr8.300 @sf m (+1) s cmmi10.329 @sf 743 2367 p m (y) s cmmi8.300 @sf 765 2374 p m (p) s cmr8.300 @sf m (+1) s cmr10.329 @sf 840 2367 p m (+) s cmsy10.329 @sf 11 r m (\001) 8 (\001) 8 (\001) ks cmr10.329 @sf 949 2367 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1010 2374 p m (p;q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1102 2367 p m (y) s cmmi8.300 @sf 1124 2374 p m (q) s cmsy8.300 @sf 1 r m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1199 2367 p m (+) s cmmi10.329 @sf 10 r m (t) s cmmi8.300 @sf 1260 2374 p m (p;q) s cmr10.329 @sf 1320 2367 p m (=) s cmmi10.329 @sf 13 r m (d;) s cmr10.329 @sf 74 2477 p m (then) s 899 2560 p m (0) s cmmi10.329 @sf m (y) s cmmi8.300 @sf 944 2567 p m (p) s cmr10.329 @sf 974 2560 p m (+) s cmmi10.329 @sf 11 r m (d) s cmr10.329 @sf 12 r m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 2670 p m (If) s cmmi10.329 @sf 16 r m (d) s cmr10.329 @sf 14 r m (happ) 1 (ens) 17 (to) 14 (b) 1 (e) 16 (zero,) 15 (then) ks cmmi10.329 @sf 15 r m (y) s cmmi8.300 @sf 688 2677 p m (p) s cmr10.329 @sf 724 2670 p m (is) 15 (arbitrary) -3 (.) ks 996 2850 p m (16) s @eop 15 @bop0 cmmi10.329 @sf [ 33 31 -2 0 38] 68 @dc cmr10.329 @sf [<381C7C3EFC7EFC7EB85C804080408040402040202010201010080804> 15 14 -5 -18 23] 92 @dc [<402020101008100808040804040204020402743AFC7EFC7EF87C7038> 15 14 -2 -18 23] 34 @dc cmmi10.329 @sf [ 37 31 -1 0 38] 88 @dc cmsy10.329 @sf [<03E0001F000FF800FFC0100C01F8E0200603E030400307C01840018F80084000DF000880007E00 0480007E000480007C00048000F800048001F800048001F800044003EC00084007C60008600F83 0008301F0180101C7E00C0200FFC007FC003E0001F00> 38 20 -3 0 45] 49 @dc cmr10.329 @sf [<07000F800F800F8007000000000000000000000002000200020002000200020003000100018001 8000C000E000700038603CF03CF03CE03C4038403830700FC0> 14 32 -3 0 21] 63 @dc cmbx10.432 @sf [<007FFFF8007FFFF8007FFFF80000FE000000FE000000FE000000FE000000FE000000FE000000FE 000000FE00FFFFFFF8FFFFFFF8FFFFFFF8E0007E0070007E0038007E001C007E000E007E000E00 7E0007007E0003807E0001C07E0000E07E0000E07E0000707E0000387E00001C7E00000E7E0000 0E7E0000077E000003FE000001FE000000FE000000FE0000007E0000003E0000001E0000000E00> 29 39 -2 0 34] 52 @dc [ 47 41 -2 0 52] 65 @dc [ 45 41 -3 0 53] 68 @dc [<001F8000FFC001F86003F87003F03807F03807F03807F03807F03807F03807F03807F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000FFFFF0FFFFF0 1FFFF007F00003F00003F00001F00000F00000F00000F000007000007000007000007000> 21 38 -1 0 27] 116 @dc [ 33 42 -3 0 38] 104 @dc [<003FC3FF8000FFF3FF8003F03BFF8007C00FF8000F8007F8001F8003F8003F8003F8007F0003F8 007F0003F8007F0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003 F800FF0003F8007F0003F8007F0003F8007F0003F8003F8003F8001F8003F8000FC007F80007E0 0FF80003F03FF80000FFFBF800001FE3F800000003F800000003F800000003F800000003F80000 0003F800000003F800000003F800000003F800000003F800000003F800000003F800000003F800 00003FF80000003FF80000003FF800> 33 42 -2 0 38] 100 @dc [ 53 27 -3 0 60] 109 @dc [ 33 39 -2 12 38] 112 @dc [<007FF00003FFFE000FC01F801F0007C03C0001E07C0001F0F80000F8F80000F8F80000F8F80000 F87C0001F83E0007F01FFFFFF007FFFFE00FFFFFC01FFFFF801FFFFF003FFFF8003E0000003C00 0000380000003800000018FF80001FFFE0000FC1F8001F80FC001F007C003F007E007F007F007F 007F007F007F007F007F007F007F007F007F003F007E101F007C381F80FC7C0FC1FE7C03FFE7F8 00FF81F0> 30 40 -2 13 34] 103 @dc [ 22 41 -2 0 26] 73 @dc [<0001C000000003E000000003E000000007F000000007F00000000FF80000000FF80000000FF800 00001FDC0000001FDC0000003FDE0000003F8E0000007F8F0000007F070000007F07000000FE03 800000FE03800001FC01C00001FC01C00003FC01E00003F800E00007F800F00007F000700007F0 007000FFFE03FF80FFFE03FF80FFFE03FF80> 33 27 -1 0 36] 118 @dc [<01FC03FC0FFF0FFC3F839FFC7F00DF807E007F80FE003F80FE003F80FE003F80FE003F807F003F 803F003F803F803F800FE03F8007FC3F8000FFFF80000FFF8000003F8000003F8000003F800700 3F800F803F801FC03F001FC07E001FC07E000F81F80007FFF00001FF8000> 30 27 -2 0 33] 97 @dc [ 29 41 -4 0 38] 83 @dc [<0E01FC00000F07FF80000F9E07E0000FF803F0000FF001F8000FE000FC000FE000FE000FE0007F 000FE0007F000FE0007F000FE0007F800FE0007F800FE0007F800FE0007F800FE0007F800FE000 7F800FE0007F800FE0007F000FE0007F000FE0007F000FE000FE000FE000FC000FF000F8000FF8 01F0000FFE07E0000FE7FF80000FE1FE00000FE00000000FE00000000FE00000000FE00000000F E00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE0000000 FFE0000000FFE0000000FFE0000000> 33 42 -2 0 38] 98 @dc 15 @bop1 cmr10.329 @sf 74 67 p m (i.e.,) s cmmi10.329 @sf 670 150 p m (T) s cmex10.300 @sf 710 78 p m ( ) s cmmi10.329 @sf 764 123 p m (Q) s cmmi8.300 @sf 800 107 p m (T) s cmmi10.329 @sf 827 123 p m (D) s cmsy8.300 @sf 866 107 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 827 178 p m (0) s cmex10.300 @sf 934 78 p m (!) s cmr10.329 @sf 980 150 p m (=) s cmex10.300 @sf 1028 78 p m ( ) s cmmi10.329 @sf 1081 123 p m (Q) s cmmi8.300 @sf 1117 107 p m (T) s cmmi10.329 @sf 1145 123 p m (D) s cmsy8.300 @sf 1184 107 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1145 178 p m (0) s cmex10.300 @sf 1251 78 p m (!) s cmmi10.329 @sf 1292 150 p m (T) s cmr8.300 @sf 1319 157 p m (22) s cmmi10.329 @sf 1356 150 p m (:) s cmr10.329 @sf 74 294 p m (The) 16 (columns) 16 (of) ks cmex10.300 @sf 397 222 p m ( ) s cmmi10.329 @sf 451 267 p m (Q) s cmmi8.300 @sf 487 251 p m (T) s cmmi10.329 @sf 514 267 p m (D) s cmsy8.300 @sf 553 251 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 514 322 p m (0) s cmex10.300 @sf 621 222 p m (!) s cmr10.329 @sf 670 294 p m (are) 15 (orthogonal,) 14 (but) 16 (not) 15 (orthonormal.) 20 (It) 16 (lo) 1 (oks) 16 (as) 15 (though) 15 (w) 0 (e) 14 (ha) 0 (v) 0 (e) ks 74 377 p m (an) 14 (orthogonal) 13 (basis) 15 (of) 13 (an) 14 (in) 0 (v) -2 (arian) 0 (t) 12 (subspace) 14 (\\b) 1 (elonging) 16 (to) ks cmmi10.329 @sf 13 r m (T) s cmr8.300 @sf 1272 384 p m (22) s cmr10.329 @sf 1309 377 p m (,") 13 (but) 15 (w) 0 (e) 12 (should) 15 (not) 14 (really) 15 (sp) 1 (eak) ks 74 460 p m (in) 16 (these) 16 (terms.) ks 124 584 p m (Nev) 0 (ertheless,) 15 (if) 15 (w) 0 (e) 14 (consider) ks cmmi10.329 @sf 169 762 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) 12 (=) ks cmex10.300 @sf 316 628 p m (0) s 316 701 p m (B) s 316 726 p m (B) s 316 751 p m (B) s 316 776 p m (B) s 316 803 p m (@) s cmr10.329 @sf 401 659 p m (1) s cmsy10.329 @sf 473 659 p m (\000) s cmr10.329 @sf m (1) s 556 688 p 2 69 ru 607 659 p m (0) s 705 659 p m (0) s 401 728 p m (1) s cmsy10.329 @sf 473 728 p m (\000) s cmr10.329 @sf m (1) s 556 757 p 2 69 ru 607 728 p m (1) s 705 728 p m (0) s 376 759 p 377 2 ru 556 826 p 2 69 ru 607 797 p m (0) s 705 797 p m (1) s 556 882 p 2 57 ru cmsy10.329 @sf 581 865 p m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 634 850 p m (2) s cmr10.329 @sf 705 865 p m (0) s cmex10.300 @sf 776 628 p m (1) s 776 701 p m (C) s 776 726 p m (C) s 776 751 p m (C) s 776 776 p m (C) s 776 803 p m (A) s cmr10.329 @sf 825 762 p m (=) s cmex10.300 @sf 873 690 p m ( ) s cmmi10.329 @sf 954 735 p m (T) s cmr8.300 @sf 981 742 p m (11) s 1042 752 p 2 57 ru cmmi10.329 @sf 1094 735 p m (T) s cmr8.300 @sf 1121 742 p m (12) s 929 754 p 282 2 ru 1042 807 p 2 57 ru cmmi10.329 @sf 1068 790 p m (T) s cmr8.300 @sf 1095 797 p m (22) s cmr10.329 @sf 1131 790 p m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s cmex10.300 @sf 1234 690 p m (!) s cmmi10.329 @sf 1274 762 p m (;) 65 (\025) ks cmr8.300 @sf 1379 769 p m (1) s cmmi10.329 @sf 1399 762 p m (;) 8 (\025) ks cmr8.300 @sf 1447 769 p m (2) s cmr10.329 @sf 1478 762 p m (=) 13 (0) ks cmmi10.329 @sf m (;) 52 (\025) ks cmr8.300 @sf 1641 769 p m (3) s cmmi10.329 @sf 1660 762 p m (;) 8 (\025) ks cmr8.300 @sf 1708 769 p m (4) s cmr10.329 @sf 1739 762 p m (=) s cmsy10.329 @sf 13 r m (\006) s cmmi10.329 @sf m (i\017;) s cmr10.329 @sf 74 982 p m (then) 13 (there) 13 (is) 14 (a) 13 (subspace) 13 (of) 13 (the) 13 (form) ks cmex10.300 @sf 787 910 p m ( ) s cmmi10.329 @sf 841 955 p m (X) s cmr10.329 @sf 4 r m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s cmmi10.329 @sf 876 1010 p m (I) s cmex10.300 @sf 956 910 p m (!) s cmr10.329 @sf 1002 982 p m (whic) 0 (h) 13 (w) 0 (e) 11 (could) 14 (justi\014ably) 15 (describ) 1 (e) 14 (as) 13 (\\b) 1 (elonging) ks 74 1071 p m (to) s cmmi10.329 @sf 12 r m (T) s cmr8.300 @sf 154 1078 p m (22) s cmr10.329 @sf 191 1071 p m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\),") 11 (pro) 0 (vided) ks cmmi10.329 @sf 12 r m (\017) s cmsy10.329 @sf 13 r m (6) s cmr10.329 @sf m (=) 13 (0.) 19 (The) 12 (elemen) 0 (ts) 12 (of) ks cmmi10.329 @sf 12 r m (X) s cmr10.329 @sf 10 r m (\() s cmmi10.329 @sf 998 1071 p m (\017) s cmr10.329 @sf m (\)) s 1046 1071 p m (will) 14 (tend) 12 (to) ks cmsy10.329 @sf 12 r m (1) s cmr10.329 @sf 12 r m (as) s cmmi10.329 @sf 12 r m (\017) s cmsy10.329 @sf 13 r m (!) s cmr10.329 @sf 13 r m (0) 12 (so) 12 (that) 11 (an) 0 (y) 11 (normalized) ks 74 1154 p m (v) 0 (ersion) 14 (of) 15 (this) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (subspace) 16 (will) 16 (ha) 0 (v) 0 (e) 13 (v) 0 (ery) 14 (small) 15 (comp) 1 (onen) 0 (ts) 14 (in) 16 (its) 15 (lo) 0 (w) 0 (er) 13 (2) ks cmsy10.329 @sf 9 r m (\002) s cmr10.329 @sf 10 r m (2) 15 (matrix.) 19 (In) ks 74 1237 p m (fact,) 14 (since) ks cmmi10.329 @sf 17 r m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s cmex10.300 @sf 375 1168 p m ( ) s cmmi10.329 @sf 428 1213 p m (Q) s cmmi8.300 @sf 464 1197 p m (T) s cmmi10.329 @sf 492 1213 p m (D) s cmsy8.300 @sf 531 1197 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 491 1268 p m (0) s cmex10.300 @sf 598 1168 p m (!) s cmsy10.329 @sf 644 1237 p m (\021) s cmmi10.329 @sf 13 r m (T) s cmex10.300 @sf 725 1168 p m ( ) s cmmi10.329 @sf 778 1213 p m (Q) s cmmi8.300 @sf 814 1197 p m (T) s cmmi10.329 @sf 842 1213 p m (D) s cmsy8.300 @sf 881 1197 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 842 1268 p m (0) s cmex10.300 @sf 949 1168 p m (!) s cmr10.329 @sf 981 1237 p m (,) 15 (w) 0 (e) 14 (observ) 0 (e) 14 (that) ks cmmi10.329 @sf 447 1390 p m (T) s cmr10.329 @sf 6 r m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s cmex10.300 @sf 542 1318 p m ( ) s cmmi10.329 @sf 595 1363 p m (Q) s cmmi8.300 @sf 631 1348 p m (T) s cmmi10.329 @sf 658 1363 p m (D) s cmsy8.300 @sf 697 1348 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 658 1419 p m (0) s cmex10.300 @sf 765 1318 p m (!) s cmsy10.329 @sf 808 1390 p m (\000) s cmex10.300 @sf 853 1318 p m ( ) s cmmi10.329 @sf 907 1363 p m (Q) s cmmi8.300 @sf 943 1348 p m (T) s cmmi10.329 @sf 970 1363 p m (D) s cmsy8.300 @sf 1009 1348 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 970 1419 p m (0) s cmex10.300 @sf 1077 1318 p m (!) s cmmi10.329 @sf 1118 1390 p m (T) s cmr8.300 @sf 1145 1397 p m (22) s cmr10.329 @sf 1182 1390 p m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s 538 1515 p m (=) s cmmi10.329 @sf 13 r m (T) s cmex10.300 @sf 626 1443 p m ( ) s cmmi10.329 @sf 680 1488 p m (Q) s cmmi8.300 @sf 716 1472 p m (T) s cmmi10.329 @sf 743 1488 p m (D) s cmsy8.300 @sf 782 1472 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 743 1543 p m (0) s cmex10.300 @sf 850 1443 p m (!) s cmsy10.329 @sf 893 1515 p m (\000) s cmex10.300 @sf 938 1443 p m ( ) s cmmi10.329 @sf 992 1488 p m (Q) s cmmi8.300 @sf 1028 1472 p m (T) s cmmi10.329 @sf 1056 1488 p m (D) s cmsy8.300 @sf 1095 1472 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1055 1543 p m (0) s cmex10.300 @sf 1162 1443 p m (!) s cmmi10.329 @sf 1203 1515 p m (T) s cmr8.300 @sf 1230 1522 p m (22) s cmr10.329 @sf 1267 1515 p m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s 538 1639 p m (=) s cmmi10.329 @sf 13 r m (T) s cmex10.300 @sf 626 1567 p m ( ) s cmmi10.329 @sf 680 1612 p m (Q) s cmmi8.300 @sf 716 1597 p m (T) s cmmi10.329 @sf 743 1612 p m (D) s cmsy8.300 @sf 782 1597 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 743 1668 p m (0) s cmex10.300 @sf 850 1567 p m (!) s cmsy10.329 @sf 893 1639 p m (\000) s cmex10.300 @sf 938 1567 p m ( ) s cmmi10.329 @sf 992 1612 p m (Q) s cmmi8.300 @sf 1028 1597 p m (T) s cmmi10.329 @sf 1056 1612 p m (D) s cmsy8.300 @sf 1095 1597 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1055 1668 p m (0) s cmex10.300 @sf 1162 1567 p m (!) s 1203 1567 p m ( ) s cmmi10.329 @sf 1236 1639 p m (T) s cmr8.300 @sf 1263 1646 p m (22) s cmr10.329 @sf 1310 1639 p m (+) s cmex10.300 @sf 1355 1567 p m ( ) s cmr10.329 @sf 1434 1612 p m (0) s 1524 1612 p m (0) s cmsy10.329 @sf 1409 1668 p m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 1462 1653 p m (2) s cmr10.329 @sf 1524 1668 p m (0) s cmex10.300 @sf 1567 1567 p m (!) s 1600 1567 p m (!) s cmr10.329 @sf 538 1764 p m (=) s cmex10.300 @sf 586 1692 p m ( ) s cmmi10.329 @sf 639 1737 p m (Q) s cmmi8.300 @sf 675 1721 p m (T) s cmmi10.329 @sf 703 1737 p m (D) s cmsy8.300 @sf 742 1721 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 703 1792 p m (0) s cmex10.300 @sf 810 1692 p m (!) s 850 1692 p m ( ) s cmr10.329 @sf 929 1737 p m (0) s 1019 1737 p m (0) s cmsy10.329 @sf 904 1792 p m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 957 1777 p m (2) s cmr10.329 @sf 1019 1792 p m (0) s cmex10.300 @sf 1062 1692 p m (!) s cmmi10.329 @sf 1103 1764 p m (:) s cmr10.329 @sf 74 1875 p m (When) s cmmi10.329 @sf 16 r m (\017) s cmr10.329 @sf 15 r m (is) 16 (small,) 16 (this) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (subspace) 16 (giv) 0 (es) 14 (negligibl) 1 (e) 17 (residuals) 17 (\\corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 14 r m (T) s cmr8.300 @sf 1785 1882 p m (22) s cmr10.329 @sf 1830 1875 p m (\() s cmmi10.329 @sf 1847 1875 p m (\017) s cmr10.329 @sf m (\)) s 1884 1875 p m (".) s 124 1999 p m (Can) 17 (w) 0 (e) 15 (exp) 1 (ect) ks cmmi10.329 @sf 18 r m (X) s cmr10.329 @sf 11 r m (\() s cmmi10.329 @sf 502 1999 p m (\017) s cmr10.329 @sf m (\)) s 555 1999 p m (to) 16 (b) 1 (e) ks cmmi10.329 @sf 18 r m (Q) s cmmi8.300 @sf 712 1984 p m (T) s cmmi10.329 @sf 739 1999 p m (D) s cmsy8.300 @sf 778 1984 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 842 1999 p m (apart) 16 (from) 16 (a) 17 (scale) 17 (factor?) 25 (Unfortunately) 17 (w) 0 (e) 16 (cannot.) 24 (In) ks 74 2082 p m (fact,) 14 (w) 0 (e) 14 (ha) 0 (v) 0 (e) ks cmmi10.329 @sf 729 2220 p m (\017) s cmr8.300 @sf 747 2202 p m (2) s cmex10.300 @sf 775 2148 p m ( ) s cmmi10.329 @sf 828 2193 p m (X) s cmr10.329 @sf 4 r m (\() s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (\)) s cmmi10.329 @sf 864 2248 p m (I) s cmex10.300 @sf 944 2148 p m (!) s cmr10.329 @sf 989 2220 p m (=) s cmex10.300 @sf 1037 2098 p m (0) s 1037 2171 p m (B) s 1037 2196 p m (B) s 1037 2221 p m (B) s 1037 2248 p m (@) s cmr10.329 @sf 1102 2135 p m (1) s 1192 2135 p m (0) s 1102 2191 p m (1) s cmsy10.329 @sf 1174 2191 p m (\000) s cmr10.329 @sf m (1) s cmmi10.329 @sf 1094 2248 p m (\017) s cmr8.300 @sf 1112 2233 p m (2) s cmr10.329 @sf 1192 2248 p m (0) s 1102 2304 p m (0) s cmmi10.329 @sf 1184 2304 p m (\017) s cmr8.300 @sf 1202 2289 p m (2) s cmex10.300 @sf 1253 2098 p m (1) s 1253 2171 p m (C) s 1253 2196 p m (C) s 1253 2221 p m (C) s 1253 2248 p m (A) s cmmi10.329 @sf 1297 2220 p m (:) s cmbx10.432 @sf 74 2473 p m (4) s 177 2473 p m (A) 23 (Direct) ks 21 r save [ 58 41 -3 0 65] 77 @pc restore 65 r m (etho) 2 (d) 23 (for) 24 (Com) -1 (puti) -1 (ng) 22 (In) -1 (v) -3 (arian) -1 (t) 21 (Subspaces) ks cmr10.329 @sf 74 2643 p m (In) 15 (this) 14 (section) 15 (w) 0 (e) 13 (consider) 15 (the) 14 (construction) 14 (of) 14 (an) 14 (in) 0 (v) -2 (arian) 0 (t) 11 (subspace) 15 (b) 0 (y) 13 (a) 14 (direct) 15 (computation) 14 (of) ks 74 2726 p m (the) 14 (v) 0 (ectors,) 13 (rather) 13 (than) 14 (b) 0 (y) 13 (applying) 16 (transformations) 12 (to) 14 (mo) 0 (v) -1 (e) 12 (the) 14 (desired) 16 (eigen) 0 (v) -2 (alues) 14 (to) 13 (the) 14 (top) ks 996 2850 p m (15) s @eop 14 @bop0 cmr10.329 @sf [ 44 1 0 -12 45] 124 @dc cmsy10.329 @sf [<00060000000006000000000F000000000F000000001F800000001F800000003CC00000003CC000 000078600000007860000000F030000000F030000001E018000001E018000003C00C000003C00C 000003C0060000078006000087800300004F000300002F000180001E00018000060000C0000000 00C0000000006000000000600000000030000000003000000000180000000018000000000C0000 00000C0000000006000000000600000000030000000003000000000180000000018000000000C0 00000000C0000000006000000000600000000030000000003000000000180000000008> 37 46 -2 44 38] 112 @dc 14 @bop1 cmr10.329 @sf 74 67 p m (in) 15 (the) 14 (lo) 0 (w) -1 (er) 12 (pair) 14 (to) 13 (agree) 13 (with) 14 (one) 14 (in) 15 (the) 14 (upp) 1 (er) 14 (pair.) 20 (If,) 14 (for) 13 (con) 0 (v) 0 (enience,) 13 (w) 0 (e) 13 (denote) 14 (the) 13 (relev) -2 (an) 0 (t) ks 74 150 p m (4) s cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (4) 15 (matrix) 15 (and) 16 (the) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (subspace) 16 (b) 0 (y) ks cmex10.300 @sf 707 201 p m ( ) s cmmi10.329 @sf 788 246 p m (T) s cmr8.300 @sf 815 253 p m (11) s 876 263 p 2 57 ru cmmi10.329 @sf 902 246 p m (T) s cmr8.300 @sf 929 253 p m (12) s 763 265 p 228 2 ru cmr10.329 @sf 808 301 p m (0) s 876 318 p 2 57 ru cmmi10.329 @sf 902 301 p m (T) s cmr8.300 @sf 929 308 p m (22) s cmex10.300 @sf 1014 201 p m (!) s cmr10.329 @sf 1070 273 p m (and) s cmex10.300 @sf 1163 201 p m ( ) s cmmi10.329 @sf 1217 246 p m (X) s 1225 301 p m (I) s cmex10.300 @sf 1279 201 p m (!) s cmmi10.329 @sf 1319 273 p m (;) s cmr10.329 @sf 74 396 p m (resp) 1 (ectiv) 0 (ely) -3 (,) 15 (where) ks cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 489 403 p m (11) s cmr10.329 @sf 526 396 p m (,) s cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 581 403 p m (12) s cmr10.329 @sf 618 396 p m (,) s cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 673 403 p m (22) s cmr10.329 @sf 725 396 p m (and) s cmmi10.329 @sf 15 r m (X) s cmr10.329 @sf 19 r m (are) 14 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 11 r m (2) 15 (matrices,) 14 (then) 16 (w) 0 (e) 14 (ha) 0 (v) -1 (e) ks cmmi10.329 @sf 818 519 p m (T) s cmr8.300 @sf 845 526 p m (11) s cmmi10.329 @sf 882 519 p m (X) s cmr10.329 @sf 13 r m (+) s cmmi10.329 @sf 10 r m (T) s cmr8.300 @sf 1005 526 p m (12) s cmr10.329 @sf 1055 519 p m (=) s cmmi10.329 @sf 13 r m (X) 4 (T) ks cmr8.300 @sf 1172 526 p m (22) s cmmi10.329 @sf 1208 519 p m (:) s cmr10.329 @sf 74 642 p m (It) 16 (is) 17 (w) 0 (ell) 16 (kno) 0 (wn) 15 (that) 16 (if) ks cmmi10.329 @sf 16 r m (T) s cmr8.300 @sf 578 649 p m (11) s cmr10.329 @sf 632 642 p m (and) s cmmi10.329 @sf 16 r m (T) s cmr8.300 @sf 748 649 p m (22) s cmr10.329 @sf 801 642 p m (ha) 0 (v) -1 (e) 15 (no) 16 (eigen) 0 (v) -2 (alue) 16 (in) 17 (common,) 16 (then) 17 (this) 16 (is) 17 (a) 16 (non-singular) ks 74 725 p m (system.) s 124 850 p m (F) -3 (or) 14 (the) 15 (case) 15 (when) ks cmmi10.329 @sf 16 r m (T) s cmr8.300 @sf 526 857 p m (11) s cmr10.329 @sf 578 850 p m (and) s cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 693 857 p m (22) s cmr10.329 @sf 745 850 p m (share) 15 (an) 15 (eigen) 0 (v) -2 (alue,) 15 (consider) 16 (the) 16 (matrix) ks cmmi10.329 @sf 360 1046 p m (T) s cmr10.329 @sf 18 r m (=) s cmex10.300 @sf 453 974 p m ( ) s cmmi10.329 @sf 507 1019 p m (T) s cmr8.300 @sf 534 1026 p m (11) s cmmi10.329 @sf 612 1019 p m (T) s cmr8.300 @sf 639 1026 p m (12) s cmmi10.329 @sf 612 1074 p m (T) s cmr8.300 @sf 639 1081 p m (22) s cmex10.300 @sf 697 974 p m (!) s cmr10.329 @sf 743 1046 p m (=) s cmex10.300 @sf 791 912 p m (0) s 791 985 p m (B) s 791 1010 p m (B) s 791 1035 p m (B) s 791 1060 p m (B) s 791 1087 p m (@) s cmr10.329 @sf 875 943 p m (1) s cmsy10.329 @sf 948 943 p m (\000) s cmr10.329 @sf m (1) s 1030 972 p 2 69 ru 1056 943 p m (0) s 1128 943 p m (0) s 875 1011 p m (1) s cmsy10.329 @sf 948 1011 p m (\000) s cmr10.329 @sf m (1) s 1030 1040 p 2 69 ru 1056 1011 p m (1) s 1128 1011 p m (0) s 850 1042 p 326 2 ru 1030 1109 p 2 69 ru 1056 1080 p m (0) s 1128 1080 p m (1) s 1030 1166 p 2 57 ru 1056 1149 p m (0) s 1128 1149 p m (0) s cmex10.300 @sf 1199 912 p m (1) s 1199 985 p m (C) s 1199 1010 p m (C) s 1199 1035 p m (C) s 1199 1060 p m (C) s 1199 1087 p m (A) s cmmi10.329 @sf 1289 1046 p m (\025) s cmmi8.300 @sf 1316 1053 p m (i) s cmr10.329 @sf 1342 1046 p m (=) 13 (0) ks cmmi10.329 @sf m (;) 17 (i) ks cmr10.329 @sf 12 r m (=) 13 (1) ks cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (.) 8 (.) 8 (.) ks cmmi10.329 @sf 1623 1046 p m (;) s cmr10.329 @sf 8 r m (4) s cmmi10.329 @sf m (:) s cmr10.329 @sf 74 1284 p m (If) 20 (w) 0 (e) 19 (try) 19 (to) 19 (\014nd) 20 (an) 20 (in) 0 (v) -2 (arian) 0 (t) 17 (subspace) 20 (of) 20 (the) 19 (form) ks cmex10.300 @sf 1138 1212 p m ( ) s cmmi10.329 @sf 1192 1257 p m (X) s 1201 1312 p m (I) s cmex10.300 @sf 1254 1212 p m (!) s cmr10.329 @sf 1287 1284 p m (,) 20 (w) 0 (e) 19 (fail;) 22 (the) 20 (elemen) 0 (ts) 19 (of) ks cmmi10.329 @sf 19 r m (X) s cmr10.329 @sf 23 r m (turn) s 74 1373 p m (out) 16 (to) 15 (b) 1 (e) 17 (in\014nite.) 24 (There) 17 (is) 16 (no) 16 (in) 0 (v) -2 (arian) 0 (t) 14 (subspace) 16 (of) 16 (dimension) 18 (t) 0 (w) -1 (o) 14 (of) 15 (the) 16 (required) 17 (form.) 22 (\(The) ks 74 1456 p m (particular) 16 (form) 15 (c) 0 (hosen) 15 (for) ks cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 635 1463 p m (12) s cmr10.329 @sf 687 1456 p m (is) 16 (not) 16 (critical|though,) 17 (of) 15 (course,) 15 (if) 16 (w) 0 (e) 15 (tak) 0 (e) ks cmmi10.329 @sf 13 r m (T) s cmr8.300 @sf 1603 1463 p m (12) s cmr10.329 @sf 1656 1456 p m (to) 15 (b) 1 (e) 16 (n) 0 (ull,) 16 (suc) 0 (h) ks 74 1539 p m (an) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (subspace) 16 (do) 1 (es) 15 (exist) 16 (with) ks cmmi10.329 @sf 16 r m (X) s cmr10.329 @sf 15 r m (=) 13 (0;) ks cmmi10.329 @sf 15 r m (T) s cmr10.329 @sf 21 r m (is) 16 (then) 15 (derogatory) -3 (.\)) 18 (Ho) 0 (w) -1 (ev) -1 (er,) ks cmmi10.329 @sf 680 1668 p m (T) s cmex10.300 @sf 720 1596 p m ( ) s cmmi10.329 @sf 774 1641 p m (I) s cmr10.329 @sf 774 1697 p m (0) s cmex10.300 @sf 818 1596 p m (!) s cmr10.329 @sf 864 1668 p m (=) s cmex10.300 @sf 912 1596 p m ( ) s cmmi10.329 @sf 965 1641 p m (T) s cmr8.300 @sf 992 1648 p m (11) s cmr10.329 @sf 986 1697 p m (0) s cmex10.300 @sf 1050 1596 p m (!) s cmr10.329 @sf 1096 1668 p m (=) s cmex10.300 @sf 1143 1596 p m ( ) s cmmi10.329 @sf 1197 1641 p m (I) s cmr10.329 @sf 1198 1697 p m (0) s cmex10.300 @sf 1241 1596 p m (!) s cmmi10.329 @sf 1282 1668 p m (T) s cmr8.300 @sf 1309 1675 p m (11) s cmmi10.329 @sf 1346 1668 p m (;) s cmr10.329 @sf 74 1792 p m (and) 16 (hence) 16 (w) 0 (e) 14 (no) 0 (w) 13 (ha) 0 (v) 0 (e) 13 (an) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (subspace) 16 (whic) 0 (h) 15 (w) 0 (e) 14 (think) 16 (of) 15 (as) 14 (b) 1 (elonging) 17 (to) ks cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 1712 1799 p m (11) s cmr10.329 @sf 1749 1792 p m (.) 20 (But) ks cmmi10.329 @sf 381 1915 p m (QT) s cmr8.300 @sf 444 1922 p m (11) s cmmi10.329 @sf 481 1915 p m (Q) s cmmi8.300 @sf 517 1897 p m (T) s cmr10.329 @sf 557 1915 p m (=) s cmex10.300 @sf 605 1843 p m ( ) s cmr10.329 @sf 658 1888 p m (0) s cmsy10.329 @sf 722 1888 p m (\000) s cmr10.329 @sf m (2) s 658 1943 p m (0) s 740 1943 p m (0) s cmex10.300 @sf 801 1843 p m (!) s cmr10.329 @sf 857 1915 p m (when) s cmmi10.329 @sf 976 1915 p m (Q) s cmr10.329 @sf 12 r m (=) s 1096 1885 p m (1) s 1077 1905 p 61 2 ru cmsy10.329 @sf 1077 1913 p m (p) s 1115 1913 p 23 2 ru cmr10.329 @sf 1115 1951 p m (2) s cmex10.300 @sf 1155 1843 p m ( ) s cmr10.329 @sf 1227 1888 p m (1) s 1308 1888 p m (1) s cmsy10.329 @sf 1209 1943 p m (\000) s cmr10.329 @sf m (1) s 1308 1943 p m (1) s cmex10.300 @sf 1352 1843 p m (!) s cmr10.329 @sf 1412 1915 p m (\(a) 15 (rotation\).) ks 74 2038 p m (Hence) s cmmi10.329 @sf 661 2121 p m (T) s cmex10.300 @sf 702 2049 p m ( ) s cmmi10.329 @sf 755 2094 p m (I) s cmr10.329 @sf 756 2149 p m (0) s cmex10.300 @sf 800 2049 p m (!) s cmmi10.329 @sf 840 2121 p m (Q) s cmmi8.300 @sf 876 2103 p m (T) s cmr10.329 @sf 916 2121 p m (=) s cmex10.300 @sf 964 2049 p m ( ) s cmmi10.329 @sf 1018 2094 p m (I) s cmr10.329 @sf 1018 2149 p m (0) s cmex10.300 @sf 1062 2049 p m (!) s cmmi10.329 @sf 1102 2121 p m (Q) s cmmi8.300 @sf 1138 2103 p m (T) s cmr10.329 @sf 1166 2121 p m (\() s cmmi10.329 @sf m (QT) s cmr8.300 @sf 1247 2128 p m (11) s cmmi10.329 @sf 1283 2121 p m (Q) s cmmi8.300 @sf 1319 2103 p m (T) s cmr10.329 @sf 1347 2121 p m (\)) s cmmi10.329 @sf m (;) s cmr10.329 @sf 74 2228 p m (i.e.,) s cmmi10.329 @sf 530 2311 p m (T) s cmex10.300 @sf 571 2239 p m ( ) s cmmi10.329 @sf 624 2284 p m (Q) s cmmi8.300 @sf 660 2268 p m (T) s cmr10.329 @sf 645 2339 p m (0) s cmex10.300 @sf 709 2239 p m (!) s cmr10.329 @sf 754 2311 p m (=) s cmex10.300 @sf 802 2239 p m ( ) s cmmi10.329 @sf 856 2284 p m (Q) s cmmi8.300 @sf 892 2268 p m (T) s cmr10.329 @sf 876 2339 p m (0) s cmex10.300 @sf 940 2239 p m (!) s 980 2239 p m ( ) s cmr10.329 @sf 1034 2284 p m (0) s cmsy10.329 @sf 1098 2284 p m (\000) s cmr10.329 @sf m (2) s 1034 2339 p m (0) s 1116 2339 p m (0) s cmex10.300 @sf 1177 2239 p m (!) s cmsy10.329 @sf 1223 2311 p m (\021) s cmex10.300 @sf 1271 2239 p m ( ) s cmmi10.329 @sf 1324 2284 p m (Q) s cmmi8.300 @sf 1360 2268 p m (T) s cmr10.329 @sf 1344 2339 p m (0) s cmex10.300 @sf 1408 2239 p m (!) s cmmi10.329 @sf 1449 2311 p m (M) 2 (:) ks cmr10.329 @sf 74 2418 p m (But) s cmex10.300 @sf 356 2429 p m ( ) s cmsy10.329 @sf 409 2474 p m (\000) s cmr8.300 @sf 449 2457 p m (1) s 449 2464 p 18 2 ru 449 2490 p m (2) s cmr10.329 @sf 514 2529 p m (1) s cmex10.300 @sf 557 2429 p m (!) s cmmi10.329 @sf 598 2501 p m (M) s cmex10.300 @sf 654 2429 p m ( ) s cmsy10.329 @sf 708 2474 p m (\000) s cmr10.329 @sf m (2) s 807 2529 p m (1) s cmex10.300 @sf 851 2429 p m (!) s cmr10.329 @sf 896 2501 p m (=) s cmex10.300 @sf 944 2429 p m ( ) s cmr10.329 @sf 998 2474 p m (0) s 1062 2474 p m (1) s 998 2529 p m (0) s 1062 2529 p m (0) s cmex10.300 @sf 1106 2429 p m (!) s cmr10.329 @sf 1151 2501 p m (=) s cmmi10.329 @sf 13 r m (T) s cmr8.300 @sf 1226 2508 p m (22) s cmmi10.329 @sf 1263 2501 p m (;) s cmr10.329 @sf 22 r m (i.e.,) s cmmi10.329 @sf 1384 2501 p m (D) 1 (M) 5 (D) ks cmsy8.300 @sf 1511 2483 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1571 2501 p m (=) s cmmi10.329 @sf 13 r m (T) s cmr8.300 @sf 1646 2508 p m (22) s cmr10.329 @sf 74 2608 p m (and) 16 (hence) ks cmmi10.329 @sf 399 2691 p m (T) s cmex10.300 @sf 439 2619 p m ( ) s cmmi10.329 @sf 493 2664 p m (Q) s cmmi8.300 @sf 529 2649 p m (T) s cmr10.329 @sf 513 2719 p m (0) s cmex10.300 @sf 577 2619 p m (!) s cmmi10.329 @sf 618 2691 p m (D) s cmsy8.300 @sf 657 2673 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 716 2691 p m (=) s cmex10.300 @sf 764 2619 p m ( ) s cmmi10.329 @sf 818 2664 p m (Q) s cmmi8.300 @sf 854 2649 p m (T) s cmr10.329 @sf 838 2719 p m (0) s cmex10.300 @sf 902 2619 p m (!) s cmmi10.329 @sf 943 2691 p m (D) s cmsy8.300 @sf 982 2673 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1029 2691 p m (\() s cmmi10.329 @sf m (D) 1 (M) 5 (D) ks cmsy8.300 @sf 1174 2673 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1220 2691 p m (\)) 13 (=) ks cmex10.300 @sf 1299 2619 p m ( ) s cmmi10.329 @sf 1352 2664 p m (Q) s cmmi8.300 @sf 1388 2649 p m (T) s cmr10.329 @sf 1373 2719 p m (0) s cmex10.300 @sf 1436 2619 p m (!) s cmmi10.329 @sf 1477 2691 p m (D) s cmsy8.300 @sf 1516 2673 p m (\000) s cmr8.300 @sf m (1) s cmmi10.329 @sf 1563 2691 p m (T) s cmr8.300 @sf 1590 2698 p m (22) s cmmi10.329 @sf 1627 2691 p m (;) s cmr10.329 @sf 996 2850 p m (14) s @eop 13 @bop0 cmr10.329 @sf [<7FE7FE0700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E0 0700E00700E00700E00700E00700E00700E0FFFFE00700E00700E00700E00700E00700E00700E0 0700E00700E00381E001C1E000E0E0003FE0> 23 32 0 0 25] 13 @dc cmmi10.329 @sf [ 32 31 -2 0 34] 69 @dc [ 24 30 -1 10 27] 22 @dc [<180600380F001C0E001C0E000C0E000E0C00060C00060C00020C00030C00030C00010C00010400 018400808400C084006082003FFFFC1FFFFC0FFFFC> 22 20 -2 0 26] 25 @dc 1.000000 /cmbx12.300 @newfont cmbx12.300 @sf [<01FE000FFFC01E07F07801F87E01FCFF00FCFF00FEFF00FEFF00FE7E00FE3C00FE0000FC0000FC 0001F80001F00007C001FF0001FE00001F800007C00003E00003F01F03F01F03F83F81F83F81F8 3F81F83F03F01E03F00F07E007FFC000FE00> 23 32 -2 0 28] 51 @dc [<387CFEFEFE7C38> 7 7 -4 0 16] 46 @dc [ 37 34 -2 0 43] 68 @dc [<00FE0007FFC00F83E01F01F03E00F87C007C7C007CFC007EFC007EFC007EFC007EFC007EFC007E FC007E7C007C7C007C7C007C3E00F81E00F00F83E007FFC000FE00> 23 22 -2 0 28] 111 @dc [<01FC3FC007FF3FC00F81BE001F00FE001F007E001F003E001F003E001F003E001F003E001F003E 001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F00 3E00FF01FE00FF01FE00> 26 22 -2 0 31] 117 @dc [<180FC0001C3FF8001EE07C001FC03E001F801F001F001F801F000F801F000FC01F000FC01F000F C01F000FC01F000FC01F000FC01F000FC01F000FC01F000F801F000F801F001F001F801E001FF0 7C001F3FF8001F0FE0001F0000001F0000001F0000001F0000001F0000001F0000001F0000001F 0000001F0000001F0000001F000000FF000000FF000000> 26 35 -2 0 31] 98 @dc [ 11 35 -2 0 16] 108 @dc [<00FF0003FFC00FC0701F00303E00187E00007C00007C0000FC0000FC0000FC0000FFFFF8FFFFF8 FC00F8FC00F87C00F87C00F03E01F01E01E00F87C007FF8000FE00> 21 22 -2 0 26] 101 @dc [ 26 32 -2 10 31] 112 @dc [<07E03F1FF87F7E0CF87E02F0FC01F0FC01F0FC01F0FC01F07E01F03F01F01FC1F003FDF0003FF0 0001F00001F01E01F03F01F03F01E03F03E03F07C01FFF8007FC00> 24 22 -2 0 27] 97 @dc [ 17 22 -2 0 22] 115 @dc [<00F80003FE0007C3000F83000F81800F81800F81800F81800F81800F80000F80000F80000F8000 0F80000F80000F80000F80000F80000F80000F8000FFFF00FFFF003F80000F8000078000078000 038000038000018000018000018000018000> 17 32 -1 0 22] 116 @dc [<01FC3FC007FF3FC00F83BE001E00FE003E007E007C003E007C003E00FC003E00FC003E00FC003E 00FC003E00FC003E00FC003E00FC003E00FC003E007C003E007E003E003E003E001F007E000F81 FE0007FFBE0001FC3E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E000001FE000001FE00> 26 35 -2 0 31] 100 @dc 13 @bop1 cmr10.329 @sf 74 67 p m (9) 20 (of) 20 ([) 0 (11) ks 232 67 p m (].) 34 (This) 20 (is) 21 (a) 20 (stable) 20 (de\015ation) 21 (in) 21 (that) 19 (pro) 0 (vided) 20 (the) 20 (eigen) 0 (v) 0 (ector) 18 (has) 20 (negligibl) 1 (e) 22 (residuals) ks 74 150 p m (\(indep) 1 (ende) 1 (n) 0 (t) 16 (of) 16 (its) 16 (absolute) 16 (accuracy\);) 16 (the) 16 (de\015ated) 16 (matrix) 16 (is) 16 (exactly) 16 (orthogonally) 16 (similar) 17 (to) 15 (a) ks 74 233 p m (matrix) 14 (that) 13 (di\013ers) 15 (from) 13 (the) 15 (original) 15 (b) 0 (y) 13 (a) 14 (matrix) ks cmmi10.329 @sf 14 r m (E) s cmr10.329 @sf 3 r m (,) 13 (whic) 0 (h) 14 (is) 15 (at) 13 (noise) 15 (lev) 0 (el) 15 (relativ) 0 (e) 13 (to) 14 (it.) 20 (This) 15 (is) ks 74 316 p m (true) 12 (ev) 0 (en) 11 (when) 13 (w) 0 (e) 10 (insert) 13 (\(without) 11 (computation\)) 12 (the) 12 (computed) 12 (eigen) 0 (v) -2 (alue) 12 (in) 13 (the) 12 (leading) 13 (p) 1 (osition) ks 74 399 p m (and) 18 (zero) 18 (in) 19 (the) 18 (rest) 18 (of) 18 (the) 18 (\014rst) 17 (column.) 30 (Suc) 0 (h) 17 (a) 18 (result) 19 (is) 18 (the) 18 (most) 18 (w) 0 (e) 16 (can) 18 (reasonably) 19 (exp) 1 (ect,) ks 74 482 p m (though) 15 (it) 16 (falls) 16 (somewhat) 14 (short) 15 (of) 14 (the) 16 (sup) 1 (er-stabilit) 0 (y) 15 (of) 15 (the) 15 (single) 17 (past) 15 (single) 16 (case.) ks 124 607 p m (W) -3 (e) 11 (ha) 0 (v) 0 (e) 10 (naturally) 13 (concen) 0 (trated) 11 (on) 13 (the) 12 (case) 12 (when) 13 (w) 0 (e) 11 (are) 12 (attempting) 12 (to) 12 (mo) 0 (v) -1 (e) 10 (a) 12 (real) 13 (eigen) 0 (v) -2 (alue) ks cmmi10.329 @sf 74 690 p m (\025) s cmr8.300 @sf 101 697 p m (3) s cmr10.329 @sf 139 690 p m (past) 18 (a) 19 (complex) 20 (conjugate) 18 (pair) 19 (eac) 0 (h) 18 (of) 18 (whic) 0 (h) 19 (is) 19 (near) ks cmmi10.329 @sf 19 r m (\025) s cmr8.300 @sf 1247 697 p m (3) s cmr10.329 @sf 1266 690 p m (,) 20 (b) 1 (ecause) 19 (n) 0 (umerical) 19 (stabilit) 0 (y) 19 (there) ks 74 773 p m (needs) 22 (serious) 22 (in) 0 (v) 0 (estigation.) 36 (Of) 22 (course,) 22 (when) ks cmmi10.329 @sf 22 r m (\025) s cmr8.300 @sf 1044 780 p m (3) s cmr10.329 @sf 1085 773 p m (is) 21 (\\to) 1 (o) 21 (close,") 23 (w) 0 (e) 20 (usually) 22 (include) 24 (all) 22 (three) ks 74 856 p m (eigen) 0 (v) -2 (alues) 17 (in) 18 (the) 17 (same) 16 (space.) 25 (Ho) 0 (w) 0 (ev) -1 (er,) 14 (when) 18 (w) 0 (e) 15 (mo) 0 (v) -1 (e) 16 (a) 16 (single) 18 (eigen) 0 (v) -2 (alue) ks cmmi10.329 @sf 17 r m (\025) s cmr8.300 @sf 1625 863 p m (3) s cmr10.329 @sf 1661 856 p m (past) 17 (a) 16 (complex) ks 74 939 p m (conjugate) 21 (pair) ks cmmi10.329 @sf 21 r m (\025) s cmsy10.329 @sf 13 r m (\006) s cmmi10.329 @sf 14 r m (i\026) s cmr10.329 @sf 21 r m (suc) 0 (h) 20 (that) ks cmmi10.329 @sf 20 r m (\025) s cmsy10.329 @sf 13 r m (\000) s cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 867 946 p m (3) s cmr10.329 @sf 907 939 p m (is) 21 (not) 20 (small) 22 (but) ks cmmi10.329 @sf 21 r m (\026) s cmr10.329 @sf 21 r m (is) 21 (small,) 23 (that) 20 (pair) 21 (will) 22 (b) 1 (e) 22 (close,) ks 74 1022 p m (and) 17 (hence,) 18 (in) 17 (general,) 17 (v) 0 (ery) 15 (sensitiv) 0 (e) 17 (to) 16 (p) 1 (erturbations.) 24 (The) 17 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (2) 16 (blo) 1 (c) 0 (k) 17 (will) 18 (itself) 17 (b) 1 (e) 18 (sub) 3 (jected) ks 74 1105 p m (to) 16 (a) 17 (similarit) 0 (y) 17 (transformation,) 16 (and) 17 (small) 17 (rounding) 18 (errors) 16 (will) 19 (mak) 0 (e) 15 (substan) 0 (tial) 16 (c) 0 (hanges) 16 (in) 18 (the) ks 74 1188 p m (eigen) 0 (v) -2 (alues.) 20 (Th) 0 (us,) 14 (if) 15 (w) 0 (e) 14 (ha) 0 (v) 0 (e) 13 (the) 15 (matrix) ks cmex10.300 @sf 778 1245 p m ( ) s cmmi10.329 @sf 849 1290 p m (:) s cmr10.329 @sf m (431263) s cmmi10.329 @sf 1058 1290 p m (:) s cmr10.329 @sf m (516325) s cmsy10.329 @sf 832 1345 p m (\000) s cmmi10.329 @sf m (:) s cmr10.329 @sf m (000003) s cmmi10.329 @sf 1058 1345 p m (:) s cmr10.329 @sf m (431937) s cmex10.300 @sf 1227 1245 p m (!) s cmr10.329 @sf 74 1451 p m (with) 14 (the) 14 (ill-condi) 1 (tioned) 16 (eigen) 0 (v) -2 (alues) ks cmmi10.329 @sf 13 r m (:) s cmr10.329 @sf m (431600) s cmsy10.329 @sf 7 r m (\006) s cmmi10.329 @sf 7 r m (i) s cmr10.329 @sf m (\() s cmmi10.329 @sf -1 r m (:) s cmr10.329 @sf m (001) -1 (198) -1 (\),) 12 (and) 13 (sub) 3 (ject) 14 (it) 13 (to) 13 (a) 14 (plane) 14 (rotation) 13 (with) ks 74 1534 p m (angle) s cmmi10.329 @sf 16 r m (\031) 2 (=) ks cmr10.329 @sf m (4,) 14 (the) 15 (exact) 15 (transform) 14 (giv) 0 (es) ks cmex10.300 @sf 778 1597 p m ( ) s cmmi10.329 @sf 849 1642 p m (:) s cmr10.329 @sf m (689761) s cmmi10.329 @sf 1058 1642 p m (:) s cmr10.329 @sf m (258501) s cmsy10.329 @sf 832 1697 p m (\000) s cmmi10.329 @sf m (:) s cmr10.329 @sf m (257827) s cmmi10.329 @sf 1058 1697 p m (:) s cmr10.329 @sf m (173439) s cmex10.300 @sf 1227 1597 p m (!) s cmr10.329 @sf 74 1798 p m (with,) 15 (of) 15 (course,) 15 (precisely) 17 (the) 15 (same) 15 (eigen) 0 (v) -2 (alues.) 20 (If) 15 (rounding) 16 (errors) 15 (pro) 1 (duced) ks cmex10.300 @sf 778 1855 p m ( ) s cmmi10.329 @sf 849 1900 p m (:) s cmr10.329 @sf m (689760) s cmmi10.329 @sf 1058 1900 p m (:) s cmr10.329 @sf m (258501) s cmsy10.329 @sf 832 1955 p m (\000) s cmmi10.329 @sf m (:) s cmr10.329 @sf m (257827) s cmmi10.329 @sf 1058 1955 p m (:) s cmr10.329 @sf m (173440) s cmex10.300 @sf 1227 1855 p m (!) s cmr10.329 @sf 74 2061 p m (\(i.e.,) 18 (c) 0 (hanges) 17 (of) 17 ({1) 17 (and) 18 (+1) 18 (in) 19 (the) 18 (last) 17 (\014gures) 18 (of) 18 (the) 18 (\(1,1\)) 16 (and) 18 (\(2,2\)) 16 (elemen) 0 (ts\)) 17 (the) 18 (eigen) 0 (v) -2 (alues) ks 74 2144 p m (b) 1 (ecome) ks cmmi10.329 @sf 17 r m (:) s cmr10.329 @sf m (431600) s cmsy10.329 @sf 8 r m (\006) s cmmi10.329 @sf 11 r m (i) s cmr10.329 @sf m (\() s cmmi10.329 @sf m (:) s cmr10.329 @sf m (00139) -1 (7\),) 13 (a) 16 (substan) 0 (tial) 15 (c) 0 (hange) 14 (in) 17 (the) 16 (imaginary) 16 (parts.) 21 (Y) -3 (et) 15 (in) 16 (this) 16 (example) ks 74 2227 p m (w) 0 (e) 12 (ha) 0 (v) 0 (e) 11 (used) 14 (an) 13 (orthogonal) 13 (similarit) 0 (y) 13 (transformation) 12 (that) 13 (is) 13 (fa) 0 (v) -1 (orable) 12 (to) 13 (n) 0 (umerical) 13 (stabilit) 0 (y) -3 (.) 18 (In) ks 74 2310 p m (general,) 15 (the) 16 (b) 0 (ypassed) 14 (matrix) 15 (will) 17 (b) 1 (e) 16 (sub) 3 (jected) 15 (to) 15 (a) 15 (non-orthogonal) 14 (similarit) 0 (y) 16 (transformation.) ks cmbx12.300 @sf 74 2474 p m (3.3) s 202 2474 p m (Double) 18 (past) 19 (double) ks cmr10.329 @sf 74 2628 p m (Finally) -3 (,) 17 (w) 0 (e) 15 (turn) 16 (to) 16 (the) 16 (problem) 17 (of) 16 (mo) 0 (ving) 15 (a) 16 (double) 18 (past) 16 (a) 16 (double.) 24 (Since) 18 (t) 0 (w) -1 (o) 14 (pairs) 16 (of) 16 (complex) ks 74 2711 p m (conjugate) 13 (eigen) 0 (v) -2 (alues) ks cmmi10.329 @sf 14 r m (\025) s cmr8.300 @sf 539 2718 p m (1) s cmsy10.329 @sf 565 2711 p m (\006) s cmmi10.329 @sf 7 r m (\026) s cmr8.300 @sf 634 2718 p m (1) s cmr10.329 @sf 667 2711 p m (and) s cmmi10.329 @sf 14 r m (\025) s cmr8.300 @sf 781 2718 p m (2) s cmsy10.329 @sf 807 2711 p m (\006) s cmmi10.329 @sf 7 r m (i\026) s cmr8.300 @sf 892 2718 p m (2) s cmr10.329 @sf 925 2711 p m (are) 13 (in) 0 (v) 0 (olv) 0 (ed,) 11 (it) 14 (is) 14 (not) 13 (p) 1 (ossible) 15 (for) 13 (just) 13 (one) 14 (eigen) 0 (v) -2 (alue) ks 996 2850 p m (13) s @eop 12 @bop0 cmr10.329 @sf [<80E063F03F181C04> 14 4 -4 -26 23] 126 @dc 12 @bop1 cmr10.329 @sf 74 67 p m (and) 16 (2,) 14 (and) ks cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 328 74 p m (3) s cmr10.329 @sf 363 67 p m (is) 16 (not) 14 (in) 0 (v) 0 (olv) 0 (ed.) 18 (Nev) 0 (ertheless,) 15 (the) 15 (transformed) 14 (matrix) 15 (is) ks cmex10.300 @sf 823 115 p m (0) s 823 188 p m (B) s 823 215 p m (@) s cmr10.329 @sf 907 156 p m (0) s 954 173 p 2 57 ru cmsy10.329 @sf 980 156 p m (\000) s cmr10.329 @sf m (1) s 1088 156 p m (0) s 883 175 p 253 2 ru 907 213 p m (0) s 954 230 p 2 57 ru 998 213 p m (0) s 1088 213 p m (1) s 907 268 p m (0) s 954 285 p 2 57 ru 998 268 p m (0) s 1088 268 p m (0) s cmex10.300 @sf 1159 115 p m (1) s 1159 188 p m (C) s 1159 215 p m (A) s cmmi10.329 @sf 1203 212 p m (;) s cmr10.329 @sf 74 359 p m (and) 16 (our) 14 (\\ob) 3 (jectiv) 0 (e") 14 (\(inappropriate) 16 (though) 15 (it) 15 (is\)) 16 (has) 15 (b) 1 (een) 16 (ac) 0 (hiev) 0 (ed.) ks 124 484 p m (The) 12 (relev) -2 (ance) 12 (of) 11 (this) 11 (discussion) 14 (to) 10 (the) 12 (p) 1 (erformance) 11 (of) 11 (our) 11 (algorithm) 12 (is) 12 (the) 11 (follo) 0 (wing.) 19 (When) 11 (w) 0 (e) ks 74 567 p m (attempt) 13 (to) 14 (bring) 15 (a) 14 (single) 16 (past) 13 (a) 14 (double) 16 (ha) 0 (ving) 13 (eigen) 0 (v) -2 (alues) 15 (that) 13 (are) 14 (fairly) 15 (close) 15 (to) 13 (it,) 15 (the) 14 (danger) ks 74 650 p m (arises) 14 (that) 13 (to) 1 (o) 13 (m) 0 (uc) 0 (h) 12 (reliance) 15 (is) 15 (placed) 15 (on) 13 (the) 14 (e\013ect) 14 (ac) 0 (hiev) 0 (ed) 12 (b) 0 (y) 13 (the) 14 (v) 0 (ery) 12 (small) 15 (third) 14 (comp) 1 (onen) 0 (t) ks 74 733 p m (in) 14 (the) 14 (normalized) 14 (v) 0 (ersion) 13 (of) 13 (the) 13 (unique) 15 (eigen) 0 (v) 0 (ector) 11 (corresp) 1 (onding) 14 (to) ks cmmi10.329 @sf 13 r m (\025) s cmr8.300 @sf 1462 740 p m (3) s cmr10.329 @sf 1481 733 p m (.) 20 (In) 13 (the) 14 (analogous) 13 (single) ks 74 816 p m (past) 18 (single) 20 (case,) 19 (the) 18 (solution) 19 (w) 0 (as) 17 (determined) 19 (with) 19 (considerable) 20 (accuracy) -3 (.) 29 (Here,) 19 (ho) 0 (w) -1 (ev) -1 (er,) 17 (the) ks 74 899 p m (solution) 20 (is) 19 (not) 19 (nearly) 19 (as) 19 (simple.) 32 (Moreo) 0 (v) -1 (er,) 17 (when) 19 (the) 19 (transformation) 18 (has) 19 (b) 1 (een) 20 (computed,) 19 (w) 0 (e) ks 74 982 p m (shall) 14 (need) 14 (to) 12 (apply) 14 (it) 13 (to) 12 (the) 13 (3) ks cmsy10.329 @sf 5 r m (\002) s cmr10.329 @sf 5 r m (3) 13 (matrix) 13 (itself,) 13 (as) 13 (w) 0 (ell) 13 (as) 12 (to) 12 (the) 13 (remainder) 14 (of) 12 (those) 13 (relev) -2 (an) 0 (t) 11 (ro) 0 (ws) ks 74 1065 p m (and) 15 (columns,) 16 (since) 16 (the) 15 (new) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2) 14 (is) 16 (not) 14 (determined) 17 (in) 16 (a) 14 (trivial) 16 (manner) 15 (as) 15 (w) 0 (ere) 14 (the) 15 (elemen) 0 (ts) 14 (in) ks 74 1148 p m (the) 15 (single) 17 (past) 15 (single) 16 (case.) ks 124 1273 p m (Clearly) 14 (the) 13 (set) 12 (of) 13 (equations) 13 (m) 0 (ust) 12 (b) 1 (e) 13 (solv) 0 (ed) 12 (with) 14 (some) 12 (care.) 19 (It) 13 (is) 14 (essen) 0 (tial) 13 (that) 12 (the) 13 (normalized) ks 74 1356 p m (v) 0 (ersion) 15 (of) ks 767 1439 p m (\() s cmmi10.329 @sf m (x) s cmr8.300 @sf 811 1446 p m (1) s cmmi10.329 @sf 831 1439 p m (;) 8 (x) ks cmr8.300 @sf 878 1446 p m (2) s cmmi10.329 @sf 897 1439 p m (;) s cmr10.329 @sf 8 r m (1\)) 13 (i.e.,) ks 1058 1439 p m (\() 3 (~) ks cmmi10.329 @sf 1076 1439 p m (x) s cmr8.300 @sf 1102 1446 p m (1) s cmmi10.329 @sf 1122 1439 p m (;) s cmr10.329 @sf 10 r m (~) s cmmi10.329 @sf 1143 1439 p m (x) s cmr8.300 @sf 1169 1446 p m (2) s cmmi10.329 @sf 1187 1439 p m (;) s cmr10.329 @sf 11 r m (~) s cmmi10.329 @sf 1208 1439 p m (x) s cmr8.300 @sf 1234 1446 p m (3) s cmr10.329 @sf 1253 1439 p m (\)) s 74 1543 p m (should) 16 (b) 1 (e) 16 (suc) 0 (h) 15 (that) ks 668 1599 p m (\() s cmmi10.329 @sf m (t) s cmr8.300 @sf 702 1606 p m (11) s cmsy10.329 @sf 749 1599 p m (\000) s cmmi10.329 @sf 11 r m (\025) s cmr8.300 @sf 822 1606 p m (3) s cmr10.329 @sf 841 1599 p m (\)) 3 (~) ks cmmi10.329 @sf 859 1599 p m (x) s cmr8.300 @sf 885 1606 p m (1) s cmr10.329 @sf 914 1599 p m (+) s cmmi10.329 @sf 11 r m (t) s cmr8.300 @sf 976 1606 p m (12) s cmr10.329 @sf 1016 1599 p m (~) s cmmi10.329 @sf 1013 1599 p m (x) s cmr8.300 @sf 1039 1606 p m (2) s cmr10.329 @sf 1069 1599 p m (+) s cmmi10.329 @sf 11 r m (t) s cmr8.300 @sf 1131 1606 p m (13) s cmr10.329 @sf 1171 1599 p m (~) s cmmi10.329 @sf 1168 1599 p m (x) s cmr8.300 @sf 1194 1606 p m (3) s cmr10.329 @sf 1256 1599 p m (=) s cmmi10.329 @sf 1333 1599 p m (\017) s cmr8.300 @sf 1351 1606 p m (1) s cmmi10.329 @sf 805 1654 p m (t) s cmr8.300 @sf 821 1661 p m (21) s cmr10.329 @sf 861 1654 p m (~) s cmmi10.329 @sf 858 1654 p m (x) s cmr8.300 @sf 884 1661 p m (1) s cmr10.329 @sf 914 1654 p m (+) s cmmi10.329 @sf 11 r m (t) s cmr8.300 @sf 976 1661 p m (22) s cmr10.329 @sf 1016 1654 p m (~) s cmmi10.329 @sf 1013 1654 p m (x) s cmr8.300 @sf 1039 1661 p m (2) s cmr10.329 @sf 1069 1654 p m (+) s cmmi10.329 @sf 11 r m (t) s cmr8.300 @sf 1131 1661 p m (23) s cmr10.329 @sf 1171 1654 p m (~) s cmmi10.329 @sf 1168 1654 p m (x) s cmr8.300 @sf 1194 1661 p m (3) s cmr10.329 @sf 1256 1654 p m (=) s cmmi10.329 @sf 1333 1654 p m (\017) s cmr8.300 @sf 1351 1661 p m (2) s cmr10.329 @sf 74 1729 p m (b) 1 (e) 16 (true) 16 (with) ks cmmi10.329 @sf 16 r m (\017) s cmr8.300 @sf 355 1736 p m (1) s cmr10.329 @sf 390 1729 p m (and) s cmmi10.329 @sf 16 r m (\017) s cmr8.300 @sf 497 1736 p m (2) s cmr10.329 @sf 517 1729 p m (,) 15 (whic) 0 (h) 16 (are) 15 (at) 15 (noise) 16 (lev) 0 (el) 16 (relativ) 0 (e) 15 (to) 15 (the) 15 (co) 1 (e\016cien) 0 (ts) 16 (on) 15 (the) 16 (left-hand) 16 (side) ks 74 1812 p m (\() s cmmi10.329 @sf m (\017) s cmr8.300 @sf 110 1819 p m (1) s cmr10.329 @sf 149 1812 p m (and) s cmmi10.329 @sf 20 r m (\017) s cmr8.300 @sf 260 1819 p m (2) s cmr10.329 @sf 300 1812 p m (w) 0 (ould) 19 (b) 1 (e) 20 (zero) 19 (with) 20 (exact) 19 (computation\).) 33 (The) 20 (solution) 20 (of) 19 (the) 19 (system) 20 (b) 0 (y) 18 (Gaussian) ks 74 1895 p m (elimination) 25 (with) 23 (piv) 0 (oting) 22 (ensures) 24 (just) 22 (that;) 26 (it) 23 (pro) 1 (duces) ks cmmi10.329 @sf 23 r m (x) s cmr8.300 @sf 1279 1902 p m (1) s cmr10.329 @sf 1322 1895 p m (and) s cmmi10.329 @sf 23 r m (x) s cmr8.300 @sf 1444 1902 p m (2) s cmr10.329 @sf 1486 1895 p m (with) 23 (errors) 22 (that) 22 (are) 22 (so) ks 74 1978 p m (correlated) 16 (that) 14 (the) 15 (normalized) 17 (v) 0 (ersions) 14 (giv) 0 (e) 15 (residuals) 16 (at) 15 (noise) 15 (lev) 0 (el.) ks 124 2103 p m (In) 21 (place) 21 (of) 20 (Gaussian) 20 (elimination) 22 (with) 21 (piv) 0 (oting,) 20 (w) 0 (e) 19 (could) 22 (use) 20 (an) 0 (y) 19 (stable) 20 (direct) 21 (metho) 1 (d) 21 (to) ks 74 2186 p m (solv) 0 (e) 19 (the) 19 (system|e.g.,) 19 (Giv) 0 (ens) 19 (triangulation.) 32 (Ho) 0 (w) -1 (ev) -1 (er,) 18 (if) 20 (w) 0 (e) 18 (w) 0 (ere) 18 (to) 18 (solv) 0 (e) 19 (the) 19 (system) 19 (b) 0 (y) 18 (an) ks 74 2269 p m (unstable) 17 (metho) 1 (d) 17 (suc) 0 (h) 16 (as) 16 (Cramer's) 15 (rule) 17 (in) 17 (standard) 16 (\015oating-p) 1 (oin) 0 (t) 16 (arithmetic,) 17 (w) 0 (e) 15 (w) 0 (ould) 16 (obtain) ks 74 2352 p m (a) 15 (computed) ks cmmi10.329 @sf 16 r m (x) s cmr8.300 @sf 348 2359 p m (1) s cmr10.329 @sf 383 2352 p m (and) s cmmi10.329 @sf 16 r m (x) s cmr8.300 @sf 498 2359 p m (2) s cmr10.329 @sf 533 2352 p m (with) 15 (errors) 15 (that) 14 (are) 16 (uncorrelated,) 15 (and) 16 (the) 15 (residual) 17 (corresp) 1 (onding) 16 (to) 15 (the) ks 74 2435 p m (normalized) 17 (v) 0 (ector) 13 (w) 0 (ould) 15 (not) 14 (then) 16 (b) 1 (e) 16 (at) 14 (noise) 16 (lev) 0 (el.) ks 124 2560 p m (Assuming,) 13 (then,) 13 (that) 12 (w) 0 (e) 11 (ha) 0 (v) -1 (e) 10 (a) 12 (normalized) 14 (eigen) 0 (v) 0 (ector) 10 (giving) 14 (negligible) 15 (residuals,) 13 (the) 13 (pro) 1 (cess) ks 74 2643 p m (is) 14 (satisfactory) -3 (.) 18 (Indeed,) 15 (it) 14 (is) 14 (merely) 14 (the) 14 (metho) 1 (d) 14 (of) 13 (de\015ation) 14 (b) 0 (y) 13 (orthogonal) 13 (similarit) 0 (y) 14 (transforma-) ks 74 2726 p m (tions) 15 (that) 13 (is) 15 (used) 14 (after) 14 (\014nding) 15 (an) 14 (eigen) 0 (v) 0 (ector) 13 (of) 13 (a) 14 (general) 15 (matrix) 14 (\(see,) 14 (e.g.,) 13 (Section) 15 (20,) 13 (Chapter) ks 996 2850 p m (12) s @eop 11 @bop0 cmmi10.329 @sf [<00FE00000381C000060020000C0018001800040038000200380001007000010070000080700000 80F0000080F0000000F0000000F0000000F0000000780000007800000078000000780000003C00 00003C0000001E0000101E0000100F000018070000180380001801C0001800C0001C0060003C00 38003C001C004C000781860000FE02> 31 33 -2 1 33] 67 @dc 11 @bop1 cmr10.329 @sf 74 67 p m (The) 18 (matrix) 17 (is) 18 (in) 18 (the) 18 (required) 18 (form,) 17 (with) ks cmmi10.329 @sf 18 r m (\025) s cmr8.300 @sf 940 74 p m (3) s cmr10.329 @sf 976 67 p m (in) 19 (the) 17 (leading) 19 (p) 1 (osition,) 19 (zeros) 17 (in) 18 (the) 18 (\014rst) 17 (column,) ks 74 150 p m (and) s cmmi10.329 @sf 16 r m (C) s cmr10.329 @sf 17 r m (giv) 0 (en) 15 (b) 0 (y) ks cmmi10.329 @sf 781 273 p m (C) s cmr10.329 @sf 15 r m (=) s cmex10.300 @sf 877 201 p m ( ) s cmr10.329 @sf 931 247 p m (0) s 995 247 p m (1) s cmmi10.329 @sf m (=) s cmsy10.329 @sf 1041 210 p m (p) s 1078 210 p 23 2 ru cmr10.329 @sf 1078 247 p m (2) s 931 302 p m (0) s 1036 302 p m (0) s cmex10.300 @sf 1122 201 p m (!) s cmmi10.329 @sf 1162 273 p m (;) s cmr10.329 @sf 1883 273 p m (\(10\)) s 74 395 p m (whic) 0 (h) 12 (is) 12 (similar) 13 (to) 11 (the) 12 (original) 13 (2) ks cmsy10.329 @sf 3 r m (\002) s cmr10.329 @sf 3 r m (2,) 12 (but) 12 (certainly) 13 (not) 11 (orthogonally) 12 (similar) 13 (since) 13 (it) 12 (has) 11 (a) 12 (di\013eren) 0 (t) ks 74 478 p m (Euclidean) 18 (norm.) 21 (Ho) 0 (w) 0 (ev) -1 (er,) 13 (when) 16 (one) 16 (considers) 17 (ho) 0 (w) 15 (it) 16 (has) 15 (come) 16 (ab) 1 (out,) 16 (it) 16 (w) 0 (ould) 15 (b) 1 (e) 17 (p) 1 (erv) 0 (erse) 15 (to) ks 74 561 p m (describ) 1 (e) 17 (it) 15 (as) 15 (\\bringing) ks cmmi10.329 @sf 16 r m (\025) s cmr8.300 @sf 580 568 p m (3) s cmr10.329 @sf 615 561 p m (past) 15 (the) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2.") s 124 686 p m (Supp) 1 (ose) 16 (no) 0 (w) 14 (w) 0 (e) 14 (p) 1 (ertub) 16 (the) 15 (\(2,1\)) 14 (en) 0 (try) 13 (of) 15 (the) 15 (matrix) 15 (b) 0 (y) ks cmmi10.329 @sf 14 r m (\017) s cmr8.300 @sf 1276 671 p m (2) s cmr10.329 @sf 1312 686 p m (to) 14 (giv) 0 (e) ks cmex10.300 @sf 595 742 p m (0) s 595 815 p m (B) s 595 842 p m (@) s cmr10.329 @sf 699 783 p m (1) s cmsy10.329 @sf 810 783 p m (\000) s cmr10.329 @sf m (1) s 910 783 p m (0) s 652 839 p m (1) 10 (+) ks cmmi10.329 @sf 11 r m (\017) s cmr8.300 @sf 749 824 p m (2) s cmsy10.329 @sf 810 839 p m (\000) s cmr10.329 @sf m (1) s 910 839 p m (1) s 699 896 p m (0) s 828 896 p m (0) s 910 896 p m (0) s cmex10.300 @sf 953 742 p m (1) s 953 815 p m (C) s 953 842 p m (A) s cmmi10.329 @sf 997 839 p m (;) 18 (\025) ks cmr8.300 @sf 1055 846 p m (1) s cmmi10.329 @sf 1074 839 p m (;) 8 (\025) ks cmr8.300 @sf 1122 846 p m (2) s cmr10.329 @sf 1153 839 p m (=) s cmsy10.329 @sf 13 r m (\006) s cmmi10.329 @sf m (i\017;) 18 (\025) ks cmr8.300 @sf 1328 846 p m (3) s cmr10.329 @sf 1360 839 p m (=) 13 (0) ks cmmi10.329 @sf m (:) s cmr10.329 @sf 74 990 p m (Then) 16 (there) 15 (is) 16 (an) 15 (eigen) 0 (v) 0 (ector) ks cmmi10.329 @sf 13 r m (x) s cmr10.329 @sf 15 r m (corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 1070 997 p m (3) s cmr10.329 @sf 1104 990 p m (of) 15 (the) 15 (form) ks cmmi10.329 @sf 752 1086 p m (x) s cmmi8.300 @sf 778 1070 p m (T) s cmr10.329 @sf 847 1086 p m (=) s 924 1086 p m (\() s cmsy10.329 @sf m (\000) s cmr10.329 @sf m (1) s cmmi10.329 @sf m (=\017) s cmr8.300 @sf 1041 1070 p m (2) s cmmi10.329 @sf 1061 1086 p m (;) s cmsy10.329 @sf 8 r m (\000) s cmr10.329 @sf m (1) s cmmi10.329 @sf m (=\017) s cmr8.300 @sf 1181 1070 p m (2) s cmmi10.329 @sf 1200 1086 p m (;) s cmr10.329 @sf 8 r m (1\)) s 847 1141 p m (=) s 924 1141 p m (\() s cmsy10.329 @sf m (\000) s cmr10.329 @sf m (1) s cmmi10.329 @sf m (=\017) s cmr8.300 @sf 1041 1126 p m (2) s cmr10.329 @sf 1061 1141 p m (\)\(1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 7 r m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 1237 1126 p m (2) s cmr10.329 @sf 1256 1141 p m (\)) s cmmi10.329 @sf m (:) s cmr10.329 @sf 74 1235 p m (The) 20 (normalized) 22 (v) 0 (ersion) 19 (of) 19 (this) 21 (v) 0 (ector) 18 (has) 20 (a) 19 (v) 0 (ery) 19 (small) 21 (third) 20 (comp) 1 (onen) 0 (t.) 34 (If) 20 (w) 0 (e) 19 (p) 1 (erform) 19 (our) ks 74 1318 p m (algorithm) 16 (exactly) -3 (,) 15 (it) 16 (giv) 0 (es) 15 (a) 16 (\(2,3\)) 14 (rotation) 15 (with) 16 (an) 16 (angle) 16 (of) 16 (order) ks cmmi10.329 @sf 15 r m (\017) s cmr8.300 @sf 1406 1303 p m (2) s cmr10.329 @sf 1442 1318 p m (\(the) 16 (corresp) 1 (onding) 16 (matrix) ks 74 1401 p m (is) 16 (almost) 15 (the) 16 (iden) 0 (tit) 0 (y) 14 (matrix\)) 14 (while) 17 (the) 16 (\(1,2\)) 14 (rotation) 14 (has) 15 (an) 16 (angle) 16 (of) 15 (almost) 15 (exactly) ks cmmi10.329 @sf 15 r m (\031) 2 (=) ks cmr10.329 @sf m (4.) 20 (The) ks 74 1484 p m (resulting) 16 (matrix) 15 (has) ks cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 517 1491 p m (3) s cmr10.329 @sf 549 1484 p m (=) 13 (0) 14 (in) 16 (the) 15 (leading) 17 (p) 1 (osition) 16 (and) 15 (the) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2) 15 (matrix) ks cmmi10.329 @sf 14 r m (C) s cmr10.329 @sf 18 r m (is) 16 (almost) 14 (exactly) 15 (as) ks 74 1567 p m (in) 16 (\(10\),) 14 (but) 15 (has) 15 (small) 16 (p) 1 (erturbations) 16 (that) 14 (mak) 0 (e) 14 (its) 15 (eigen) 0 (v) -2 (alues) ks cmsy10.329 @sf 15 r m (\006) s cmmi10.329 @sf m (\017) s cmr10.329 @sf m (.) s 124 1692 p m (The) 17 (simplici) 1 (t) 0 (y) 17 (of) 16 (this) 18 (discussion) 18 (is) 17 (sligh) 0 (tly) 17 (obscured) 18 (b) 0 (y) 15 (the) 17 (use) 17 (of) 17 (plane) 18 (rotations) 16 (and) 17 (their) ks 74 1775 p m (in) 0 (tro) 1 (duction) 22 (of) 21 (irrationals.) 40 (If) 22 (w) 0 (e) 21 (think) 23 (in) 22 (terms) 21 (of) 22 (nonorthogonal) 21 (transformations,) 22 (then) 22 (to) ks 74 1858 p m (con) 0 (v) 0 (ert) ks 808 1941 p m (\(1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmsy10.329 @sf 8 r m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 967 1923 p m (2) s cmr10.329 @sf 985 1941 p m (\)) 15 (to) ks 1074 1941 p m (\(1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) s cmr10.329 @sf 7 r m (0\)) s cmmi10.329 @sf m (;) s cmr10.329 @sf 74 2048 p m (w) 0 (e) 14 (p) 1 (erform) 15 (a) 15 (similarit) 0 (y) 15 (with) 16 (the) 15 (unit) 16 (lo) 0 (w) 0 (er) 13 (triangular) 15 (matrix) ks cmmi10.329 @sf 814 2199 p m (M) s cmr10.329 @sf 18 r m (=) s cmex10.300 @sf 924 2102 p m (0) s 924 2175 p m (B) s 924 2202 p m (@) s cmr10.329 @sf 998 2142 p m (1) s cmsy10.329 @sf 981 2198 p m (\000) s cmr10.329 @sf m (1) s 1080 2198 p m (1) s cmmi10.329 @sf 991 2255 p m (\017) s cmr8.300 @sf 1009 2240 p m (2) s cmr10.329 @sf 1080 2255 p m (0) s 1145 2255 p m (1) s cmex10.300 @sf 1188 2102 p m (1) s 1188 2175 p m (C) s 1188 2202 p m (A) s cmr10.329 @sf 74 2350 p m (and) 16 (obtain) 15 (as) 15 (our) 15 (transformed) 14 (matrix) ks cmex10.300 @sf 815 2395 p m (0) s 815 2468 p m (B) s 815 2495 p m (@) s cmr10.329 @sf 900 2435 p m (0) s 947 2452 p 2 57 ru cmsy10.329 @sf 980 2435 p m (\000) s cmr10.329 @sf m (1) s 1096 2435 p m (0) s 875 2454 p 269 2 ru 900 2492 p m (0) s 947 2509 p 2 57 ru 998 2492 p m (0) s 1096 2492 p m (1) s 900 2548 p m (0) s 947 2565 p 2 57 ru cmsy10.329 @sf 972 2548 p m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 1025 2533 p m (2) s cmr10.329 @sf 1096 2548 p m (0) s cmex10.300 @sf 1167 2395 p m (1) s 1167 2468 p m (C) s 1167 2495 p m (A) s cmmi10.329 @sf 1211 2492 p m (:) s cmr10.329 @sf 74 2643 p m (The) 19 (zero) 18 (eigen) 0 (v) -2 (alue) 18 (is) 19 (brough) 0 (t) 17 (to) 18 (the) 18 (top) 18 (and) 18 (the) 19 (eigen) 0 (v) -2 (alues) ks cmsy10.329 @sf 18 r m (\006) s cmmi10.329 @sf m (i\017) s cmr10.329 @sf 19 r m (mo) 0 (v) -1 (ed) 17 (to) 17 (the) 19 (b) 1 (ottom) 17 (in) 19 (a) ks 74 2726 p m (transparen) 0 (tly) 13 (ob) 0 (vious) 13 (w) 0 (a) -1 (y) -4 (.) 18 (When) ks cmmi10.329 @sf 14 r m (\017) s cmr10.329 @sf 14 r m (=) 13 (0,) 13 (the) 14 (transformation) 13 (op) 1 (erates) 14 (only) 15 (on) 14 (ro) 0 (w) 12 (and) 14 (column) 15 (1) ks 996 2850 p m (11) s @eop 10 @bop0 cmex10.300 @sf [ 22 7 0 -24 23] 98 @dc cmr10.329 @sf [<70F8F8F8700000000000202020202070707070707070707070F8F8F8F8F8F8F870> 5 33 -4 0 13] 33 @dc cmmi10.329 @sf [<1E000031800070C00060E000607000E03000E03800E03C00E01C00F01E00F00E00F00F00F00F00 F00F007807807FFF807FFF807807803C03C03C03C03C03C01C03C01E03C00E01C00F01C00701C0 0301C003818001C18000C380006300001E00> 18 32 -2 0 21] 18 @dc 10 @bop1 cmr10.329 @sf 74 67 p m (then) s cmmi10.329 @sf 16 r m (T) s cmr8.300 @sf 205 74 p m (3) s cmmi10.329 @sf 224 67 p m (x) s cmr10.329 @sf 13 r m (=) s cmmi10.329 @sf 13 r m (\025) s cmr8.300 @sf 338 74 p m (3) s cmmi10.329 @sf 357 67 p m (x) s cmr10.329 @sf 15 r m (giv) 0 (es) ks 692 187 p m (\() s cmmi10.329 @sf m (t) s cmr8.300 @sf 726 194 p m (11) s cmsy10.329 @sf 773 187 p m (\000) s cmmi10.329 @sf 11 r m (\025) s cmr8.300 @sf 846 194 p m (3) s cmr10.329 @sf 865 187 p m (\)) s cmmi10.329 @sf m (x) s cmr8.300 @sf 909 194 p m (1) s cmr10.329 @sf 939 187 p m (+) s cmmi10.329 @sf 10 r m (t) s cmr8.300 @sf 1000 194 p m (12) s cmmi10.329 @sf 1038 187 p m (x) s cmr8.300 @sf 1064 194 p m (2) s cmr10.329 @sf 1094 187 p m (+) s cmmi10.329 @sf 10 r m (t) s cmr8.300 @sf 1155 194 p m (13) s cmr10.329 @sf 1234 187 p m (=) s 1311 187 p m (0) s 1906 187 p m (\(9\)) s cmmi10.329 @sf 692 283 p m (t) s cmr8.300 @sf 708 290 p m (21) s cmmi10.329 @sf 746 283 p m (x) s cmr8.300 @sf 772 290 p m (1) s cmr10.329 @sf 801 283 p m (+) 11 (\() ks cmmi10.329 @sf m (t) s cmr8.300 @sf 881 290 p m (22) s cmsy10.329 @sf 928 283 p m (\000) s cmmi10.329 @sf 11 r m (\025) s cmr8.300 @sf 1001 290 p m (3) s cmr10.329 @sf 1020 283 p m (\)) s cmmi10.329 @sf m (x) s cmr8.300 @sf 1064 290 p m (2) s cmr10.329 @sf 1094 283 p m (+) s cmmi10.329 @sf 10 r m (t) s cmr8.300 @sf 1155 290 p m (23) s cmr10.329 @sf 1234 283 p m (=) s 1311 283 p m (0) s cmmi10.329 @sf m (:) s cmr10.329 @sf 74 403 p m (The) 16 (matrix) 14 (of) 15 (co) 1 (e\016cien) 0 (ts) ks cmex10.300 @sf 606 391 p m (b) s cmmi10.329 @sf 597 404 p m (T) s cmr10.329 @sf 645 403 p m (of) 15 (this) 16 (system) 14 (of) 15 (equations) 16 (is) ks cmex10.300 @sf 741 511 p m (b) s cmmi10.329 @sf 732 524 p m (T) s cmr10.329 @sf 778 523 p m (=) s cmex10.300 @sf 826 451 p m ( ) s cmmi10.329 @sf 879 496 p m (t) s cmr8.300 @sf 895 503 p m (11) s cmsy10.329 @sf 943 496 p m (\000) s cmmi10.329 @sf 11 r m (\025) s cmr8.300 @sf 1016 503 p m (3) s cmmi10.329 @sf 1127 496 p m (t) s cmr8.300 @sf 1143 503 p m (12) s cmmi10.329 @sf 930 551 p m (t) s cmr8.300 @sf 946 558 p m (21) s cmmi10.329 @sf 1077 551 p m (t) s cmr8.300 @sf 1093 558 p m (22) s cmsy10.329 @sf 1140 551 p m (\000) s cmmi10.329 @sf 11 r m (\025) s cmr8.300 @sf 1213 558 p m (3) s cmex10.300 @sf 1253 451 p m (!) s cmmi10.329 @sf 1293 523 p m (;) s cmr10.329 @sf 74 643 p m (whic) 0 (h) 12 (can) 12 (b) 1 (e) 12 (singular) 13 (only) 12 (if) ks cmmi10.329 @sf 12 r m (\025) s cmr8.300 @sf 669 650 p m (3) s cmr10.329 @sf 700 643 p m (is) 13 (an) 11 (eigen) 0 (v) -2 (alue) 12 (of) 12 (the) 11 (leading) 14 (2) ks cmsy10.329 @sf 3 r m (\002) s cmr10.329 @sf 3 r m (2) 12 (matrix) 11 (of) ks cmmi10.329 @sf 12 r m (T) s cmr8.300 @sf 1612 650 p m (3) s cmr10.329 @sf 1631 643 p m (.) 18 (This) 13 (p) 1 (ossibili) 1 (t) 0 (y) ks 74 726 p m (is) 13 (sp) 1 (eci\014cally) 14 (excluded) 14 (since) ks cmmi10.329 @sf 13 r m (\025) s cmr8.300 @sf 663 733 p m (3) s cmr10.329 @sf 694 726 p m (is) 13 (real) 12 (and) 12 (the) 12 (2) ks cmsy10.329 @sf 4 r m (\002) s cmr10.329 @sf 4 r m (2) 11 (has) 12 (complex) 12 (eigen) 0 (v) -2 (alues) 12 (\(otherwise) 12 (w) 0 (e) 11 (w) 0 (ould) ks 74 809 p m (ha) 0 (v) 0 (e) 16 (triangularized) 19 (it\).) 29 (When) ks cmmi10.329 @sf 18 r m (\025) s cmr8.300 @sf 727 816 p m (3) s cmr10.329 @sf 765 809 p m (is) 18 (v) 0 (ery) 17 (w) 0 (ell) 18 (separated) 18 (from) 17 (the) 18 (t) 0 (w) -1 (o) 16 (complex) 19 (eigen) 0 (v) -2 (alues,) ks cmex10.300 @sf 1940 797 p m (b) s cmmi10.329 @sf 1931 810 p m (T) s cmr10.329 @sf 74 892 p m (will) 18 (b) 1 (e) 16 (v) 0 (ery) 15 (w) 0 (ell) 16 (conditioned) 17 (and) ks cmmi10.329 @sf 16 r m (x) s cmr8.300 @sf 777 899 p m (1) s cmr10.329 @sf 813 892 p m (and) s cmmi10.329 @sf 16 r m (x) s cmr8.300 @sf 928 899 p m (2) s cmr10.329 @sf 964 892 p m (will) 17 (not) 16 (b) 1 (e) 16 (large;) 16 (hence,) 17 (in) 17 (the) 16 (normalized) 17 (v) 0 (ersion) ks 74 975 p m (of) s cmmi10.329 @sf 16 r m (x) s cmr10.329 @sf 17 r m (the) 16 (third) 17 (comp) 1 (onen) 0 (t) 16 (will) 18 (not) 16 (b) 1 (e) 17 (small.) 24 (If) 17 (w) 0 (e) 15 (compute) 17 (the) 16 (transformation) 15 (and) 17 (apply) 17 (it) 17 (to) ks 74 1058 p m (the) 16 (full) 16 (3) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 11 r m (3) 15 (matrix,) 15 (the) 15 (top) 15 (elemen) 0 (t) 15 (will) 17 (b) 1 (e) ks cmmi10.329 @sf 16 r m (\025) s cmr8.300 @sf 1012 1065 p m (3) s cmr10.329 @sf 1047 1058 p m (to) 15 (high) 16 (accuracy) -3 (,) 14 (the) 16 (t) 0 (w) -1 (o) 13 (complex) 16 (eigen) 0 (v) -2 (alues) ks 74 1141 p m (will) 17 (b) 1 (e) 15 (accurately) 15 (preserv) 0 (ed,) 14 (and) 15 (the) 15 (\(3,1\)) 13 (and) 15 (\(3,2\)) 14 (elemen) 0 (ts) 14 (will) 17 (b) 1 (e) 15 (negligibl) 1 (e.) 22 (The) 15 (computed) ks 74 1224 p m (results) 16 (will) 17 (b) 1 (e) 15 (v) 0 (ery) 14 (close) 16 (to) 15 (those) 15 (deriv) 0 (ed) 15 (b) 0 (y) 14 (exact) 15 (arithmetic.) ks 124 1349 p m (As) s cmmi10.329 @sf 20 r m (\025) s cmr8.300 @sf 223 1356 p m (3) s cmr10.329 @sf 263 1349 p m (approac) 0 (hes) 19 (an) 21 (eigen) 0 (v) -2 (alue) 20 (of) 21 (the) 20 (2) ks cmsy10.329 @sf 14 r m (\002) s cmr10.329 @sf 14 r m (2) 20 (blo) 1 (c) 0 (k,) 21 (ho) 0 (w) -1 (ev) -1 (er) 19 (\(notice) 21 (that) 19 (this) 21 (means) 21 (that) ks 74 1432 p m (the) 19 (imaginary) 19 (parts) 18 (of) 19 (the) 19 (complex) 19 (eigen) 0 (v) -2 (alues) 19 (m) 0 (ust) 17 (b) 1 (e) 20 (small) 19 (since) ks cmmi10.329 @sf 20 r m (\025) s cmr8.300 @sf 1497 1439 p m (3) s cmr10.329 @sf 1535 1432 p m (is) 20 (real,) 19 (and) 19 (hence) 20 (w) 0 (e) ks 74 1515 p m (are) 16 (really) 16 (mo) 0 (ving) 15 (to) 0 (w) -1 (ar) -1 (ds) 14 (a) 16 (triple) 17 (eigen) 0 (v) -2 (alue\),) 15 (the) 16 (matrix) ks cmex10.300 @sf 1251 1503 p m (b) s cmmi10.329 @sf 1243 1516 p m (T) s cmr10.329 @sf 1291 1515 p m (will) 18 (b) 1 (ecome) 16 (progressiv) 0 (ely) 16 (more) 15 (ill) ks 74 1598 p m (conditioned,) 18 (and) 17 (in) 18 (general) ks cmmi10.329 @sf 17 r m (x) s cmr8.300 @sf 662 1605 p m (1) s cmr10.329 @sf 698 1598 p m (and) s cmmi10.329 @sf 17 r m (x) s cmr8.300 @sf 814 1605 p m (2) s cmr10.329 @sf 850 1598 p m (will) 18 (b) 1 (e) 17 (larger.) 24 (In) 17 (the) 17 (limiting) 18 (situation,) 17 (the) 17 (eigen) 0 (v) 0 (ector) ks 74 1681 p m (will) 18 (ha) 0 (v) -1 (e) 14 (a) 16 (zero) 15 (third) 17 (comp) 1 (onen) 0 (t) 15 (and) 16 (will) 17 (b) 1 (e) 17 (an) 16 (eigen) 0 (v) 0 (ector) 14 (of) 15 (the) 16 (leading) 17 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (2) 15 (matrix) 16 (rather) ks 74 1764 p m (than) 16 (one) 17 (corresp) 1 (onding) 17 (to) ks cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 642 1771 p m (3) s cmr10.329 @sf 678 1764 p m (in) 17 (the) 16 (3) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (3) 16 (matrix.) 23 (The) 16 (matrix) ks cmmi10.329 @sf 16 r m (Q) s cmr10.329 @sf 16 r m (is) 17 (merely) 17 (a) 16 (plane) 17 (rotation) 15 (in) ks 74 1847 p m (the) 16 (\(1,2\)) 14 (plane) 17 (and) 17 (do) 1 (es) 16 (not) 15 (a\013ect) ks cmmi10.329 @sf 16 r m (\025) s cmr8.300 @sf 809 1854 p m (3) s cmr10.329 @sf 828 1847 p m (.) 22 (It) 16 (is) 16 (di\016cult) 18 (to) 15 (view) 16 (this) 17 (in) 17 (terms) 15 (of) 16 (bringing) 17 (the) 16 (\(3,3\)) ks 74 1930 p m (elemen) 0 (t) 19 (in) 0 (to) 17 (the) 19 (leading) 20 (p) 1 (osition!) 31 (Indeed,) 21 (w) 0 (e) 17 (are) 18 (merely) 20 (recognizing) 20 (the) 18 (fact) 18 (that) 18 (the) 19 (upp) 1 (er) ks 74 2013 p m (2) s cmsy10.329 @sf 13 r m (\002) s cmr10.329 @sf 13 r m (2) 19 (no) 0 (w) 18 (has) 19 (a) 19 (double) 21 (real) 19 (ro) 1 (ot,) 20 (and) 19 (w) 0 (e) 18 (are) 19 (triangularizing) 21 (it.) 33 (Since) 21 (the) 19 (real) 20 (ro) 1 (ots) 18 (that) 19 (it) ks 74 2096 p m (has) 18 (are) 18 (the) 19 (same) 18 (as) ks cmmi10.329 @sf 18 r m (\025) s cmr8.300 @sf 522 2103 p m (3) s cmr10.329 @sf 541 2096 p m (,) 19 (ho) 0 (w) -1 (ev) -1 (er,) 17 (the) 18 (illu) 1 (sion) 20 (of) 18 (ha) 0 (ving) 18 (mo) 0 (v) -1 (ed) ks cmmi10.329 @sf 17 r m (\025) s cmr8.300 @sf 1383 2103 p m (3) s cmr10.329 @sf 1421 2096 p m (in) 0 (to) 17 (the) 18 (leading) 20 (p) 1 (osition) 20 (is) ks 74 2179 p m (preserv) 0 (ed.) 19 (Th) 0 (us,) 14 (if) ks cmmi10.329 @sf 615 2290 p m (T) s cmr8.300 @sf 642 2297 p m (3) s cmr10.329 @sf 674 2290 p m (=) s cmex10.300 @sf 722 2194 p m (0) s 722 2267 p m (B) s 722 2294 p m (@) s cmr10.329 @sf 783 2234 p m (1) s 856 2234 p m (-1) s 917 2251 p 2 57 ru 943 2234 p m (0) s 783 2290 p m (1) s 856 2290 p m (-1) s 917 2307 p 2 57 ru 943 2290 p m (1) s 758 2309 p 233 2 ru 783 2347 p m (0) s 863 2347 p m (0) s 917 2364 p 2 57 ru 943 2347 p m (0) s cmex10.300 @sf 991 2194 p m (1) s 991 2267 p m (C) s 991 2294 p m (A) s cmmi10.329 @sf 1035 2290 p m (;) 20 (\025) ks cmr8.300 @sf 1095 2297 p m (1) s cmr10.329 @sf 1126 2290 p m (=) s cmmi10.329 @sf 13 r m (\025) s cmr8.300 @sf 1201 2297 p m (2) s cmr10.329 @sf 1233 2290 p m (=) s cmmi10.329 @sf 13 r m (\025) s cmr8.300 @sf 1308 2297 p m (3) s cmr10.329 @sf 1340 2290 p m (=) 13 (0) ks cmmi10.329 @sf m (;) s cmr10.329 @sf 74 2432 p m (the) 16 (only) 15 (eigen) 0 (v) 0 (ector) 14 (is) 16 (\(1) ks cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (1) s cmmi10.329 @sf m (;) s cmr10.329 @sf 7 r m (0\)) s cmmi8.300 @sf 681 2417 p m (T) s cmr10.329 @sf 706 2432 p m (;) 15 (there) 15 (is) 16 (no) 15 (eigen) 0 (v) 0 (ector) 14 (of) 15 (the) 15 (form) 15 (\() ks cmmi10.329 @sf m (x;) 8 (x;) ks cmr10.329 @sf 8 r m (1) -1 (\)) ks cmmi8.300 @sf 1586 2417 p m (T) s cmr10.329 @sf 1612 2432 p m (.) 20 (F) -3 (or) 14 (the) 15 (rotation) ks 74 2515 p m (in) 16 (the) 15 (\(1,2\)) 14 (plane) ks cmmi10.329 @sf 16 r m (\022) s cmr10.329 @sf 14 r m (=) s cmmi10.329 @sf 13 r m (\031) 2 (=) ks cmr10.329 @sf m (4) 15 (and) 15 (the) 15 (transformed) 15 (matrix) 14 (is) ks cmex10.300 @sf 596 2570 p m (0) s 596 2643 p m (B) s 596 2670 p m (@) s cmr10.329 @sf 653 2610 p m (0) s cmsy10.329 @sf 718 2610 p m (\000) s cmr10.329 @sf m (2) s 817 2610 p m (1) s cmmi10.329 @sf m (=) s cmsy10.329 @sf 863 2573 p m (p) s 901 2573 p 23 2 ru cmr10.329 @sf 901 2611 p m (2) s 653 2668 p m (0) s 735 2668 p m (0) s 817 2668 p m (1) s cmmi10.329 @sf m (=) s cmsy10.329 @sf 863 2631 p m (p) s 901 2631 p 23 2 ru cmr10.329 @sf 901 2669 p m (2) s 653 2724 p m (0) s 735 2724 p m (0) s 859 2724 p m (0) s cmex10.300 @sf 944 2570 p m (1) s 944 2643 p m (C) s 944 2670 p m (A) s cmr10.329 @sf 993 2666 p m (=) s cmex10.300 @sf 1041 2570 p m (0) s 1041 2643 p m (B) s 1041 2670 p m (@) s cmmi10.329 @sf 1135 2610 p m (\025) s cmr8.300 @sf 1162 2617 p m (3) s 1214 2627 p 2 57 ru cmmi10.329 @sf 1240 2610 p m (x) 45 (x) ks 1101 2629 p 262 2 ru cmr10.329 @sf 1146 2668 p m (0) s 1146 2723 p m (0) s 1214 2740 p 2 113 ru 1265 2695 p save [<000FF8000000003FFF00000000FC03C0000003F00060000007C0001800000F80000C00001F0000 0600001F00000300003E00000100003E00000080007C00000040007C00000040007C0000002000 7C00000020007C0000002000FC0000000000FC0000000000FC00000000007C00000000007E0000 0000007E00000000007E00000000007E00000000003F00000000003F00000000003F0000000000 1F80000000001F80000002000FC00000020007C00000020007E00000020003E00000030001F000 00030000F800000700007C00000700003E00000780001F00000F80000F80000F800003E0001780 0001F80063C000007F01C1C000001FFF80C0000001FE0040> 42 43 -2 1 43] 67 @pc restore cmex10.300 @sf 1385 2570 p m (1) s 1385 2643 p m (C) s 1385 2670 p m (A) s cmmi10.329 @sf 1429 2666 p m (:) s cmr10.329 @sf 996 2850 p m (10) s @eop 9 @bop0 cmex10.300 @sf [<00200040008001000300060006000C000C00180018003800300030007000700070006000E000E0 00E000E000E000E000E000E000E000E000E000E000E00060007000700070003000300038001800 18000C000C000600060003000100008000400020> 11 49 -6 47 19] 0 @dc [<800040002000100018000C000C00060006000300030003800180018001C001C001C000C000E000 E000E000E000E000E000E000E000E000E000E000E000E000C001C001C001C00180018003800300 0300060006000C000C0018001000200040008000> 11 49 -1 47 19] 1 @dc cmbx12.300 @sf [ 21 32 -3 0 28] 50 @dc [<80FF80C7FFE0FF00F8FC003CF0003CE0001EE0001EC0001FC0001FC0001F00003F00003F00007F 0003FF003FFE03FFFE0FFFFC1FFFF83FFFF07FFFC07FFF00FFE000FF0000FC0000FC000CF8000C F8000C78001C78001C7C003C3C007C1F03FC07FF8C01FC04> 24 34 -3 0 31] 83 @dc [ 11 36 -2 0 16] 105 @dc [ 26 22 -2 0 31] 110 @dc [<01FF000FFFE03F01F87C007C78003CF0001EF0001EF0001EF0001E70003E3800FC1FFFFC0FFFF8 1FFFF03FFF8038000030000030000013FC001FFF001F0F803E07C03C03C07C03E07C03E07C03E0 7C03E07C03E03C03C03E07CF1F0F8F0FFF7F03FC1E> 24 33 -2 11 28] 103 @dc [ 19 22 -2 0 23] 114 @dc 9 @bop1 cmmi10.329 @sf 74 67 p m (\025) s cmr8.300 @sf 101 74 p m (1) s cmr10.329 @sf 136 67 p m (=) 16 (1) ks cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 11 r m (\017) s cmr10.329 @sf m (,) s cmmi10.329 @sf 18 r m (\025) s cmr8.300 @sf 343 74 p m (2) s cmr10.329 @sf 378 67 p m (=) 15 (1,) ks cmmi10.329 @sf 17 r m (\025) s cmr8.300 @sf 508 74 p m (1) s cmr10.329 @sf 543 67 p m (=) 16 (1) 11 (+) ks cmmi10.329 @sf 11 r m (\017) s cmr10.329 @sf m (,) 18 (and) ks cmmi10.329 @sf 17 r m (\017) s cmr10.329 @sf 16 r m (=) 16 (10) ks cmsy8.300 @sf 944 52 p m (\000) s cmr8.300 @sf m (6) s cmr10.329 @sf 990 67 p m (.) 25 (A) 17 (p) 1 (erturbation) 18 (ev) 0 (en) 16 (as) 17 (small) 17 (as) 17 (10) ks cmsy8.300 @sf 1735 52 p m (\000) s cmr8.300 @sf m (12) s cmr10.329 @sf 1816 67 p m (in) 18 (\(3,1\)) ks 74 150 p m (giv) 0 (es) 12 (three) 13 (eigen) 0 (v) -2 (alues) 12 (of) 12 (the) 13 (form) 12 (1) 5 (+) ks cmmi10.329 @sf 5 r m (O) s cmex10.300 @sf 871 116 p m (\000) s cmr10.329 @sf 890 150 p m (10) s cmsy8.300 @sf 936 135 p m (\000) s cmr8.300 @sf m (4) s cmex10.300 @sf 982 116 p m (\001) s cmr10.329 @sf 1001 150 p m (.) 19 (This) 13 (problem) 14 (is) 13 (discussed) 14 (in) 14 (considerable) 14 (detail) ks 74 233 p m (in) 17 ([) 0 (10) ks 186 233 p m (,) 15 (12) ks 260 233 p m (,) 15 (13) ks 334 233 p m (].) 21 (Clearly) -3 (,) 16 (deciding) 18 (whic) 0 (h) 15 (eigen) 0 (v) -2 (alues) 16 (should) 17 (b) 1 (e) 16 (group) 1 (ed) 17 (together) 15 (cannot) 15 (b) 1 (e) 17 (done) ks 74 316 p m (on) 15 (the) 16 (sup) 1 (er\014cial) 17 (basis) 15 (of) 15 (\\lo) 1 (oking) 16 (at) 15 (the) 15 (separations.") ks 124 441 p m (The) 22 (remark) -2 (able) 21 (fact) 22 (is) 22 (that) 21 (in) 23 (the) 21 (single) 24 (past) 21 (single) 23 (case,) 23 (the) 22 (cos) ks cmmi10.329 @sf 1504 441 p m (\022) s cmr10.329 @sf 24 r m (and) 22 (sin) ks cmmi10.329 @sf 1707 441 p m (\022) s cmr10.329 @sf 23 r m (are) 22 (alw) 0 (a) -1 (ys) ks 74 524 p m (giv) 0 (en) 16 (with) 16 (v) 0 (ery) 15 (lo) 0 (w) 15 (relativ) 0 (e) 15 (errors) 16 (on) 16 (a) 15 (computer) 16 (with) 17 (correct) 15 (rounding) 17 (or) 16 (c) 0 (hopping.) 22 (On) 17 (suc) 0 (h) ks 74 607 p m (computers,) s cmmi10.329 @sf 20 r m (\026) s cmsy10.329 @sf 13 r m (\000) s cmmi10.329 @sf 13 r m (\025) s cmr10.329 @sf 19 r m (is) 19 (alw) 0 (a) 0 (ys) 17 (computed) 19 (without) 20 (rounding) 20 (errors) 18 (ev) 0 (en) 18 (when) 20 (sev) 0 (ere) 18 (cancellation) ks 74 690 p m (tak) 0 (es) 14 (place.) 20 (Th) 0 (us,) 14 (if) ks cmex10.300 @sf 786 701 p m ( ) s cmmi10.329 @sf 839 746 p m (:) s cmr10.329 @sf m (832567) s cmmi10.329 @sf 1030 746 p m (:) s cmr10.329 @sf m (912863) s 902 801 p m (0) s cmmi10.329 @sf 1030 801 p m (:) s cmr10.329 @sf m (832569) s cmex10.300 @sf 1200 701 p m (!) s cmmi10.329 @sf 1240 773 p m (;) s cmr10.329 @sf 74 889 p m (w) 0 (e) 18 (ha) 0 (v) 0 (e) 17 (on) 20 (a) 19 (six-digit) 21 (computer) ks cmmi10.329 @sf 19 r m (\026) s cmsy10.329 @sf 13 r m (\000) s cmmi10.329 @sf 14 r m (\025) s cmr10.329 @sf 19 r m (=) s cmmi10.329 @sf 20 r m (:) s cmr10.329 @sf m (000002,) 18 (and) 19 (this) 20 (has) 19 (no) 20 (error.) 31 (\(This) 20 (will) 21 (b) 1 (e) 20 (true) ks 74 972 p m (ev) 0 (en) 16 (when,) 17 (e.g.,) ks cmmi10.329 @sf 16 r m (\025) s cmr10.329 @sf 14 r m (=) s cmmi10.329 @sf 15 r m (:) s cmr10.329 @sf m (999999) 14 (and) ks cmmi10.329 @sf 17 r m (\026) s cmr10.329 @sf 15 r m (=) 15 (10) ks cmr8.300 @sf 894 957 p m (1) s cmr10.329 @sf 914 972 p m (\() s cmmi10.329 @sf m (:) s cmr10.329 @sf m (100001) -1 (\),) 14 (that) 16 (is,) 17 (when) 17 (close) ks cmmi10.329 @sf 17 r m (\025) s cmr10.329 @sf 16 r m (and) s cmmi10.329 @sf 17 r m (\026) s cmr10.329 @sf 17 r m (ha) 0 (v) -1 (e) 15 (di\013eren) 0 (t) ks 74 1055 p m (exp) 1 (onen) 0 (ts.\)) 19 (Six-\014gure) 17 (\015oating-p) 1 (oin) 0 (t) 14 (computation) 16 (using) 15 (\(3\)) 15 (giv) 0 (es) ks 590 1184 p m (cos) s cmmi10.329 @sf 659 1184 p m (\022) s cmr10.329 @sf 14 r m (=) 13 (10) ks cmr8.300 @sf 788 1166 p m (1) s cmr10.329 @sf 807 1184 p m (\() s cmmi10.329 @sf m (:) s cmr10.329 @sf m (100000\)) s cmmi10.329 @sf m (;) s cmr10.329 @sf 7 r m (sin) s cmmi10.329 @sf 1075 1184 p m (\022) s cmr10.329 @sf 14 r m (=) 13 (10) ks cmsy8.300 @sf 1204 1166 p m (\000) s cmr8.300 @sf m (5) s cmr10.329 @sf 1251 1184 p m (\() s cmmi10.329 @sf m (:) s cmr10.329 @sf m (219091\)) s cmmi10.329 @sf -1 r m (;) s cmr10.329 @sf 74 1312 p m (and) 13 (b) 1 (oth) 13 (of) 12 (these) 13 (ha) 0 (v) -1 (e) 11 (relativ) 0 (e) 12 (errors) 12 (on) 13 (the) 13 (order) 12 (of) 12 (mac) 0 (hine) 13 (precision) 14 (\(10) ks cmsy8.300 @sf 1545 1297 p m (\000) s cmr8.300 @sf m (12) s cmr10.329 @sf 1609 1312 p m (\)) 12 (in) 14 (spite) 13 (of) 12 (sev) 0 (ere) ks 74 1395 p m (cancellation) 18 (ha) 0 (ving) 16 (tak) 0 (en) 15 (place.) 24 (Hence,) 18 (if) 17 (w) 0 (e) 15 (actually) 17 (do) 16 (the) 17 (computation) 16 (of) 16 (the) 16 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (2) 16 (matrix) ks 74 1478 p m (\(in) 14 (practice) 14 (w) 0 (e) 12 (w) 0 (ould) 12 (not,) 13 (w) 0 (e) 12 (could) 14 (merely) 14 (insert) ks cmmi10.329 @sf 14 r m (\026) s cmr10.329 @sf m (,) s cmmi10.329 @sf 13 r m (\025) s cmr10.329 @sf m (,) 13 (and) ks cmmi10.329 @sf 14 r m (\013) s cmr10.329 @sf 13 r m (in) 14 (the) 13 (appropriate) 14 (places\),) 13 (w) 0 (e) 12 (\014nd) ks 74 1561 p m (that) 16 (the) 18 (coupled) 18 (\(1,1\),) 15 (\(1,2\),) 16 (and) 17 (\(2,2\)) 16 (elemen) 0 (ts) 16 (are) 17 (correct) 16 (to) 17 (w) 0 (orking) 15 (accuracy) 18 (and) 17 (that) 16 (the) ks 74 1644 p m (\(2,1\)) 13 (elemen) 0 (t) 15 (is) 15 (w) 0 (ell) 15 (b) 1 (elo) 0 (w) 14 (the) 15 (negligibl) 1 (e) 16 (lev) 0 (el.) 20 (This) 16 (is) 15 (comforting) 14 (b) 1 (ecause) 16 (w) 0 (e) 13 (shall) 16 (b) 1 (e) 16 (applying) ks 74 1727 p m (the) 15 (transformation) 14 (to) 15 (the) 15 (rest) 15 (of) 15 (the) 15 (matrix.) ks 124 1852 p m (This) 17 (is) 17 (an) 16 (impressiv) 0 (ely) 17 (go) 1 (o) 1 (d) 16 (result.) 24 (In) 17 (man) 0 (y) 15 (situations,) 17 (not) 16 (dissimilar) 18 (from) 16 (this,) 16 (one) 17 (w) 0 (ould) ks 74 1935 p m (ha) 0 (v) 0 (e) 14 (to) 15 (b) 1 (e) 17 (satis\014ed) 17 (with) 16 (a) 16 (matrix) 16 (whic) 0 (h) 16 (is) 17 (exactly) 16 (similar) 17 (to) 16 (a) ks cmmi10.329 @sf 16 r m (T) s cmr10.329 @sf 22 r m (with) 16 (a) 16 (p) 1 (erturbation) 17 (of) 15 (order) ks 74 2018 p m (10) s cmsy8.300 @sf 120 2003 p m (\000) s cmr8.300 @sf m (6) s cmr10.329 @sf 180 2018 p m (in) 15 (its) 13 (elemen) 0 (ts) 14 (and) 13 (suc) 0 (h) 13 (a) 14 (matrix) 13 (could) 14 (ha) 0 (v) 0 (e) 12 (eigen) 0 (v) -2 (alues) 13 (agreeing) 14 (with) ks cmmi10.329 @sf 14 r m (\025) s cmr10.329 @sf 13 r m (and) s cmmi10.329 @sf 14 r m (\026) s cmr10.329 @sf 13 r m (in) 15 (only) 14 (the) ks 74 2101 p m (\014rst) 15 (three) 15 (\014gures,) 15 (a) 15 (disaster) 15 (from) 15 (the) 15 (p) 1 (oin) 0 (t) 15 (of) 15 (view) 15 (of) 15 (e\013ecting) 16 (an) 15 (in) 0 (terc) 0 (hange) 13 (of) ks cmmi10.329 @sf 15 r m (\025) s cmr10.329 @sf 15 r m (and) s cmmi10.329 @sf 15 r m (\026) s cmr10.329 @sf m (!) s cmbx12.300 @sf 74 2264 p m (3.2) s 202 2264 p m (Single) 17 (past) 19 (double) 19 (or) 18 (double) 18 (past) 19 (single) ks cmr10.329 @sf 74 2418 p m (When) 21 (w) 0 (e) 19 (turn) 21 (to) 20 (the) 21 (other) 20 (three) 21 (cases,) 21 (the) 21 (situation) 21 (is) 21 (not) 20 (so) 20 (simple.) 38 (Let) 21 (us) 20 (consider) 22 (the) ks 74 2501 p m (algorithm) 15 (for) 15 (mo) 0 (ving) 14 (a) 15 (single) 16 (past) 15 (a) 15 (double.) 21 (If) 16 (w) 0 (e) 14 (denote) 15 (the) 15 (eigen) 0 (v) 0 (ector) 13 (in) 17 (\(5\)) 14 (b) 0 (y) ks cmmi10.329 @sf 861 2630 p m (x) s cmr10.329 @sf 12 r m (=) 13 (\() ks cmmi10.329 @sf m (x) s cmr8.300 @sf 991 2637 p m (1) s cmmi10.329 @sf 1011 2630 p m (;) 8 (x) ks cmr8.300 @sf 1058 2637 p m (2) s cmmi10.329 @sf 1077 2630 p m (;) s cmr10.329 @sf 8 r m (1\)) s cmmi8.300 @sf 1139 2612 p m (T) s cmmi10.329 @sf 1165 2630 p m (;) s cmr10.329 @sf 1008 2850 p m (9) s @eop 8 @bop0 cmbx12.300 @sf [<7FFFE07FFFE001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F800FFF800FFF80001F800007800001800> 19 32 -4 0 28] 49 @dc cmmi10.329 @sf [<1801E03803101C03081C07081C07041C03840E03820E01C00E01C00E01C00700E00700E00700E0 8700E043807043807043C07023A0602318C01E0780> 23 20 -2 0 27] 110 @dc cmr10.329 @sf [ 16 1 -3 -26 23] 22 @dc cmmi8.300 @sf [ 13 33 -2 8 18] 61 @dc 8 @bop1 cmbx12.300 @sf 74 67 p m (3.1) s 202 67 p m (Single) 17 (past) 19 (single) ks cmr10.329 @sf 74 221 p m (When) 17 (taking) 15 (a) 16 (single) 17 (past) 16 (a) 16 (single,) 17 (the) 16 (form) 0 (ulae) 14 (giving) 17 (the) 16 (comp) 1 (onen) 0 (ts) 15 (of) 16 (the) 16 (v) 0 (ectors) 14 (are) 16 (of) 15 (a) ks 74 304 p m (particularly) 15 (simple) 14 (form.) 18 (F) -3 (or) 12 (consistency) 14 (with) 13 (the) 14 (other) 12 (three) 13 (cases,) 13 (the) 14 (eigen) 0 (v) 0 (ector) 11 (in) 14 (equation) ks 74 387 p m (\(1\)) 14 (should) 17 (p) 1 (erhaps) 15 (ha) 0 (v) 0 (e) 13 (b) 1 (een) 17 (expressed) 15 (in) 16 (the) 16 (form) ks 850 511 p m (\() s cmmi10.329 @sf 868 511 p m (\013) 0 (=) ks cmr10.329 @sf 8 r m (\() s cmmi10.329 @sf 945 511 p m (\026) s cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmr10.329 @sf m (\)) s cmmi10.329 @sf 1080 511 p m (;) s cmr10.329 @sf 8 r m (1) 0 (\)) ks cmmi8.300 @sf 1140 491 p m (T) s cmmi10.329 @sf 1175 511 p m (:) s cmr10.329 @sf 74 636 p m (This) 14 (emphasizes) 15 (the) 14 (fact) 13 (that) 13 (when) ks cmmi10.329 @sf 14 r m (\026) s cmsy10.329 @sf 7 r m (\000) s cmmi10.329 @sf 7 r m (\025) s cmr10.329 @sf 13 r m (is) 14 (v) 0 (ery) 12 (small) 15 (compared) 13 (with) ks cmmi10.329 @sf 14 r m (\013) s cmr10.329 @sf m (,) 14 (the) 14 (\014rst) 13 (comp) 1 (onen) 0 (t) 13 (of) ks 74 719 p m (the) 15 (eigen) 0 (v) 0 (ector) 14 (is) 15 (v) 0 (ery) 14 (large|i.e.,) 15 (in) 16 (the) 16 (normalized) 16 (form,) 14 (the) 15 (second) 16 (comp) 1 (onen) 0 (t) 14 (is) 16 (v) 0 (ery) 14 (small.) ks 74 802 p m (Ho) 0 (w) -1 (ev) -1 (er,) 17 (in) 18 (this) 19 (case) ks cmmi10.329 @sf 18 r m (\025) s cmr10.329 @sf 17 r m (and) s cmmi10.329 @sf 19 r m (\026) s cmr10.329 @sf 18 r m (should) 19 (almost) 17 (certainly) 20 (ha) 0 (v) -1 (e) 16 (b) 1 (een) 19 (asso) 1 (ciated) 19 (together,) 17 (and) 18 (w) 0 (e) ks 74 885 p m (should) 16 (not) 15 (b) 1 (e) 16 (trying) 15 (to) 15 (in) 0 (terc) 0 (hange) 13 (them!) ks 124 1009 p m (This) 20 (remark) 20 (has) 19 (more) 20 (force) 19 (than) 20 (migh) 0 (t) 19 (b) 1 (e) 20 (imagined) 22 (when) 20 (the) 20 (full) ks cmmi10.329 @sf 21 r m (n) s cmsy10.329 @sf 14 r m (\002) s cmmi10.329 @sf 13 r m (n) s cmr10.329 @sf 20 r m (quasi-triangular) s 74 1093 p m (matrix) 14 (has) 15 (b) 1 (een) 16 (pro) 1 (duced) 16 (from) 13 (a) 15 (general) 15 (matrix) ks cmmi10.329 @sf 14 r m (A) s cmr10.329 @sf 15 r m (b) 0 (y) 14 (an) 14 (orthogonal) 14 (similarit) 0 (y) 15 (transformation.) ks 74 1176 p m (In) 18 (this) 18 (case) 17 (the) 17 (elemen) 0 (ts) 17 (b) 1 (elo) 0 (w) 17 (the) 17 (diagonal) 18 (elemen) 0 (ts) 17 (are) 17 (in) 18 (no) 18 (sense) 17 (true) 17 (zeros.) 26 (They) 18 (are) 17 (at) ks 74 1259 p m (b) 1 (est) 16 (negligible) 18 (to) 14 (w) 0 (orking) 14 (accuracy) -3 (.) ks 124 1383 p m (As) 15 (an) 15 (example,) 16 (consider) 16 (the) 15 (matrix) ks cmex10.300 @sf 560 1435 p m ( ) s cmr10.329 @sf 614 1480 p m (1) s cmsy10.329 @sf 9 r m (\000) s cmmi10.329 @sf 11 r m (\017) s cmr10.329 @sf 789 1480 p m (1) s 651 1536 p m (0) s 752 1536 p m (1) 10 (+) ks cmmi10.329 @sf 10 r m (\017) s cmex10.300 @sf 869 1435 p m (!) s cmmi10.329 @sf 910 1507 p m (;) 20 (\025) ks cmr8.300 @sf 970 1514 p m (1) s cmr10.329 @sf 1001 1507 p m (=) 13 (1) ks cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 11 r m (\017;) 20 (\025) ks cmr8.300 @sf 1206 1514 p m (2) s cmr10.329 @sf 1238 1507 p m (=) 13 (1) 10 (+) ks cmmi10.329 @sf 10 r m (\017:) s cmr10.329 @sf 1906 1507 p m (\(8\)) s 74 1642 p m (A) 14 (p) 1 (erturbation) ks cmsy10.329 @sf 14 r m (\000) s cmmi10.329 @sf m (\017) s cmr8.300 @sf 440 1627 p m (2) s cmr10.329 @sf 475 1642 p m (in) 14 (the) 14 (\(2,1\)) 12 (elemen) 0 (t) 14 (giv) 0 (es) 13 (mo) 1 (di\014ed) 15 (eigen) 0 (v) -2 (alues) ks 1406 1631 p m (\026) s cmmi10.329 @sf 1405 1643 p m (\025) s cmr8.300 @sf 1432 1650 p m (1) s cmr10.329 @sf 1463 1642 p m (=) s 1513 1631 p m (\026) s cmmi10.329 @sf 1511 1643 p m (\025) s cmr8.300 @sf 1538 1650 p m (2) s cmr10.329 @sf 1570 1642 p m (=) 13 (1,) 14 (and) 14 (the) 13 (matrix) ks 74 1725 p m (is) 17 (defectiv) 0 (e.) 22 (Supp) 1 (ose) 18 (w) 0 (e) 14 (are) 16 (w) 0 (orking) 15 (on) 16 (a) 16 (10-digit) 16 (computer) 16 (and) ks cmmi10.329 @sf 17 r m (\017) s cmr10.329 @sf 14 r m (=) 15 (10) ks cmsy8.300 @sf 1517 1710 p m (\000) s cmr8.300 @sf m (6) s cmr10.329 @sf 1563 1725 p m (.) 23 (W) -3 (e) 15 (ma) 0 (y) 15 (not) 15 (think) ks 74 1808 p m (of) 14 (1) ks cmsy10.329 @sf 8 r m (\006) s cmr10.329 @sf 9 r m (10) s cmsy8.300 @sf 246 1793 p m (\000) s cmr8.300 @sf m (6) s cmr10.329 @sf 307 1808 p m (as) 14 (unduly) 16 (close,) 14 (but) 15 (a) 14 (p) 1 (erturbation) 15 (of) ks cmsy10.329 @sf 14 r m (\000) s cmr10.329 @sf m (10) s cmsy8.300 @sf 1152 1793 p m (\000) s cmr8.300 @sf m (12) s cmr10.329 @sf 1231 1808 p m (giv) 0 (es) 13 (coinciden) 0 (t) 15 (eigen) 0 (v) -2 (alues,) 14 (and) 15 (this) ks 74 1891 p m (p) 1 (erturbation) 19 (is) 19 (w) 0 (ell) 18 (b) 1 (elo) 0 (w) 18 (the) 18 (negligible) 21 (lev) 0 (el.) 29 (If) 19 (w) 0 (e) 17 (think) 18 (in) 20 (terms) 17 (of) 18 (p) 1 (erturbations) 19 (of) 17 (order) ks 74 1974 p m (10) s cmsy8.300 @sf 120 1959 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 200 1974 p m (\(i.e.,) 16 (computer) 16 (noise) 16 (lev) 0 (el\),) 16 (all) 17 (w) 0 (e) 15 (can) 16 (sa) 0 (y) 14 (is) 16 (that) 16 (the) 16 (true) 16 (eigen) 0 (v) -2 (alues) 15 (are) 16 (\(roughly\)) 16 (in) 16 (a) ks 74 2057 p m (disk) 13 (cen) 0 (tered) 11 (on) ks cmmi10.329 @sf 12 r m (\025) s cmr10.329 @sf 12 r m (=) 13 (1) 11 (and) 12 (of) 11 (radius) 13 (10) ks cmsy8.300 @sf 837 2042 p m (\000) s cmr8.300 @sf m (5) s cmr10.329 @sf 883 2057 p m (.) 19 (Th) 0 (us) 11 (a) 11 (p) 1 (erturbation) 12 (+10) ks cmsy8.300 @sf 1405 2042 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 1482 2057 p m (in) 12 (\(2,1\)) 10 (giv) 0 (es) 11 (eigen) 0 (v) -2 (alues) ks 74 2140 p m (1) s cmsy10.329 @sf 4 r m (\006) s cmmi10.329 @sf 4 r m (i) s cmr10.329 @sf m (\() s cmmi10.329 @sf m (:) s cmr10.329 @sf m (99\)) s cmr8.300 @sf 250 2125 p m (1) s cmmi8.300 @sf m (=) s cmr8.300 @sf m (2) s cmr10.329 @sf 303 2140 p m (10) s cmsy8.300 @sf 349 2125 p m (\000) s cmr8.300 @sf m (5) s cmr10.329 @sf 396 2140 p m (,) 12 (while) 14 (a) 11 (p) 1 (erturbation) 13 (of) ks cmsy10.329 @sf 11 r m (\000) s cmr10.329 @sf m (10) s cmsy8.300 @sf 964 2125 p m (\000) s cmr8.300 @sf m (10) s cmr10.329 @sf 1041 2140 p m (giv) 0 (es) 11 (eigen) 0 (v) -2 (alues) 12 (1) 4 (+) 4 (\(1) ks cmmi10.329 @sf m (:) s cmr10.329 @sf m (01) -1 (\)) ks cmr8.300 @sf 1564 2125 p m (1) s cmmi8.300 @sf m (=) s cmr8.300 @sf -1 r m (2) s cmr10.329 @sf 1617 2140 p m (10) s cmsy8.300 @sf 1663 2125 p m (\000) s cmr8.300 @sf m (5) s cmr10.329 @sf 1710 2140 p m (.) 18 (T) -3 (o) 11 (attempt) ks 74 2223 p m (to) 15 (distinguish) 17 (b) 1 (et) 0 (w) 0 (een) 14 (1) 10 (+) 11 (10) ks cmsy8.300 @sf 661 2208 p m (\000) s cmr8.300 @sf m (6) s cmr10.329 @sf 723 2223 p m (and) 15 (1) ks cmsy10.329 @sf 10 r m (\000) s cmr10.329 @sf 11 r m (10) s cmsy8.300 @sf 936 2208 p m (\000) s cmr8.300 @sf m (6) s cmr10.329 @sf 982 2223 p m (,) 16 (and) 15 (to) 15 (in) 0 (terc) 0 (hange) 14 (them,) 15 (mak) 0 (es) 14 (no) 15 (sense.) 21 (They) ks 74 2306 p m (ha) 0 (v) 0 (e) 13 (no) 15 (separate) 14 (iden) 0 (tit) 0 (y) -3 (,) 12 (and) 16 (di\013eren) 0 (t) 14 (rounding) 16 (errors) 14 (in) 15 (the) 15 (triangularization) 16 (program) 14 (giving) ks cmmi10.329 @sf 74 2389 p m (T) s cmr10.329 @sf 21 r m (migh) 0 (t) 14 (w) 0 (ell) 15 (ha) 0 (v) 0 (e) 13 (led) 16 (to) 15 (complex) 16 (eigen) 0 (v) -2 (alues) 15 (and) 15 (ha) 0 (v) 0 (e) 13 (a) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2) 15 (blo) 1 (c) 0 (k) 15 (rather) 15 (than) 15 (that) 14 (in) 16 (\(8\).) ks 124 2514 p m (F) -3 (or) 14 (sev) 0 (eral) 14 (mo) 1 (derately) 15 (close) 16 (eigen) 0 (v) -2 (alues,) 15 (the) 15 (remark) 15 (has) 15 (ev) 0 (en) 14 (greater) 15 (force.) 19 (Th) 0 (us,) 14 (if) ks cmmi10.329 @sf 756 2666 p m (T) s cmr10.329 @sf 18 r m (=) s cmex10.300 @sf 849 2570 p m (0) s 849 2643 p m (B) s 849 2670 p m (@) s cmr10.329 @sf 906 2610 p m (1) s cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 10 r m (\017) s cmr10.329 @sf 1044 2610 p m (1) s 1146 2610 p m (0) s 1044 2666 p m (1) s 1146 2666 p m (1) s 1109 2723 p m (1) 9 (+) ks cmmi10.329 @sf 11 r m (\017) s cmex10.300 @sf 1226 2570 p m (1) s 1226 2643 p m (C) s 1226 2670 p m (A) s cmmi10.329 @sf 1270 2666 p m (;) s cmr10.329 @sf 1008 2850 p m (8) s @eop 7 @bop0 cmmi10.329 @sf [<0FC01838300470027001F000F000F000F000F0007800780078003C041C0E1E0F0E0F070101C200 7C> 16 20 -2 0 20] 99 @dc cmex10.300 @sf [<800F80601FE03078301FE01807C004> 22 5 0 -25 23] 101 @dc 7 @bop1 cmr10.329 @sf 124 67 p m (The) 17 (same) 17 (general) 18 (principle) 20 (ma) 0 (y) 15 (b) 1 (e) 18 (used.) 26 (W) -3 (e) 16 (compute) 18 (generators) 16 (of) 16 (the) 18 (in) 0 (v) -2 (arian) 0 (t) 14 (subspace) ks 74 150 p m (corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 15 r m (C) s cmr10.329 @sf 18 r m (in) 16 (the) 15 (form) ks 836 287 p m (\() s cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) 11 (=) ks cmex10.300 @sf 1002 166 p m (0) s 1002 239 p m (B) s 1002 264 p m (B) s 1002 289 p m (B) s 1002 316 p m (@) s cmsy10.329 @sf 1059 203 p m (\003) s 1123 203 p m (\003) s 1059 259 p m (\003) s 1123 259 p m (\003) s cmr10.329 @sf 1059 316 p m (1) s 1123 316 p m (0) s 1059 372 p m (0) s 1123 372 p m (1) s cmex10.300 @sf 1166 166 p m (1) s 1166 239 p m (C) s 1166 264 p m (C) s 1166 289 p m (C) s 1166 316 p m (A) s cmr10.329 @sf 74 452 p m (b) 0 (y) 14 (solving) ks cmmi10.329 @sf 590 580 p m (T) s cmr8.300 @sf 617 587 p m (4) s cmr10.329 @sf 636 580 p m (\() s cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) 12 (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) s cmmi10.329 @sf m (C) s cmr10.329 @sf 13 r m (=) 13 (\() ks cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) s cmex10.300 @sf 1119 508 p m ( ) s cmmi10.329 @sf 1171 553 p m (c) s cmr8.300 @sf 1191 560 p m (1) s cmmi10.329 @sf 1252 553 p m (c) s cmr8.300 @sf 1272 560 p m (2) s cmmi10.329 @sf 1171 609 p m (c) s cmr8.300 @sf 1191 616 p m (3) s cmmi10.329 @sf 1252 609 p m (c) s cmr8.300 @sf 1272 616 p m (4) s cmex10.300 @sf 1312 508 p m (!) s cmmi10.329 @sf 1352 580 p m (:) s cmr10.329 @sf 1906 580 p m (\(7\)) s 74 715 p m (This) 19 (giv) 0 (es) 18 (us) 18 (four) 18 (equations) 19 (for) 17 (the) 19 (four) 18 (top) 18 (comp) 1 (onen) 0 (ts) 17 (in) 19 (\() ks cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\).) 28 (If) 19 (w) 0 (e) 17 (no) 0 (w) 17 (determine) 19 (a) ks cmmi10.329 @sf 18 r m (Q) s cmr10.329 @sf 74 798 p m (suc) 0 (h) 15 (that) ks cmmi10.329 @sf 706 927 p m (Q) s cmr10.329 @sf m (\() s cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) 11 (=) ks cmex10.300 @sf 908 805 p m (0) s 908 878 p m (B) s 908 903 p m (B) s 908 928 p m (B) s 908 955 p m (@) s cmsy10.329 @sf 965 842 p m (\003) s 1029 842 p m (\003) s cmr10.329 @sf 965 898 p m (0) s cmsy10.329 @sf 1029 898 p m (\003) s cmr10.329 @sf 965 955 p m (0) s 1029 955 p m (0) s 965 1011 p m (0) s 1029 1011 p m (0) s cmex10.300 @sf 1073 805 p m (1) s 1073 878 p m (C) s 1073 903 p m (C) s 1073 928 p m (C) s 1073 955 p m (A) s cmr10.329 @sf 1122 927 p m (=) s cmex10.300 @sf 1170 855 p m ( ) s cmmi10.329 @sf 1223 900 p m (R) s cmr10.329 @sf 1230 955 p m (0) s cmex10.300 @sf 1279 855 p m (!) s cmmi10.329 @sf 1320 927 p m (;) s cmr10.329 @sf 74 1098 p m (then) s cmmi10.329 @sf 17 r m (QT) s cmr8.300 @sf 242 1105 p m (4) s cmmi10.329 @sf 261 1098 p m (Q) s cmmi8.300 @sf 297 1083 p m (T) s cmr10.329 @sf 341 1098 p m (will) 17 (b) 1 (e) 17 (of) 16 (the) 16 (required) 18 (form.) 22 (Suc) 0 (h) 16 (a) ks cmmi10.329 @sf 16 r m (Q) s cmr10.329 @sf 16 r m (ma) 0 (y) 14 (b) 1 (e) 17 (determined) 18 (as) 16 (the) 16 (pro) 1 (duct) 17 (of) 15 (t) 0 (w) -1 (o) ks 74 1182 p m (Householder) 16 (matrices) 16 (or) 14 (four) 15 (Giv) 0 (ens) 15 (rotations.) ks 124 1306 p m (T) -3 (o) 14 (see) 15 (ho) 0 (w) ks cmex10.300 @sf 369 1294 p m (e) s cmmi10.329 @sf 359 1307 p m (C) s cmr10.329 @sf 410 1306 p m (is) 16 (related) 15 (to) ks cmmi10.329 @sf 15 r m (C) s cmr10.329 @sf 3 r m (,) 14 (w) 0 (e) 14 (observ) 0 (e) 14 (that) 15 (\(7\)) 14 (implies) 17 (that) ks cmmi10.329 @sf 752 1435 p m (QT) s cmr8.300 @sf 815 1442 p m (4) s cmmi10.329 @sf 834 1435 p m (Q) s cmmi8.300 @sf 870 1417 p m (T) s cmmi10.329 @sf 897 1435 p m (Q) s cmr10.329 @sf m (\() s cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) 11 (=) ks cmmi10.329 @sf 13 r m (Q) s cmr10.329 @sf m (\() s cmmi10.329 @sf m (x;) 8 (y) ks cmr10.329 @sf 2 r m (\)) s cmmi10.329 @sf m (C) 1 (;) ks cmr10.329 @sf 74 1563 p m (giving) s cmmi10.329 @sf 742 1646 p m (QT) s cmr8.300 @sf 805 1653 p m (4) s cmmi10.329 @sf 824 1646 p m (Q) s cmmi8.300 @sf 860 1628 p m (T) s cmex10.300 @sf 895 1574 p m ( ) s cmmi10.329 @sf 949 1619 p m (R) s cmr10.329 @sf 955 1675 p m (0) s cmex10.300 @sf 1005 1574 p m (!) s cmr10.329 @sf 1050 1646 p m (=) s cmex10.300 @sf 1098 1574 p m ( ) s cmmi10.329 @sf 1152 1619 p m (R) s cmr10.329 @sf 1158 1675 p m (0) s cmex10.300 @sf 1207 1574 p m (!) s cmmi10.329 @sf 1248 1646 p m (C) s cmr10.329 @sf 3 r m (;) s 74 1756 p m (that) 15 (is,) ks cmmi10.329 @sf 677 1839 p m (QT) s cmr8.300 @sf 740 1846 p m (4) s cmmi10.329 @sf 760 1839 p m (Q) s cmmi8.300 @sf 796 1821 p m (T) s cmex10.300 @sf 831 1767 p m ( ) s cmmi10.329 @sf 884 1812 p m (I) s cmr10.329 @sf 885 1868 p m (0) s cmex10.300 @sf 929 1767 p m (!) s cmr10.329 @sf 974 1839 p m (=) s cmex10.300 @sf 1022 1767 p m ( ) s cmmi10.329 @sf 1076 1812 p m (I) s cmr10.329 @sf 1076 1868 p m (0) s cmex10.300 @sf 1120 1767 p m (!) s cmr10.329 @sf 1160 1839 p m (\() s cmmi10.329 @sf m (R) 0 (C) 3 (R) ks cmsy8.300 @sf 1284 1821 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 1331 1839 p m (\)) s cmmi10.329 @sf m (:) s cmr10.329 @sf 74 1961 p m (This) 16 (last) 15 (equation) 16 (states) 14 (that) 14 (the) 16 (\014rst) 15 (t) 0 (w) -2 (o) 13 (columns) 16 (of) ks cmmi10.329 @sf 15 r m (QT) 6 (Q) ks cmmi8.300 @sf 1269 1946 p m (T) s cmr10.329 @sf 1312 1961 p m (are) s cmex10.300 @sf 879 2017 p m ( ) s cmmi10.329 @sf 933 2062 p m (R) 0 (C) 3 (R) ks cmsy8.300 @sf 1039 2047 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 998 2118 p m (0) s cmex10.300 @sf 1106 2017 p m (!) s cmmi10.329 @sf 1147 2089 p m (;) s cmr10.329 @sf 74 2233 p m (and) 16 (hence) ks cmex10.300 @sf 299 2221 p m (e) s cmmi10.329 @sf 289 2234 p m (C) s cmr10.329 @sf 337 2233 p m (=) s cmmi10.329 @sf 13 r m (R) 0 (C) 3 (R) ks cmsy8.300 @sf 491 2218 p m (\000) s cmr8.300 @sf m (1) s cmr10.329 @sf 538 2233 p m (.) 20 (W) -3 (e) 14 (shall) 16 (not,) 15 (of) 14 (course,) 15 (compute) ks cmex10.300 @sf 1247 2221 p m (e) s cmmi10.329 @sf 1237 2234 p m (C) s cmr10.329 @sf 1288 2233 p m (via) s cmmi10.329 @sf 15 r m (R) s cmr10.329 @sf m (!) s cmbx10.432 @sf 74 2418 p save [<00FF800007FFF0001FFFFC003F01FE007C007F007E007F80FF007FC0FF003FC0FF003FE0FF003F E07E003FE03C003FE000003FE000003FE000003FC000003FC000007F8000007F0000007E000001 FC0000FFF00000FFC0000007F0000001F8000001FC000000FE000000FF000000FF000F007F801F 807F803F807F803F807F803F807F803F80FF001F00FF000F81FE0007FFFC0003FFF000007F8000> 27 39 -3 0 34] 51 @pc restore 177 2418 p save [ 47 41 -3 0 54] 78 @pc restore 54 r m (ume) -1 (ri) -1 (cal) 20 (Considerations) ks cmr10.329 @sf 74 2587 p m (In) 14 (eac) 0 (h) 13 (of) 13 (the) 13 (four) 13 (cases) 14 (discussed) 15 (ab) 1 (o) 0 (v) -1 (e) 12 (w) 0 (e) 12 (determine) 15 (either) 14 (an) 13 (eigen) 0 (v) 0 (ector) 12 (or) 13 (t) 0 (w) -1 (o) 11 (indep) 1 (enden) 1 (t) ks 74 2671 p m (generators) 14 (of) 15 (an) 15 (in) 0 (v) -2 (arian) 0 (t) 13 (subspace.) ks 1008 2850 p m (7) s @eop 6 @bop0 cmmi10.329 @sf [ 31 31 -2 0 34] 66 @dc cmbx12.300 @sf [<00FFFE00FFFE0007E00007E00007E00007E00007E00007E00007E0FFFFFEFFFFFEE007E07007E0 3807E01807E00C07E00E07E00707E00307E00187E001C7E000E7E00077E00037E0001FE0001FE0 000FE00007E00003E00003E00001E00000E0> 23 32 -2 0 28] 52 @dc cmmi10.329 @sf [<1F0031C060E06070E038E03CE01CE01EE01EE01E700F700F700F700F380F380F3C0E3A0E1D0C1C F01C001C000E000E000E000E0007000700070007003F800780> 16 32 -2 0 20] 98 @dc 1.000000 /cmr12.300 @newfont cmr12.300 @sf [<01F000071C000C06001C07003803803803807803C07001C07001C07001C0F001E0F001E0F001E0 F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E07001C07001C0 7001C07001C03803803803801803000C0600071C0001F000> 19 34 -2 1 24] 48 @dc 6 @bop1 cmbx12.300 @sf 74 67 p m (2.3) s 202 67 p m (Double) 18 (past) 19 (single) ks cmr10.329 @sf 74 221 p m (When) 16 (a) 15 (pair) 16 (of) 16 (complex) 16 (conjugate) 15 (eigen) 0 (v) -2 (alues) 16 (is) 16 (included) 18 (in) 17 (the) 15 (selected) 17 (group,) 15 (the) 16 (asso) 1 (ciated) ks 74 304 p m (2) s cmsy10.329 @sf 13 r m (\002) s cmr10.329 @sf 14 r m (2) 20 (diagonal) 21 (blo) 1 (c) 0 (k) 20 (has) 20 (to) 20 (b) 1 (e) 21 (mo) 0 (v) -1 (ed) 19 (in) 0 (to) 19 (a) 20 (leading) 22 (p) 1 (osition) 21 (on) 21 (the) 20 (diagonal.) 36 (On) 21 (the) 20 (w) 0 (a) -1 (y) ks 74 387 p m (up) 18 (it) 17 (will,) 19 (in) 18 (general,) 18 (pass) 17 (b) 1 (oth) 17 (single) 19 (eigen) 0 (v) -2 (alues) 17 (and) 18 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (2) 17 (blo) 1 (c) 0 (ks) 17 (with) 18 (whic) 0 (h) 17 (it) 17 (is) 18 (not) 17 (to) 16 (b) 1 (e) ks 74 470 p m (asso) 1 (ciated.) 27 (W) -3 (e) 16 (consider) 18 (\014rst) 17 (taking) 17 (a) 17 (complex) 18 (pair) 18 (past) 17 (a) 17 (real) 18 (eigen) 0 (v) -2 (alue.) 26 (In) 18 (other) 17 (w) 0 (ords,) 15 (in) ks 74 553 p m (terms) 15 (of) 15 (the) 15 (relev) -2 (an) 0 (t) 14 (3) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (3) 15 (matrix,) 14 (w) 0 (e) 14 (require) 16 (an) 15 (orthogonal) ks cmmi10.329 @sf 15 r m (Q) s cmr10.329 @sf 15 r m (suc) 0 (h) 14 (that) ks cmmi10.329 @sf 460 710 p m (QT) 6 (Q) ks cmmi8.300 @sf 565 692 p m (T) s cmr10.329 @sf 605 710 p m (=) s cmex10.300 @sf 653 613 p m (0) s 653 687 p m (B) s 653 714 p m (@) s cmmi10.329 @sf 747 654 p m (\025) s cmr8.300 @sf 774 661 p m (1) s 826 671 p 2 57 ru cmmi10.329 @sf 852 654 p m (x) 45 (x) ks 713 673 p 262 2 ru cmr10.329 @sf 758 712 p m (0) s 758 767 p m (0) s 826 784 p 2 113 ru 876 739 p save [ 42 41 -2 0 45] 66 @pc restore cmex10.300 @sf 997 613 p m (1) s 997 687 p m (C) s 997 714 p m (A) s cmmi10.329 @sf 1041 710 p m (Q) s cmmi8.300 @sf 1077 692 p m (T) s cmr10.329 @sf 1117 710 p m (=) s cmex10.300 @sf 1165 613 p m (0) s 1165 687 p m (B) s 1165 714 p m (@) s 1280 683 p save [<000FF8000000003FFF00000000FC03C0000003F00060000007C0001800000F80000C00001F0000 0600001F00000300003E00000100003E00000080007C00000040007C00000040007C0000002000 7C00000020007C0000002000FC0000000000FC0000000000FC00000000007C00000000007E0000 0000007E00000000007E00000000007E00000000003F00000000003F00000000003F0000000000 1F80000000001F80000002000FC00000020007C00000020007E00000020003E00000030001F000 00030000F800000700007C00000700003E00000780001F00000F80000F80000F800003E0001780 0001F80063C000007F01C1C000001FFF80C0000001FE0040> 42 43 -2 1 43] 67 @pc restore 1380 728 p 2 113 ru cmmi10.329 @sf 1427 656 p m (x) s 1427 710 p m (x) s 1225 729 p 274 2 ru cmr10.329 @sf 1250 767 p m (0) 60 (0) ks 1380 784 p 2 57 ru cmmi10.329 @sf 1424 767 p m (\025) s cmr8.300 @sf 1451 774 p m (1) s cmex10.300 @sf 1522 613 p m (1) s 1522 687 p m (C) s 1522 714 p m (A) s cmmi10.329 @sf 1566 710 p m (:) s cmr10.329 @sf 74 868 p m (Here) 13 (the) 12 (selected) 13 (eigen) 0 (v) -2 (alues) 12 (are) 12 (those) 12 (of) ks cmmi10.329 @sf 12 r m (B) s cmr10.329 @sf 2 r m (,) 13 (a) 12 (complex) 12 (conjugate) 12 (pair.) 19 (The) 13 (eigen) 0 (v) -2 (alues) 12 (of) ks cmmi10.329 @sf 12 r m (C) s cmr10.329 @sf 14 r m (will) s 74 951 p m (b) 1 (e) 14 (the) 14 (same) 13 (pair,) 14 (but) 14 (in) 14 (general) ks cmmi10.329 @sf 14 r m (C) s cmr10.329 @sf 16 r m (and) s cmmi10.329 @sf 14 r m (B) s cmr10.329 @sf 16 r m (will) 16 (b) 1 (e) 14 (di\013eren) 0 (t) 12 (matrices) 14 (and) 14 (will) 15 (not) 13 (b) 1 (e) 14 (orthogonally) ks 74 1034 p m (similar.) 24 (If) 16 (w) 0 (e) 14 (think) 17 (in) 17 (terms) 15 (of) 16 (mo) 0 (ving) ks cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 899 1041 p m (1) s cmr10.329 @sf 934 1034 p m (to) 15 (the) 16 (b) 1 (ottom) 15 (w) 0 (e) 15 (ma) 0 (y) 14 (use) 17 (m) 0 (uc) -1 (h) 15 (the) 16 (same) 15 (principl) 1 (e) ks 74 1117 p m (as) 15 (b) 1 (efore) 15 (but) 16 (no) 0 (w) 13 (w) 0 (e) 14 (w) 0 (ork) 14 (in) 16 (terms) 14 (of) 15 (a) 15 (left-hand) 16 (eigen) 0 (v) 0 (ector.) 18 (If) ks cmmi10.329 @sf 678 1245 p m (y) s cmmi8.300 @sf 702 1227 p m (T) s cmmi10.329 @sf 729 1245 p m (T) s cmr8.300 @sf 756 1252 p m (3) s cmr10.329 @sf 788 1245 p m (=) s cmmi10.329 @sf 13 r m (\025y) s cmmi8.300 @sf 887 1227 p m (T) s cmmi10.329 @sf 914 1245 p m (;) s cmr10.329 @sf 23 r m (with) s cmmi10.329 @sf 1053 1245 p m (y) s cmmi8.300 @sf 1077 1227 p m (T) s cmr10.329 @sf 1117 1245 p m (=) 13 (\(1) ks cmmi10.329 @sf m (;) 8 (y) ks cmr8.300 @sf 1249 1252 p m (2) s cmmi10.329 @sf 1268 1245 p m (;) 8 (y) ks cmr8.300 @sf 1311 1252 p m (3) s cmr10.329 @sf 1330 1245 p m (\)) s cmmi10.329 @sf m (;) s cmr10.329 @sf 74 1374 p m (w) 0 (e) 14 (determine) 16 (a) ks cmmi10.329 @sf 15 r m (Q) s cmr10.329 @sf 15 r m (suc) 0 (h) 15 (that) ks cmmi10.329 @sf 865 1457 p m (y) s cmmi8.300 @sf 889 1439 p m (T) s cmmi10.329 @sf 917 1457 p m (Q) s cmr10.329 @sf 12 r m (=) 13 (\(0) ks cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) 8 (x) ks cmr10.329 @sf m (\)) s cmmi10.329 @sf -1 r m (:) s cmr10.329 @sf 74 1567 p m (Then) s cmmi10.329 @sf 16 r m (Q) s cmmi8.300 @sf 229 1552 p m (T) s cmmi10.329 @sf 264 1567 p m (T) s cmr8.300 @sf 291 1574 p m (3) s cmmi10.329 @sf 318 1567 p m (Q) s cmr10.329 @sf 15 r m (has) 15 (\(0) ks cmmi10.329 @sf m (;) s cmr10.329 @sf 8 r m (0) s cmmi10.329 @sf m (;) 7 (\025) ks cmr8.300 @sf 582 1574 p m (1) s cmr10.329 @sf 600 1567 p m (\)) 14 (as) 15 (its) 16 (last) 15 (ro) 0 (w,) 13 (and) 15 (the) 15 (ob) 3 (jectiv) 0 (e) 14 (has) 16 (b) 1 (een) 16 (ac) 0 (hiev) 0 (ed.) ks cmbx12.300 @sf 74 1730 p m (2.4) s 202 1730 p m (Double) 18 (past) 19 (double) ks cmr10.329 @sf 74 1884 p m (Finally) -3 (,) 15 (w) 0 (e) 14 (ma) 0 (y) 13 (need) 16 (to) 14 (mo) 0 (v) -1 (e) 13 (a) 14 (selected) 16 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 9 r m (2) 15 (matrix) 14 (past) 15 (an) 14 (unrelated) 16 (2) ks cmsy10.329 @sf 9 r m (\002) s cmr10.329 @sf 10 r m (2.) 19 (If) 15 (w) 0 (e) 14 (denote) 15 (the) ks 74 1967 p m (relev) -2 (an) 0 (t) 14 (4) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (4) 15 (matrix) ks cmmi10.329 @sf 15 r m (T) s cmr8.300 @sf 537 1974 p m (4) s cmr10.329 @sf 571 1967 p m (b) 0 (y) ks cmex10.300 @sf 598 2028 p m (0) s 598 2101 p m (B) s 598 2126 p m (B) s 598 2151 p m (B) s 598 2176 p m (B) s 598 2201 p m (B) s 598 2228 p m (@) s cmmi10.329 @sf 682 2065 p m (b) s cmr8.300 @sf 702 2072 p m (1) s cmmi10.329 @sf 767 2065 p m (b) s cmr8.300 @sf 787 2072 p m (2) s 830 2095 p 2 69 ru cmmi10.329 @sf 863 2065 p m (x) s 952 2065 p m (x) s 682 2134 p m (b) s cmr8.300 @sf 702 2141 p m (3) s cmmi10.329 @sf 767 2134 p m (b) s cmr8.300 @sf 787 2141 p m (4) s 830 2164 p 2 69 ru cmmi10.329 @sf 863 2134 p m (x) s 952 2134 p m (x) s 658 2165 p 353 2 ru 830 2220 p 2 57 ru 856 2203 p m (c) s cmr8.300 @sf 876 2210 p m (1) s cmmi10.329 @sf 945 2203 p m (c) s cmr8.300 @sf 965 2210 p m (2) s 729 2259 p save [<007F000001C1C0000780F0000F0078000E0038001C001C003C001E003C001E003C001E0078000F 0078000F0078000F0078000F00F8000F80F8000F80F8000F80F8000F80F8000F80F8000F80F800 0F80F8000F80F8000F80F8000F80F8000F80F8000F80F8000F80F8000F80F8000F8078000F0078 000F0078000F0078000F003C001E003C001E003C001E001C001C000E0038000F0078000780F000 01C1C000007F0000> 25 41 -2 1 30] 48 @pc restore 830 2300 p 2 81 ru cmmi10.329 @sf 856 2283 p m (c) s cmr8.300 @sf 876 2290 p m (3) s cmmi10.329 @sf 945 2283 p m (c) s cmr8.300 @sf 965 2290 p m (4) s cmex10.300 @sf 1033 2028 p m (1) s 1033 2101 p m (C) s 1033 2126 p m (C) s 1033 2151 p m (C) s 1033 2176 p m (C) s 1033 2201 p m (C) s 1033 2228 p m (A) s cmr10.329 @sf 1082 2174 p m (=) s cmex10.300 @sf 1130 2102 p m ( ) s cmmi10.329 @sf 1211 2147 p m (B) s 1272 2164 p 2 57 ru 1298 2147 p m (X) s 1186 2166 p 178 2 ru cmr12.300 @sf 1217 2216 p m (0) s 1272 2221 p 2 57 ru cmmi10.329 @sf 1301 2212 p m (C) s cmex10.300 @sf 1387 2102 p m (!) s cmmi10.329 @sf 1428 2174 p m (;) s cmr10.329 @sf 74 2382 p m (then) 16 (w) 0 (e) 14 (require) 16 (an) 15 (orthogonal) ks cmmi10.329 @sf 14 r m (Q) s cmr10.329 @sf 15 r m (so) 15 (that) ks cmex10.300 @sf 727 2498 p m (e) s cmmi10.329 @sf 718 2511 p m (T) s cmr8.300 @sf 745 2518 p m (4) s cmr10.329 @sf 777 2510 p m (=) s cmmi10.329 @sf 13 r m (QT) s cmr8.300 @sf 888 2517 p m (4) s cmmi10.329 @sf 907 2510 p m (Q) s cmmi8.300 @sf 943 2492 p m (T) s cmr10.329 @sf 983 2510 p m (=) s cmex10.300 @sf 1031 2438 p m ( ) s 1122 2473 p m (e) s cmmi10.329 @sf 1112 2486 p m (C) s 1172 2502 p 2 61 ru cmex10.300 @sf 1211 2473 p m (e) s cmmi10.329 @sf 1198 2486 p m (X) s 1087 2504 p 177 2 ru cmr12.300 @sf 1118 2559 p m (0) s 1172 2559 p 2 57 ru cmex10.300 @sf 1211 2542 p m (e) s cmmi10.329 @sf 1200 2555 p m (B) s cmex10.300 @sf 1287 2438 p m (!) s cmr10.329 @sf 74 2654 p m (where) s cmmi10.329 @sf 16 r m (B) s cmr10.329 @sf 18 r m (and) s cmmi10.329 @sf 15 r m (C) s cmr10.329 @sf 18 r m (ha) 0 (v) -1 (e) 14 (the) 15 (same) 15 (eigen) 0 (v) -2 (alues) 15 (as) ks cmex10.300 @sf 995 2642 p m (e) s cmmi10.329 @sf 985 2655 p m (B) s cmr10.329 @sf 1037 2654 p m (and) s cmex10.300 @sf 1135 2642 p m (e) s cmmi10.329 @sf 1125 2655 p m (C) s cmr10.329 @sf 1161 2654 p m (,) 14 (resp) 1 (ectiv) 0 (ely) -3 (.) ks 1008 2850 p m (6) s @eop 5 @bop0 5 @bop1 cmbx12.300 @sf 74 67 p m (2.2) s 202 67 p m (Single) 17 (past) 19 (double) ks cmr10.329 @sf 74 221 p m (In) 20 (bringing) 20 (a) 19 (selected) 20 (real) 20 (eigen) 0 (v) -2 (alue) 19 (to) 18 (a) 19 (leading) 20 (p) 1 (osition) 20 (w) 0 (e) 18 (shall,) 21 (in) 20 (general,) 20 (need) 20 (to) 18 (pass) ks 74 304 p m (2) s cmsy10.329 @sf 9 r m (\002) s cmr10.329 @sf 10 r m (2) 15 (blo) 1 (c) 0 (ks) 14 (on) 15 (the) 15 (diagonal) 16 (corresp) 1 (onding) 16 (to) 14 (complex) 15 (conjugate) 15 (pairs.) 20 (Hence) 16 (w) 0 (e) 13 (m) 0 (ust) 14 (b) 1 (e) 15 (able) ks 74 387 p m (to) 18 (in) 0 (terc) 0 (hange) 16 (a) 18 (real) 19 (eigen) 0 (v) -2 (alue) 18 (with) 19 (a) 18 (real) 18 (2) ks cmsy10.329 @sf 12 r m (\002) s cmr10.329 @sf 13 r m (2) 18 (blo) 1 (c) 0 (k) 18 (b) 0 (y) 17 (means) 18 (of) 18 (an) 18 (orthogonal) 18 (similarit) 0 (y) ks 74 470 p m (transformation.) 18 (Ob) 0 (viously) -3 (,) 11 (the) 12 (transformation) 11 (is) 12 (determined) 13 (b) 0 (y) 11 (the) 12 (relev) -2 (an) 0 (t) 10 (3) ks cmsy10.329 @sf 3 r m (\002) s cmr10.329 @sf 3 r m (3) 13 (diagonal) 12 (blo) 1 (c) 0 (k) ks 74 553 p m (whic) 0 (h,) 15 (for) 14 (simplic) 1 (it) 0 (y) -3 (,) 15 (w) 0 (e) 13 (write) 16 (as) ks cmex10.300 @sf 564 613 p m (0) s 564 687 p m (B) s 564 714 p m (@) s cmsy10.329 @sf 648 654 p m (\003) s 721 654 p m (\003) s 768 671 p 2 57 ru cmmi10.329 @sf 807 654 p m (b) s cmsy10.329 @sf 648 710 p m (\003) s 721 710 p m (\003) s 768 727 p 2 57 ru cmmi10.329 @sf 807 710 p m (c) s 623 729 p 242 2 ru cmr10.329 @sf 648 767 p m (0) s 721 767 p m (0) s 768 784 p 2 57 ru cmmi10.329 @sf 793 767 p m (\025) s cmr8.300 @sf 820 774 p m (3) s cmex10.300 @sf 888 613 p m (1) s 888 687 p m (C) s 888 714 p m (A) s cmsy10.329 @sf 937 710 p m (\021) s cmex10.300 @sf 985 613 p m (0) s 985 687 p m (B) s 985 714 p m (@) s 1098 683 p save [ 42 41 -2 0 45] 66 @pc restore 1200 728 p 2 113 ru cmmi10.329 @sf 1246 656 p m (b) s 1246 710 p m (c) s 1045 729 p 268 2 ru cmr10.329 @sf 1070 767 p m (0) 60 (0) ks 1200 784 p 2 57 ru cmmi10.329 @sf 1241 767 p m (\025) s cmr8.300 @sf 1268 774 p m (3) s cmex10.300 @sf 1335 613 p m (1) s 1335 687 p m (C) s 1335 714 p m (A) s cmmi10.329 @sf 1379 710 p m (:) s cmr10.329 @sf 1906 710 p m (\(4\)) s 74 868 p m (The) 16 (same) 14 (principl) 1 (e) 17 (ma) 0 (y) 14 (b) 1 (e) 16 (used) 16 (as) 14 (in) 16 (the) 16 (single) 16 (past) 15 (single) 16 (case.) 20 (If) ks cmex10.300 @sf 898 928 p m (0) s 898 1001 p m (B) s 898 1028 p m (@) s cmmi10.329 @sf 955 968 p m (x) s cmr8.300 @sf 981 975 p m (1) s cmmi10.329 @sf 955 1025 p m (x) s cmr8.300 @sf 981 1032 p m (2) s cmr10.329 @sf 966 1081 p m (1) s cmex10.300 @sf 1021 928 p m (1) s 1021 1001 p m (C) s 1021 1028 p m (A) s cmr10.329 @sf 1906 1025 p m (\(5\)) s 74 1182 p m (denotes) 16 (the) 15 (eigen) 0 (v) 0 (ector) 13 (corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 15 r m (\025) s cmr8.300 @sf 928 1189 p m (3) s cmr10.329 @sf 962 1182 p m (then) 16 (w) 0 (e) 14 (require) 16 (a) ks cmmi10.329 @sf 15 r m (Q) s cmr10.329 @sf 15 r m (suc) 0 (h) 14 (that) ks cmmi10.329 @sf 819 1339 p m (Q) s cmex10.300 @sf 863 1242 p m (0) s 862 1315 p m (B) s 862 1342 p m (@) s cmmi10.329 @sf 919 1283 p m (x) s cmr8.300 @sf 945 1290 p m (1) s cmmi10.329 @sf 919 1339 p m (x) s cmr8.300 @sf 945 1346 p m (2) s cmr10.329 @sf 931 1395 p m (1) s cmex10.300 @sf 986 1242 p m (1) s 986 1315 p m (C) s 986 1342 p m (A) s cmr10.329 @sf 1035 1339 p m (=) s cmex10.300 @sf 1083 1242 p m (0) s 1083 1315 p m (B) s 1083 1342 p m (@) s cmmi10.329 @sf 1140 1283 p m (r) s cmr10.329 @sf 1140 1339 p m (0) s 1140 1395 p m (0) s cmex10.300 @sf 1183 1242 p m (1) s 1183 1315 p m (C) s 1183 1342 p m (A) s cmr10.329 @sf 74 1496 p m (and) 16 (then,) 15 (as) 15 (b) 1 (efore,) ks cmmi10.329 @sf 467 1653 p m (QT) 6 (Q) ks cmmi8.300 @sf 572 1635 p m (T) s cmr10.329 @sf 612 1653 p m (=) s cmex10.300 @sf 660 1557 p m (0) s 660 1630 p m (B) s 660 1657 p m (@) s cmmi10.329 @sf 744 1597 p m (\025) s cmr8.300 @sf 771 1604 p m (3) s 815 1614 p 2 57 ru cmmi10.329 @sf 840 1597 p m (x) s 916 1597 p m (x) s 719 1616 p 248 2 ru cmr10.329 @sf 756 1654 p m (0) s 815 1671 p 2 57 ru cmmi10.329 @sf 840 1654 p m (x) s 916 1654 p m (x) s cmr10.329 @sf 756 1710 p m (0) s 815 1727 p 2 57 ru cmmi10.329 @sf 840 1710 p m (x) s 916 1710 p m (x) s cmex10.300 @sf 991 1557 p m (1) s 991 1630 p m (C) s 991 1657 p m (A) s cmr10.329 @sf 1039 1653 p m (=) s cmex10.300 @sf 1087 1557 p m (0) s 1087 1630 p m (B) s 1087 1657 p m (@) s cmmi10.329 @sf 1181 1597 p m (\025) s cmr8.300 @sf 1208 1604 p m (3) s 1260 1614 p 2 57 ru cmmi10.329 @sf 1286 1597 p m (x) 46 (x) ks 1147 1616 p 262 2 ru cmr10.329 @sf 1193 1655 p m (0) s 1193 1710 p m (0) s 1260 1727 p 2 113 ru 1311 1682 p save [<000FF8000000003FFF00000000FC03C0000003F00060000007C0001800000F80000C00001F0000 0600001F00000300003E00000100003E00000080007C00000040007C00000040007C0000002000 7C00000020007C0000002000FC0000000000FC0000000000FC00000000007C00000000007E0000 0000007E00000000007E00000000007E00000000003F00000000003F00000000003F0000000000 1F80000000001F80000002000FC00000020007C00000020007E00000020003E00000030001F000 00030000F800000700007C00000700003E00000780001F00000F80000F80000F800003E0001780 0001F80063C000007F01C1C000001FFF80C0000001FE0040> 42 43 -2 1 43] 67 @pc restore cmex10.300 @sf 1432 1557 p m (1) s 1432 1630 p m (C) s 1432 1657 p m (A) s cmmi10.329 @sf 1476 1653 p m (:) s cmr10.329 @sf 1906 1653 p m (\(6\)) s 74 1811 p m (Note) 19 (that) 19 (the) 20 (general) 20 (principl) 1 (e) 21 (w) 0 (e) 19 (are) 19 (using) 20 (is) 20 (the) 20 (one) 19 (commonly) 20 (emplo) 0 (y) 0 (ed) 18 (to) 19 (establish) 21 (the) ks 74 1894 p m (Sc) 0 (h) 0 (ur) 16 (canonical) 18 (form) 17 (b) 0 (y) 17 (induction.) 28 (The) 18 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 12 r m (2) 17 (matrix) ks cmmi10.329 @sf 17 r m (C) s cmr10.329 @sf 20 r m (in) 18 (the) 18 (b) 1 (ottom) 16 (of) 17 (\(6\)) 17 (is) 18 (not) 17 (the) 17 (same) ks 74 1977 p m (as) s cmmi10.329 @sf 18 r m (B) s cmr10.329 @sf 20 r m (in) 19 (\(4\),) 17 (but) 18 (it) 18 (will,) 20 (of) 17 (course,) 18 (ha) 0 (v) 0 (e) 16 (the) 18 (same) 17 (eigen) 0 (v) -2 (alues.) 28 (Ho) 0 (w) -1 (ev) -1 (er,) ks cmmi10.329 @sf 16 r m (B) s cmr10.329 @sf 20 r m (and) s cmmi10.329 @sf 19 r m (C) s cmr10.329 @sf 20 r m (will) 20 (not,) 17 (in) ks 74 2060 p m (general,) 15 (b) 1 (e) 16 (orthogonally) 15 (similar.) ks 124 2184 p m (The) 15 (matrix) ks cmmi10.329 @sf 14 r m (Q) s cmr10.329 @sf 14 r m (can) 14 (b) 1 (e) 15 (determined) 16 (as) 14 (one) 14 (Householder) 16 (matrix) 14 (or) 14 (as) 14 (the) 14 (pro) 1 (duct) 15 (of) 14 (t) 0 (w) -1 (o) 12 (Giv) 0 (ens) ks 74 2267 p m (rotations.) 19 (Since) ks cmmi10.329 @sf 16 r m (\025) s cmr8.300 @sf 429 2274 p m (3) s cmr10.329 @sf 462 2267 p m (is) 15 (real) 15 (and) ks cmmi10.329 @sf 15 r m (B) s cmr10.329 @sf 17 r m (has) 14 (complex) 16 (conjugate) 14 (eigen) 0 (v) -2 (alues,) ks cmmi10.329 @sf 14 r m (B) s cmr10.329 @sf 17 r m (can) 15 (ha) 0 (v) -1 (e) 13 (no) 14 (eigen) 0 (v) -2 (alues) ks 74 2350 p m (in) 13 (common) 11 (with) ks cmmi10.329 @sf 12 r m (\025) s cmr8.300 @sf 429 2357 p m (3) s cmr10.329 @sf 448 2350 p m (;) 12 (hence,) 14 (a) 11 (unique) 13 (eigen) 0 (v) 0 (ector) 10 (of) 11 (the) 12 (form) 11 (\(5\)) 10 (will) 13 (exist.) 20 (As) 11 (the) 12 (t) 0 (w) -1 (o) 9 (eigen) 0 (v) -2 (alues) ks 74 2433 p m (of) s cmmi10.329 @sf 13 r m (B) s cmr10.329 @sf 15 r m (approac) 0 (h) 12 (the) 13 (real) ks cmmi10.329 @sf 13 r m (\025) s cmr8.300 @sf 556 2440 p m (3) s cmr10.329 @sf 575 2433 p m (,) 13 (their) 14 (imaginary) 13 (parts) 12 (b) 1 (ecome) 14 (small,) 14 (and) 13 (the) 13 (eigen) 0 (v) 0 (ector) 11 (\(5\)) 12 (will) 14 (ha) 0 (v) 0 (e) ks 74 2516 p m (progressiv) 0 (ely) 15 (larger) 15 (comp) 1 (onen) 0 (ts) 14 (in) 16 (the) 15 (\014rst) 15 (t) 0 (w) -1 (o) 13 (p) 1 (ositions;) 16 (i.e.,) 15 (the) 15 (normalized) 16 (v) 0 (ersion) 15 (will) 16 (ha) 0 (v) 0 (e) ks 74 2599 p m (a) 15 (progressiv) 0 (ely) 15 (smaller) 16 (third) 15 (comp) 1 (onen) 0 (t.) ks 1008 2850 p m (5) s @eop 4 @bop0 cmmi10.329 @sf [<001000001800001800001800000C00000C00000C00000C00000600000600000600000200000200 000200000300000300000300000280000280000280000240000240800420C00420400C10301C10 3FF8101FF8080FF00803E004> 22 30 -1 10 24] 13 @dc cmex10.300 @sf [ 8 74 -10 72 20] 104 @dc [ 8 74 -1 72 20] 105 @dc 4 @bop1 cmr10.329 @sf 74 69 p m (i.e.,) 15 (\() ks cmmi10.329 @sf 178 69 p m (\013) 0 (;) 8 (\026) ks cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmr10.329 @sf m (\)) s cmmi8.300 @sf 354 49 p m (T) s cmr10.329 @sf 397 69 p m (is) 16 (the) 15 (eigen) 0 (v) 0 (ector) 13 (corresp) 1 (onding) 16 (to) ks cmmi10.329 @sf 15 r m (\026) s cmr10.329 @sf m (.) 20 (If) ks cmmi10.329 @sf 16 r m (Q) s cmr10.329 @sf 15 r m (is) 15 (c) 0 (hosen) 15 (so) 15 (that) ks cmmi10.329 @sf 742 204 p m (Q) s cmex10.300 @sf 786 132 p m ( ) s cmmi10.329 @sf 879 177 p m (\013) s 839 232 p m (\026) s cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmex10.300 @sf 969 132 p m (!) s cmr10.329 @sf 1015 204 p m (=) s cmex10.300 @sf 1063 132 p m ( ) s cmmi10.329 @sf 1117 177 p m (r) s cmr10.329 @sf 1117 232 p m (0) s cmex10.300 @sf 1160 132 p m (!) s cmmi10.329 @sf 1200 204 p m (;) s cmr10.329 @sf 1906 204 p m (\(2\)) s 74 333 p m (then) s cmmi10.329 @sf 545 416 p m (Q) s cmex10.300 @sf 589 344 p m ( ) s cmmi10.329 @sf 642 389 p m (\025) s 710 389 p m (\013) s cmr10.329 @sf 644 444 p m (0) s cmmi10.329 @sf 711 444 p m (\026) s cmex10.300 @sf 760 344 p m (!) s cmmi10.329 @sf 801 416 p m (Q) s cmmi8.300 @sf 837 398 p m (T) s cmmi10.329 @sf 864 416 p m (Q) s cmex10.300 @sf 908 344 p m ( ) s cmmi10.329 @sf 1002 389 p m (\013) s 961 444 p m (\026) s cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmex10.300 @sf 1092 344 p m (!) s cmr10.329 @sf 1137 416 p m (=) s cmmi10.329 @sf 13 r m (\026Q) s cmex10.300 @sf 1256 344 p m ( ) s cmmi10.329 @sf 1350 389 p m (\013) s 1310 444 p m (\026) s cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 11 r m (\025) s cmex10.300 @sf 1440 344 p m (!) s cmmi10.329 @sf 1480 416 p m (;) s cmr10.329 @sf 74 532 p m (and) 16 (hence,) 15 (using) 16 (\(2\)) 15 (and) 15 (dividin) 1 (g) 16 (b) 0 (y) ks cmmi10.329 @sf 15 r m (r) s cmr10.329 @sf 1 r m (,) 14 (w) 0 (e) 14 (ha) 0 (v) 0 (e) ks cmmi10.329 @sf 570 667 p m (Q) s cmex10.300 @sf 614 595 p m ( ) s cmmi10.329 @sf 667 640 p m (\025) s 736 640 p m (\013) s cmr10.329 @sf 669 695 p m (0) s cmmi10.329 @sf 736 695 p m (\026) s cmex10.300 @sf 786 595 p m (!) s cmmi10.329 @sf 826 667 p m (Q) s cmmi8.300 @sf 862 649 p m (T) s cmex10.300 @sf 897 595 p m ( ) s cmr10.329 @sf 951 640 p m (1) s 951 695 p m (0) s cmex10.300 @sf 994 595 p m (!) s cmr10.329 @sf 1040 667 p m (=) s cmmi10.329 @sf 13 r m (\026) s cmex10.300 @sf 1123 595 p m ( ) s cmr10.329 @sf 1176 640 p m (1) s 1176 695 p m (0) s cmex10.300 @sf 1220 595 p m (!) s cmr10.329 @sf 1265 667 p m (=) s cmex10.300 @sf 1313 595 p m ( ) s cmmi10.329 @sf 1367 640 p m (\026) s cmr10.329 @sf 1369 695 p m (0) s cmex10.300 @sf 1415 595 p m (!) s cmmi10.329 @sf 1455 667 p m (:) s cmr10.329 @sf 74 795 p m (This) 14 (states) 12 (that) 13 (the) 13 (\014rst) 13 (column) 14 (of) 13 (the) 14 (transformed) 12 (2) ks cmsy10.329 @sf 6 r m (\002) s cmr10.329 @sf 6 r m (2) 14 (is) 13 (in) 14 (the) 14 (required) 14 (form.) 19 (Hence) 14 (w) 0 (e) 12 (ma) 0 (y) ks 74 878 p m (write) s cmmi10.329 @sf 717 961 p m (Q) s cmex10.300 @sf 761 889 p m ( ) s cmmi10.329 @sf 814 934 p m (\025) s 882 934 p m (\013) s cmr10.329 @sf 816 990 p m (0) s cmmi10.329 @sf 883 990 p m (\026) s cmex10.300 @sf 932 889 p m (!) s cmmi10.329 @sf 973 961 p m (Q) s cmmi8.300 @sf 1009 943 p m (T) s cmr10.329 @sf 1049 961 p m (=) s cmex10.300 @sf 1097 889 p m ( ) s cmmi10.329 @sf 1150 934 p m (\026) s 1219 934 p save [<800000800000800000400000400000400000400000200000200000207C002187001201C01400E0 1400F018007008007808003C08003C08003C04001E04001E04001E04001E02001E02001C02001C 0207D80104700107F801000C01000E008006008007004007004007002003001003000803000406 0003060000F8> 24 41 -1 9 26] 12 @pc restore cmr10.329 @sf 1153 990 p m (0) s cmmi10.329 @sf 1220 990 p m (\015) s cmex10.300 @sf 1268 889 p m (!) s cmmi10.329 @sf 1309 961 p m (:) s cmr10.329 @sf 74 1071 p m (Since) 17 (the) 15 (trace) 15 (and) 15 (F) -3 (rob) 1 (enius) 16 (norm) 14 (are) 15 (in) 0 (v) -2 (arian) 0 (t,) ks cmmi10.329 @sf 575 1200 p m (\025) s cmr10.329 @sf 10 r m (+) s cmmi10.329 @sf 10 r m (\026) s cmr10.329 @sf 13 r m (=) s cmmi10.329 @sf 13 r m (\026) s cmr10.329 @sf 11 r m (+) s cmmi10.329 @sf 10 r m (\015) 3 (;) 17 (\025) ks cmr8.300 @sf 912 1182 p m (2) s cmr10.329 @sf 941 1200 p m (+) s cmmi10.329 @sf 10 r m (\026) s cmr8.300 @sf 1013 1182 p m (2) s cmr10.329 @sf 1044 1200 p m (+) s cmmi10.329 @sf 10 r m (\013) s cmr8.300 @sf 1118 1182 p m (2) s cmr10.329 @sf 1151 1200 p m (=) s cmmi10.329 @sf 13 r m (\026) s cmr8.300 @sf 1226 1182 p m (2) s cmr10.329 @sf 1256 1200 p m (+) s cmmi10.329 @sf 10 r m (\015) s cmr8.300 @sf 1328 1182 p m (2) s cmr10.329 @sf 1357 1200 p m (+) s cmmi10.329 @sf 11 r save [<800000800000800000400000400000400000400000200000200000207C002187001201C01400E0 1400F018007008007808003C08003C08003C04001E04001E04001E04001E02001E02001C02001C 0207D80104700107F801000C01000E008006008007004007004007002003001003000803000406 0003060000F8> 24 41 -1 9 26] 12 @pc restore cmr8.300 @sf 1431 1182 p m (2) s cmmi10.329 @sf 1450 1200 p m (;) s cmr10.329 @sf 74 1329 p m (giving) s cmmi10.329 @sf 828 1412 p m (\015) s cmr10.329 @sf 14 r m (=) s cmmi10.329 @sf 13 r m (\025) s cmr10.329 @sf 15 r m (and) s cmmi10.329 @sf 1045 1412 p save [<800000800000800000400000400000400000400000200000200000207C002187001201C01400E0 1400F018007008007808003C08003C08003C04001E04001E04001E04001E02001E02001C02001C 0207D80104700107F801000C01000E008006008007004007004007002003001003000803000406 0003060000F8> 24 41 -1 9 26] 12 @pc restore cmr10.329 @sf 40 r m (=) s cmsy10.329 @sf 13 r m (\006) s cmmi10.329 @sf m (\013) 0 (:) ks cmr10.329 @sf 74 1522 p m (A) 15 (rotation) 15 (giving) 16 (\(2\)) 14 (is) 16 (de\014ned) 16 (b) 0 (y) ks 421 1651 p m (cos) s cmmi10.329 @sf 490 1651 p m (\022) s cmr10.329 @sf 14 r m (=) s cmmi10.329 @sf 13 r m (\013) 0 (=r) 0 (;) ks cmr10.329 @sf 7 r m (sin) s cmmi10.329 @sf 728 1651 p m (\022) s cmr10.329 @sf 14 r m (=) 13 (\() ks cmmi10.329 @sf 828 1651 p m (\026) s cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmr10.329 @sf m (\)) s cmmi10.329 @sf 963 1651 p m (=r) 0 (;) 7 (r) ks cmr10.329 @sf 12 r m (=) 13 (+) ks cmex10.300 @sf 1151 1603 p m (h) s cmmi10.329 @sf 1170 1651 p m (\013) s cmr8.300 @sf 1199 1633 p m (2) s cmr10.329 @sf 1229 1651 p m (+) 11 (\() ks cmmi10.329 @sf 1292 1651 p m (\026) s cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmr10.329 @sf m (\)) s cmr8.300 @sf 1420 1631 p m (2) s cmex10.300 @sf 1439 1603 p m (i) s cmr8.300 @sf 1459 1615 p m (1) s cmmi8.300 @sf m (=) s cmr8.300 @sf m (2) s cmmi10.329 @sf 1521 1651 p m (;) s cmr10.329 @sf 1906 1651 p m (\(3\)) s 74 1779 p m (and) 16 (it) 15 (will) 17 (readily) 16 (b) 1 (e) 16 (v) 0 (eri\014ed) 15 (that) 15 (this) 15 (giv) 0 (es) ks cmmi10.329 @sf 15 r save [<800000800000800000400000400000400000400000200000200000207C002187001201C01400E0 1400F018007008007808003C08003C08003C04001E04001E04001E04001E02001E02001C02001C 0207D80104700107F801000C01000E008006008007004007004007002003001003000803000406 0003060000F8> 24 41 -1 9 26] 12 @pc restore cmr10.329 @sf 41 r m (=) 12 (+) ks cmmi10.329 @sf m (\013) s cmr10.329 @sf m (.) s 124 1904 p m (If) 13 (the) 13 (original) ks cmmi10.329 @sf 13 r m (T) s cmr10.329 @sf 19 r m (has) 12 (b) 1 (een) 14 (determined) 14 (from) 12 (a) 13 (matrix) ks cmmi10.329 @sf 12 r m (A) s cmr10.329 @sf 13 r m (b) 0 (y) 11 (means) 13 (of) 12 (an) 13 (orthogonal) 12 (transforma-) ks 74 1987 p m (tion,) 15 (the) 14 (matrix) 15 (de\014ning) 16 (this) 15 (transformation) 13 (m) 0 (ust) 13 (b) 1 (e) 15 (up) 1 (dated) 16 (b) 0 (y) 13 (m) 0 (ultiplication) 16 (with) 15 (the) 15 (plane) ks 74 2070 p m (rotations) 17 (used) 19 (in) 19 (the) 18 (reordering) 18 (pro) 1 (cess.) 29 (Note) 17 (that) 18 (in) 19 (this) 18 (metho) 1 (d,) 19 (wherev) 0 (er) 17 (t) 0 (w) -2 (o) 16 (eigen) 0 (v) -2 (alues) ks 74 2153 p m (that) 12 (w) 0 (e) 11 (ha) 0 (v) -1 (e) 11 (decided) 14 (to) 12 (place) 13 (in) 14 (the) 12 (same) 12 (group) 12 (are) 12 (in) 0 (terc) 0 (hanged,) 12 (a) 12 (selected) 13 (eigen) 0 (v) -2 (alue) 13 (is) 13 (mo) 0 (v) -1 (ed) ks 74 2236 p m (up) 14 (only) 14 (past) 13 (eigen) 0 (v) -2 (alues) 13 (with) 14 (whic) 0 (h) 13 (it) 14 (is) 14 (not) 13 (to) 12 (b) 1 (e) 14 (asso) 1 (ciated.) 20 (Moreo) 0 (v) -2 (er,) 12 (ha) 0 (ving) 12 (determined) 15 (the) ks 74 2319 p m (rotation,) 16 (w) 0 (e) 15 (shall) 18 (apply) 17 (it) 17 (to) 16 (ro) 0 (ws) 15 (and) 17 (columns) ks cmmi10.329 @sf 17 r m (p) s cmr10.329 @sf 16 r m (and) s cmmi10.329 @sf 17 r m (p) s cmr10.329 @sf 11 r m (+) 11 (1) 17 (but) 17 (not) 16 (to) 16 (the) 16 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 12 r m (2) 16 (itself.) 25 (There) ks 74 2402 p m (w) 0 (e) 15 (shall) 17 (merely) 16 (in) 0 (terc) 0 (hange) ks cmmi10.329 @sf 14 r m (\025) s cmr10.329 @sf 15 r m (and) s cmmi10.329 @sf 16 r m (\026) s cmr10.329 @sf 17 r m (and) 16 (do) 15 (no) 16 (computation.) 22 (Mo) 0 (ving) 14 (1) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 11 r m (1) 16 (blo) 1 (c) 0 (ks) 15 (is) 17 (discussed) ks 74 2485 p m (in) 16 ([) 0 (8) ks 163 2485 p m (].) s 1008 2850 p m (4) s @eop 3 @bop0 3 @bop1 cmr10.329 @sf 124 67 p m (In) 13 (this) 13 (pap) 1 (er,) 13 (w) 0 (e) 11 (presen) 0 (t) 11 (t) 0 (w) -1 (o) 10 (other) 12 (metho) 1 (ds) 13 (for) 11 (constructing) 13 (the) 12 (in) 0 (v) -2 (arian) 0 (t) 10 (subspace.) 20 (The) 12 (\014rst) ks 74 150 p m (in) 0 (v) 0 (olv) 0 (es) 15 (applying) 18 (transformations) 15 (directly) 19 (to) 16 (in) 0 (terc) 0 (hange) 15 (the) 17 (eigen) 0 (v) -2 (alues.) 25 (The) 17 (second) 18 (metho) 1 (d) ks 74 233 p m (in) 0 (v) 0 (olv) 0 (es) 13 (direct) 16 (computation) 15 (of) 15 (the) 15 (v) 0 (ectors.) ks cmbx10.432 @sf 74 418 p save [ 27 39 -3 0 34] 50 @pc restore 177 418 p m (In) -1 (terc) -1 (hanging) ks 22 r save [ 39 41 -3 0 45] 69 @pc restore 45 r m (igen) -1 (v) -4 (alues) ks cmr10.329 @sf 74 587 p m (The) 21 (reordering) 21 (of) 20 (the) 21 (eigen) 0 (v) -2 (alues) 21 (can) 20 (b) 1 (e) 22 (ac) 0 (hiev) 0 (ed) 19 (b) 0 (y) 20 (successiv) 0 (ely) 21 (in) 0 (terc) 0 (hanging) 20 (neigh) 0 (b) 1 (oring) ks 74 671 p m (blo) 1 (c) 0 (ks) 15 (in) 16 (the) 15 (Sc) 0 (h) 0 (ur) 14 (factor) ks cmmi10.329 @sf 14 r m (T) s cmr10.329 @sf 6 r m (.) s 124 795 p m (Supp) 1 (ose,) 13 (in) 13 (a) 12 (giv) 0 (en) ks cmmi10.329 @sf 12 r m (T) s cmr10.329 @sf 6 r m (,) 12 (one) 12 (has) 12 (decided) 14 (to) 12 (group) ks cmmi10.329 @sf 11 r m (\025) s cmmi8.300 @sf 1095 802 p m (p) s cmmi10.329 @sf 1115 795 p m (;) 8 (\025) ks cmmi8.300 @sf 1163 802 p m (q) s cmmi10.329 @sf 1181 795 p m (;) 8 (\025) ks cmmi8.300 @sf 1229 802 p m (r) s cmr10.329 @sf 1258 795 p m (together.) 18 (W) -3 (e) 12 (kno) 0 (w) 10 (that) 12 (there) 12 (exists) ks 74 878 p m (a) 15 (unitary) 16 (matrix) ks cmex10.300 @sf 431 866 p m (e) s cmmi10.329 @sf 420 879 p m (Q) s cmr10.329 @sf 472 878 p m (suc) 0 (h) 14 (that) ks cmex10.300 @sf 681 866 p m (e) s cmmi10.329 @sf 673 879 p m (T) s cmr10.329 @sf 718 878 p m (=) s cmex10.300 @sf 776 866 p m (e) s cmmi10.329 @sf 766 879 p m (Q) s 802 878 p m (T) s cmex10.300 @sf 845 866 p m (e) s cmmi10.329 @sf 835 879 p m (Q) s cmmi8.300 @sf 871 864 p m (H) s cmr10.329 @sf 920 878 p m (is) 15 (still) 17 (upp) 1 (er) 16 (triangular) 16 (but) 15 (has) ks cmmi10.329 @sf 15 r m (\025) s cmmi8.300 @sf 1588 885 p m (p) s cmmi10.329 @sf 1607 878 p m (;) 8 (\025) ks cmmi8.300 @sf 1655 885 p m (q) s cmmi10.329 @sf 1673 878 p m (;) 8 (\025) ks cmmi8.300 @sf 1721 885 p m (r) s cmr10.329 @sf 1754 878 p m (in) 16 (the) 15 (\014rst) ks 74 961 p m (three) 15 (p) 1 (ositions.) 21 (Suc) 0 (h) 15 (a) ks cmmi10.329 @sf 15 r m (Q) s cmr10.329 @sf 15 r m (can) 15 (b) 1 (e) 16 (readily) 16 (determined) 17 (as) 14 (the) 16 (pro) 1 (duct) 15 (of) 15 (a) 15 (\014nite) 16 (n) 0 (um) 0 (b) 1 (er) 13 (of) 15 (plane) ks 74 1044 p m (rotations.) 19 (W) -3 (e) 15 (merely) 16 (need) 16 (an) 15 (algorithm) 15 (whic) 0 (h) 15 (will) 17 (enable) 17 (us) 15 (to) 15 (in) 0 (terc) 0 (hange) 13 (consecutiv) 0 (e) 15 (blo) 1 (c) 0 (ks) ks 74 1127 p m (on) 16 (the) 15 (diagonal) 16 (b) 0 (y) 15 (means) 15 (of) 16 (a) 15 (plane) 16 (rotation.) 20 (Rep) 1 (eated) 17 (application) 17 (of) 16 (this) 15 (algorithm) 16 (can) 16 (then) ks 74 1210 p m (bring) 16 (an) 0 (y) 14 (selected) 16 (set) 15 (of) 15 (eigen) 0 (v) -2 (alues) 15 (in) 0 (to) 14 (the) 16 (leading) 16 (p) 1 (ositions.) ks 124 1335 p m (The) 15 (algorithm) 15 (w) 0 (e) 13 (describ) 1 (e) 17 (could) 15 (b) 1 (e) 16 (used) 15 (on) 15 (a) 14 (complex) 16 (triangular) 15 (matrix.) 19 (Ho) 0 (w) -1 (ev) -1 (er,) 12 (since) 16 (w) 0 (e) ks 74 1418 p m (are) 13 (in) 0 (terested) 13 (here) 14 (in) 14 (real) 14 (matrices,) 14 (and) 13 (since) 15 (complex) 14 (conjugate) 13 (eigen) 0 (v) -2 (alues) 13 (will) 15 (b) 1 (e) 15 (represen) 0 (ted) ks 74 1501 p m (b) 0 (y) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2) 15 (real) 16 (diagonal) 16 (blo) 1 (c) 0 (ks,) 15 (w) 0 (e) 14 (describ) 1 (e) 18 (\014rst) 15 (the) 15 (algorithm) 16 (for) 15 (in) 0 (terc) 0 (hanging) 14 (t) 0 (w) -1 (o) 13 (consecutiv) 0 (e) ks 74 1584 p m (real) 16 (eigen) 0 (v) -2 (alues.) ks cmbx12.300 @sf 74 1747 p m (2.1) s 202 1747 p m (Single) 17 (past) 19 (single) ks cmr10.329 @sf 74 1901 p m (Supp) 1 (ose) ks cmmi10.329 @sf 18 r m (\025) s cmr10.329 @sf 15 r m (and) s cmmi10.329 @sf 17 r m (\026) s cmr10.329 @sf 17 r m (are) 16 (in) 17 (p) 1 (ositions) ks cmmi10.329 @sf 17 r m (p) s cmr10.329 @sf 17 r m (and) s cmmi10.329 @sf 16 r m (p) s cmr10.329 @sf 11 r m (+) 11 (1.) 23 (A) 17 (similarit) 0 (y) 16 (rotation) 16 (in) 17 (planes) ks cmmi10.329 @sf 18 r m (p) s cmr10.329 @sf 16 r m (and) s cmmi10.329 @sf 16 r m (p) s cmr10.329 @sf 11 r m (+) 11 (1) 16 (will) ks 74 1984 p m (alter) 20 (only) 20 (ro) 0 (ws) 17 (and) 20 (columns) ks cmmi10.329 @sf 20 r m (p) s cmr10.329 @sf 20 r m (and) s cmmi10.329 @sf 19 r m (p) s cmr10.329 @sf 13 r m (+) 14 (1) 19 (and) 19 (will) 22 (retain) 19 (the) 20 (triangular) 20 (form) 19 (apart) 18 (from) 19 (the) ks 74 2067 p m (p) 1 (ossible) 19 (in) 0 (tro) 1 (duction) 17 (of) 17 (a) 17 (non-zero) 17 (in) 18 (p) 1 (osition) 18 (\() ks cmmi10.329 @sf m (p) s cmr10.329 @sf 11 r m (+) 12 (1) ks cmmi10.329 @sf m (;) 8 (p) ks cmr10.329 @sf m (\).) 24 (The) 17 (rotation) 17 (can) 17 (b) 1 (e) 18 (c) 0 (hosen) 16 (so) 17 (as) 17 (to) ks 74 2150 p m (in) 0 (terc) 0 (hange) ks cmmi10.329 @sf 16 r m (\025) s cmr10.329 @sf 18 r m (and) s cmmi10.329 @sf 18 r m (\026) s cmr10.329 @sf 18 r m (while) 19 (retaining) 19 (the) 18 (zero) 17 (in) 19 (\() ks cmmi10.329 @sf m (p) s cmr10.329 @sf 12 r m (+) 12 (1) ks cmmi10.329 @sf m (;) 8 (p) ks cmr10.329 @sf m (\).) 26 (Clearly) 19 (the) 17 (rotation) 17 (is) 19 (determined) ks 74 2233 p m (solely) 16 (b) 0 (y) 15 (the) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 10 r m (2) 15 (matrix,) 14 (whic) 0 (h) 16 (w) 0 (e) 13 (denote) 16 (b) 0 (y) ks cmex10.300 @sf 907 2290 p m ( ) s cmmi10.329 @sf 960 2335 p m (\025) s 1028 2335 p m (\013) s cmr10.329 @sf 962 2390 p m (0) s cmmi10.329 @sf 1029 2390 p m (\026) s cmex10.300 @sf 1078 2290 p m (!) s cmmi10.329 @sf 1119 2362 p m (:) s cmr10.329 @sf 74 2490 p m (W) -3 (e) 14 (ha) 0 (v) 0 (e) ks cmex10.300 @sf 607 2547 p m ( ) s cmmi10.329 @sf 661 2592 p m (\025) s 729 2592 p m (\013) s cmr10.329 @sf 663 2647 p m (0) s cmmi10.329 @sf 730 2647 p m (\026) s cmex10.300 @sf 779 2547 p m (!) s 819 2547 p m ( ) s cmmi10.329 @sf 913 2592 p m (\013) s 873 2647 p m (\026) s cmsy10.329 @sf 10 r m (\000) s cmmi10.329 @sf 11 r m (\025) s cmex10.300 @sf 1003 2547 p m (!) s cmr10.329 @sf 1049 2619 p m (=) s cmmi10.329 @sf 12 r m (\026) s cmex10.300 @sf 1131 2547 p m ( ) s cmmi10.329 @sf 1225 2592 p m (\013) s 1185 2647 p m (\026) s cmsy10.329 @sf 11 r m (\000) s cmmi10.329 @sf 10 r m (\025) s cmex10.300 @sf 1315 2547 p m (!) s cmr10.329 @sf 1906 2619 p m (\(1\)) s 1008 2850 p m (3) s @eop 2 @bop0 2 @bop1 cmr10.329 @sf 74 67 p m (Let) 16 (us) 15 (denote) 15 (the) 16 (Sc) 0 (h) 0 (ur) 13 (factorization) 15 (of) 15 (the) 15 (real) 16 (matrix) ks cmmi10.329 @sf 14 r m (A) s cmr10.329 @sf 16 r m (as) s cmmi10.329 @sf 899 182 p m (A) s cmr10.329 @sf 13 r m (=) s cmmi10.329 @sf 13 r m (QT) 6 (Q) ks cmmi8.300 @sf 1099 164 p m (T) s cmmi10.329 @sf 1126 182 p m (;) s cmr10.329 @sf 74 296 p m (where) s cmmi10.329 @sf 13 r m (Q) s cmr10.329 @sf 13 r m (is) 13 (orthogonal) 12 (and) ks cmmi10.329 @sf 13 r m (T) s cmr10.329 @sf 18 r m (blo) 1 (c) 0 (k) 12 (upp) 1 (er) 14 (triangular,) 13 (with) 13 (1) ks cmsy10.329 @sf 5 r m (\002) s cmr10.329 @sf 5 r m (1) 13 (and) 13 (2) ks cmsy10.329 @sf 5 r m (\002) s cmr10.329 @sf 5 r m (2) 12 (blo) 1 (c) 0 (ks) 12 (on) 13 (the) 13 (diagonal,) ks 74 379 p m (the) 15 (2) ks cmsy10.329 @sf 10 r m (\002) s cmr10.329 @sf 11 r m (2) 15 (blo) 1 (c) 0 (ks) 14 (corresp) 1 (onding) 17 (to) 14 (complex) 16 (conjugate) 15 (pairs) 16 (of) 14 (eigen) 0 (v) -2 (alues.) 20 (Since) ks cmmi10.329 @sf 914 494 p m (AQ) s cmr10.329 @sf 13 r m (=) s cmmi10.329 @sf 13 r m (QT) 4 (;) ks 74 609 p m (Q) s cmr10.329 @sf m (,) 12 (of) 12 (course,) 12 (pro) 0 (vides) 11 (an) 12 (orthonormal) 11 (basis) 13 (for) 11 (the) 12 (in) 0 (v) -2 (arian) 0 (t) 9 (subspace) 13 (of) 11 (the) 12 (complete) 13 (eigen) 0 (v) -2 (alue) ks 74 692 p m (sp) 1 (ectrum) 20 (of) ks cmmi10.329 @sf 19 r m (A) s cmr10.329 @sf m (.) 34 (Numerically) -3 (,) ks cmmi10.329 @sf 21 r m (Q) s cmr10.329 @sf 20 r m (is) 20 (a) 19 (m) 0 (uc) 0 (h) 18 (more) 19 (satisfactory) 19 (basis) 20 (than) 20 (the) 19 (eigen) 0 (v) 0 (ectors) 18 (and) ks 74 775 p m (principal) 16 (v) 0 (ectors) 12 (of) ks cmmi10.329 @sf 13 r m (A) s cmr10.329 @sf m (,) 13 (whic) 0 (h) 14 (ma) 0 (y) 11 (w) 0 (ell) 14 (b) 1 (e) 14 (almost) 13 (linearly) 15 (dep) 1 (enden) 1 (t.) 19 (If) 14 (w) 0 (e) 12 (partition) ks cmmi10.329 @sf 14 r m (Q) s cmr10.329 @sf 13 r m (and) s cmmi10.329 @sf 14 r m (T) s cmr10.329 @sf 19 r m (as) s cmmi10.329 @sf 687 889 p m (Q) s cmr10.329 @sf 13 r m (=) 13 (\() ks cmmi10.329 @sf 801 889 p m (Q) s cmr8.300 @sf 837 896 p m (1) s cmmi10.329 @sf 870 889 p m (Q) s cmr8.300 @sf 906 896 p m (2) s cmr10.329 @sf 925 889 p m (\)) s cmmi10.329 @sf 951 889 p m (;) 17 (T) ks cmr10.329 @sf 18 r m (=) s cmex10.300 @sf 1074 817 p m ( ) s cmmi10.329 @sf 1128 862 p m (T) s cmr8.300 @sf 1155 869 p m (11) s cmmi10.329 @sf 1233 862 p m (T) s cmr8.300 @sf 1260 869 p m (12) s cmr10.329 @sf 1149 917 p m (0) s cmmi10.329 @sf 1233 917 p m (T) s cmr8.300 @sf 1260 924 p m (22) s cmex10.300 @sf 1318 817 p m (!) s cmr10.329 @sf 74 1004 p m (then) s cmmi10.329 @sf 878 1087 p m (AQ) s cmr8.300 @sf 948 1094 p m (1) s cmr10.329 @sf 980 1087 p m (=) s cmmi10.329 @sf 13 r m (Q) s cmr8.300 @sf 1064 1094 p m (1) s cmmi10.329 @sf 1084 1087 p m (T) s cmr8.300 @sf 1111 1094 p m (11) s cmmi10.329 @sf 1148 1087 p m (;) s cmr10.329 @sf 74 1190 p m (and) s cmmi10.329 @sf 16 r m (Q) s cmr8.300 @sf 199 1197 p m (1) s cmr10.329 @sf 235 1190 p m (giv) 0 (es) 15 (an) 16 (orthonormal) 15 (basis) 17 (for) 15 (the) 16 (in) 0 (v) -2 (arian) 0 (t) 14 (subspace) 17 (of) ks cmmi10.329 @sf 15 r m (A) s cmr10.329 @sf 16 r m (corresp) 1 (onding) 17 (to) 16 (the) 16 (eigen-) ks 74 1273 p m (v) -2 (alues) 20 (con) 0 (tained) 19 (in) ks cmmi10.329 @sf 21 r m (T) s cmr8.300 @sf 509 1280 p m (11) s cmr10.329 @sf 546 1273 p m (.) 33 (It) 20 (is) 20 (therefore) 19 (a) 20 (common) 19 (requiremen) 0 (t) 19 (to) 19 (reorder) ks cmmi10.329 @sf 20 r m (T) s cmr10.329 @sf 25 r m (so) 20 (that) ks cmmi10.329 @sf 19 r m (T) s cmr8.300 @sf 1842 1280 p m (11) s cmr10.329 @sf 1898 1273 p m (has) s 74 1356 p m (eigen) 0 (v) -2 (alues) 18 (with) 18 (some) 17 (desired) 19 (prop) 1 (ert) 0 (y) -3 (.) 25 (F) -3 (or) 16 (example,) 19 (w) 0 (e) 16 (migh) 0 (t) 17 (require) ks cmmi10.329 @sf 18 r m (T) s cmr8.300 @sf 1560 1363 p m (11) s cmr10.329 @sf 1615 1356 p m (to) 17 (con) 0 (tain) 16 (all) 19 (the) ks 74 1439 p m (stable) 16 (eigen) 0 (v) -2 (alues.) ks 124 1563 p m (Unfortunately) -3 (,) 15 (unless) 16 (w) 0 (e) 14 (kno) 0 (w) 15 (the) 15 (required) 17 (group) 15 (of) 15 (eigen) 0 (v) -2 (alues) 16 (in) 16 (adv) -2 (ance) 15 (and) 16 (accordingly) ks 74 1646 p m (mo) 1 (dify) 23 (the) 22 (standard) 21 (shift) 22 (strategy) 21 (of) 21 (the) ks cmmi10.329 @sf 22 r m (QR) s cmr10.329 @sf 22 r m (algorithm,) s cmmi10.329 @sf 23 r m (T) s cmr8.300 @sf 1298 1653 p m (11) s cmr10.329 @sf 1357 1646 p m (will) 23 (not) 22 (normally) 22 (con) 0 (tain) 21 (the) ks 74 1729 p m (required) 22 (eigen) 0 (v) -2 (alues) 20 (on) 20 (completion) 22 (of) 19 (the) 21 (computation) 20 (of) 20 (the) 21 (Sc) 0 (h) 0 (ur) 18 (factorization.) 35 (W) -3 (e) 20 (m) 0 (ust) ks 74 1812 p m (therefore) 14 (p) 1 (erform) 13 (some) 13 (further) 14 (computation) 14 (to) 13 (reorder) 13 (the) 14 (eigen) 0 (v) -2 (alues.) 19 (Indeed) 15 (in) 15 (most) 13 (applica-) ks 74 1895 p m (tions) 15 (w) 0 (e) 14 (p) 1 (erform) 15 (an) 15 (initial) 17 (Sc) 0 (h) 0 (ur) 14 (factorization) 15 (in) 15 (order) 15 (to) 15 (compute) 15 (the) 15 (eigen) 0 (v) -2 (alues,) 15 (whic) 0 (h) 15 (then) ks 74 1978 p m (giv) 0 (es) 15 (us) 15 (information) 15 (on) 15 (the) 16 (required) 16 (grouping.) ks 124 2103 p m (An) 19 (example) 19 (of) 18 (the) 18 (application) 20 (is) 19 (the) 18 (computation) 19 (of) 18 (matrix) 18 (functions) 19 (via) 18 (the) 19 (blo) 1 (c) 0 (k) 18 (diago-) ks 74 2186 p m (nal) 19 (form) 18 (of) 18 (a) 18 (matrix.) 30 (In) 19 (computing) 19 (the) 18 (blo) 1 (c) 0 (k) 18 (diagonal) 20 (form) 17 (it) 19 (is) 19 (essen) 0 (tial) 18 (to) 18 (include) 21 (\\close") ks 74 2269 p m (eigen) 0 (v) -2 (alues) 15 (in) 16 (the) 16 (same) 14 (diagonal) 16 (blo) 1 (c) 0 (k) 15 ([) 0 (3) ks 892 2269 p m (].) s 124 2394 p m (T) -3 (o) 18 (this) 19 (end,) 20 (Stew) 0 (art) 17 ([) 0 (9) ks 597 2394 p m (]) 19 (has) 18 (describ) 1 (ed) 21 (an) 19 (iterativ) 0 (e) 18 (algorithm) 19 (for) 18 (in) 0 (terc) 0 (hanging) 18 (consecutiv) 0 (e) ks 74 2477 p m (1) s cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (1) 17 (and) 17 (2) ks cmsy10.329 @sf 11 r m (\002) s cmr10.329 @sf 11 r m (2) 16 (blo) 1 (c) 0 (ks) 17 (of) 16 (the) 17 (blo) 1 (c) 0 (k) 16 (triangular) 17 (matrix.) 24 (The) 17 (\014rst) 16 (blo) 1 (c) 0 (k) 17 (is) 17 (used) 17 (to) 16 (determine) 18 (an) ks 74 2560 p m (implici) 1 (t) ks cmmi10.329 @sf 13 r m (QR) s cmr10.329 @sf 11 r m (shift.) 19 (An) 12 (arbitrary) ks cmmi10.329 @sf 11 r m (QR) s cmr10.329 @sf 11 r m (step) 12 (is) 12 (p) 1 (erformed) 12 (on) 11 (b) 1 (oth) 12 (blo) 1 (c) 0 (ks) 11 (to) 11 (eliminate) 13 (the) 12 (uncoupling) ks 74 2643 p m (b) 1 (et) 0 (w) 0 (een) 12 (them.) 20 (Then) 14 (a) 13 (sequence) 15 (of) ks cmmi10.329 @sf 14 r m (QR) s cmr10.329 @sf 13 r m (steps) 14 (using) 14 (the) 14 (previously) 15 (determined) 15 (shift) 15 (is) 14 (p) 1 (erformed) ks 74 2726 p m (on) 15 (b) 1 (oth) 16 (blo) 1 (c) 0 (ks.) 19 (Except) 16 (in) 16 (ill-condi) 1 (tioned) 18 (cases,) 14 (the) 16 (t) 0 (w) -1 (o) 12 (blo) 1 (c) 0 (ks) 15 (will) 17 (in) 0 (terc) 0 (hange) 14 (their) 15 (p) 1 (ositions.) ks 1008 2850 p m (2) s @eop 1 @bop0 cmti10.329 @sf [<601E00E0310070310070708070708070708038384038700038700038E0001FC0001E00001D0000 1C80000E40C00E21E00E11E00E08E00704200703C0070000070000038000038000038000038000 01C00001C00001C00001C0000FE00001E000> 19 32 -3 0 21] 107 @dc [ 8 20 -5 0 14] 58 @dc 1.000000 /cmr6.300 @newfont cmr6.300 @sf [ 9 16 -2 0 15] 49 @dc cmr9.300 @sf [ 9 26 0 0 10] 105 @dc [<8F80F040C020C0308030807000F01FE03FC07F00F000C020C020402060E01F20> 12 16 -1 0 15] 115 @dc [<010040000180C0000380E0000380E00003C1E0000741D000074190000F2398000E2388000E2308 001E170C001C1704001C160400380E06003C0E0700FF3F9F80> 25 16 -1 0 28] 119 @dc [<07E01C38381C700E6006E007E007E007E007E007E0076006700E300C1C3807E0> 16 16 -1 0 19] 111 @dc [ 12 16 -1 0 15] 114 @dc [ 18 26 -1 0 20] 107 @dc [<1E3C71FBE0F1E071E071E070707030701C7007F000700070207070E070C03F80> 16 16 -2 0 19] 97 @dc [<07CFC00C2E001C1E001C0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E00 1C0E001C0E00FC7E00> 18 16 -1 0 21] 117 @dc [ 18 23 -1 7 21] 112 @dc [<07E00C18380830046000E000E000E000E000FFFCE00C600C701830181C3007C0> 14 16 -1 0 17] 101 @dc [<07CFC01C2E00381E00700E00600E00E00E00E00E00E00E00E00E00E00E00E00E00700E00300E00 380E000C3E0003CE00000E00000E00000E00000E00000E00000E00000E00000E00000E00007E00> 18 26 -1 0 21] 100 @dc [ 18 16 -1 0 21] 110 @dc [<780000E40000E20000E1000001000001000000800000800001C00001C00001C00003A00003A000 0790000710000710000E08000E08001E0C001C04001C04003C0600FF1F80> 17 23 -1 7 20] 121 @dc [ 9 26 0 0 10] 108 @dc [ 29 16 -1 0 32] 109 @dc [<07E00C18380830047000E000E000E000E000E000E00070003008381C0C1C07F8> 14 16 -1 0 17] 99 @dc [<87E0D818E004C006C00280038003800300030007000E003E07FC1FF83FE07E007000E000C002C0 02C002C0066006200E18360FC2> 16 26 -2 0 21] 83 @dc [<07E0381C6006C003C003C003C003600F1FFE3FFC3FF07000600020002FC0186030307038703870 387038303018730FCE> 16 24 -1 8 19] 103 @dc [<7FE00E000E000E000E000E000E000E000E000E000E000E000E000E000E00FFC00E000E000E000E 000E000E000E100738031801F0> 13 26 0 0 12] 102 @dc [<8040202010101070F0F060> 4 11 -3 7 11] 44 @dc [<60F0F060> 4 4 -3 0 11] 46 @dc [ 10 2 0 -7 13] 45 @dc [<078018603030703860186018E01CE01CE01CE01CE01CE01CE01CE01CE01CE01CE01CE01C601860 183030303018600780> 14 24 -2 0 19] 48 @dc [<0F8030E0403040388018E01CE01C401C001C001C001800382030386027C0200020002000200020 003FC03FE03FF03018> 14 24 -2 0 19] 53 @dc [<0FC0186020106008C00CC00CC00CC00C401C603830F819F007C00F803E603C3070106018601860 182018303018600780> 14 24 -2 0 19] 56 @dc [<07FF007000700070007000700070FFFFC070407020703070107008700C70047002700370017000 F000F0007000300030> 16 24 -1 0 19] 52 @dc [ 14 24 -2 0 19] 50 @dc [ 12 24 -3 0 19] 49 @dc [<00800001C00001C00001C00003A00003A0000790000710000710000E08000E08001E0C001C0400 1C04003C0600FF1F80> 17 16 -1 0 20] 118 @dc cmr6.300 @sf [ 11 16 -1 0 15] 50 @dc cmr9.300 @sf [<1E00618041C0E0E0E0E040E000E000E000E000E000E000E000E000E000E000E000E000E000E000 E000E000E000E000E000E01FFC> 14 26 -3 0 20] 74 @dc [<1F002180604070607030003800180018001C0F9C185C303C603C601CE01CE01CE01CE018E01860 187030303018600780> 14 24 -2 0 19] 57 @dc [<0F8030E040704038C03CE03CE03C003C003C0038007000E00F8000C00060007000700078203870 386078207010E00F80> 14 24 -2 0 19] 51 @dc [ 5 11 -3 -15 11] 39 @dc 1 @bop1 cmbx10.432 @sf 410 67 p save [ 47 41 -3 0 54] 78 @pc restore 54 r m (um) -1 (eri) -1 (cal) 20 (Considerations) 22 (in) 22 (Computi) -1 (ng) ks 709 124 p m (In) -1 (v) -3 (arian) -1 (t) 21 (Subspaces) ks cmti10.329 @sf 836 271 p m (Jack) 17 (J.) ks 16 r save [ 31 31 -3 0 34] 68 @pc restore 34 r m (ongarr) -1 (a) ks cmr8.300 @sf 1183 256 p m (1) s cmr10.329 @sf 687 349 p m (Departmen) 0 (t) 14 (of) 14 (Computer) 15 (Science) ks 786 405 p m (Univ) 0 (ersit) 0 (y) 14 (of) 14 (T) -3 (ennessee) ks 760 462 p m (Kno) 0 (xville,) 16 (TN) 15 (37996-1301) ks 983 518 p m (and) s 714 574 p m (Mathematical) 15 (Sciences) 17 (Section) ks 701 631 p m (Oak) 16 (Ridge) 16 (National) 16 (Lab) 1 (oratory) ks 749 687 p m (Oak) 15 (Ridge,) 16 (TN) 15 (37831-8083) ks cmti10.329 @sf 842 810 p m (Sven) s 15 r save [ 35 31 -3 0 34] 72 @pc restore 34 r m (ammarling) s cmr10.329 @sf 679 887 p m (Numerical) 16 (Algorithms) 16 (Group) 15 (Ltd.) ks 666 944 p m (Wilkinson) 17 (House,) 15 (Jordan) 15 (Hill) 17 (Road) ks 664 1000 p m (Oxford) 15 (O) 0 (X2) 14 (8DR,) 15 (United) 16 (Kingdom) ks cmti10.329 @sf 808 1123 p m (James) s 16 r save [ 35 31 -3 0 34] 72 @pc restore 34 r m (.) s 16 r save [<04000800000006000800000006000C00000007001E00000007001E00000007801F00000007801F 0000000FC01E8000000F601E8000000F201E4000000F101E4000000F101E2000000F081E200000 0F081E1000000F041E0800000F041E0800000F021E0400000F021E0400000F011E0200000F009E 0200000F009E0100000F005E0100000F005E0080000F003E00C0000F003E0040000F003C002000 0F003C0020000F003C0010000F003C0010001F003C0018001F003E001C00FFF1FFC0FF80> 41 32 -9 1 45] 87 @pc restore 45 r m (ilkinson) s cmr8.300 @sf 1212 1108 p m (2) s cmr10.329 @sf 825 1250 p m (Septem) 0 (b) 1 (er) 15 (26,) 14 (1991) ks cmti10.329 @sf 224 1398 p m (A) 0 (bstr) -2 (act:) ks cmr10.329 @sf 49 r m (This) 18 (pap) 1 (er) 18 (describ) 1 (es) 19 (t) 0 (w) -1 (o) 15 (metho) 1 (ds) 18 (for) 17 (computing) 18 (the) 17 (in) 0 (v) -2 (arian) 0 (t) 16 (sub-) ks 224 1454 p m (space) 12 (of) 11 (a) 11 (matrix.) 18 (The) 12 (\014rst) 11 (metho) 1 (d) 11 (in) 0 (v) 0 (olv) 0 (es) 9 (using) 12 (transformations) 10 (to) 11 (in) 0 (terc) 0 (hange) ks 224 1511 p m (the) 14 (eigen) 0 (v) -2 (alues.) 19 (The) 14 (matrix) 14 (is) 14 (assumed) 14 (to) 13 (b) 1 (e) 15 (in) 14 (Sc) 0 (h) 0 (ur) 12 (form) 14 (and) 13 (transformations) ks 224 1567 p m (are) 16 (applied) 19 (to) 15 (in) 0 (terc) 0 (hange) 15 (neigh) 0 (b) 1 (oring) 17 (blo) 1 (c) 0 (ks.) 23 (The) 17 (blo) 1 (c) 0 (ks) 16 (can) 16 (b) 1 (e) 17 (either) 18 (one) 16 (b) 0 (y) ks 224 1624 p m (one) 18 (or) 18 (t) 0 (w) -1 (o) 15 (b) 0 (y) 17 (t) 0 (w) -1 (o.) 26 (The) 19 (second) 18 (metho) 1 (d) 18 (in) 0 (v) 0 (olv) 0 (es) 16 (the) 18 (construction) 18 (of) 18 (an) 18 (in) 0 (v) -2 (ari-) ks 224 1680 p m (an) 0 (t) 18 (subspace) 20 (b) 0 (y) 18 (a) 20 (direct) 20 (computation) 19 (of) 19 (the) 20 (v) 0 (ectors,) 18 (rather) 19 (than) 19 (b) 0 (y) 19 (applying) ks 224 1737 p m (transformations) 14 (to) 14 (mo) 0 (v) 0 (e) 13 (the) 15 (desired) 17 (eigen) 0 (v) -2 (alues) 15 (to) 14 (the) 16 (top) 14 (of) 15 (the) 15 (matrix.) ks cmbx10.432 @sf 74 1921 p save [<7FFFFE7FFFFE7FFFFE00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00 00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00 00FE0000FE0000FE0000FE0000FE0000FE00F8FE00FFFE00FFFE0007FE00007E00001E00000E00> 23 39 -5 0 34] 49 @pc restore 177 1921 p m (In) -1 (tro) 2 (duction) ks cmr10.329 @sf 74 2091 p m (In) 16 (this) 16 (pap) 1 (er) 16 (w) 0 (e) 14 (consider) 17 (the) 15 (computation) 16 (of) 15 (the) 15 (in) 0 (v) -2 (arian) 0 (t) 14 (subspace) 16 (of) 15 (a) 15 (matrix) 15 (corresp) 1 (onding) ks 74 2174 p m (to) 15 (some) 15 (giv) 0 (en) 14 (group) 15 (of) 15 (eigen) 0 (v) -2 (alues.) ks 124 2299 p m (P) 0 (oten) -1 (tially) -3 (,) 10 (the) 11 (Sc) 0 (h) 0 (ur) 10 (factorization) 10 (pro) 0 (vides) 11 (a) 11 (metho) 1 (d) 11 (for) 10 (computing) 12 (suc) 0 (h) 10 (in) 0 (v) -2 (arian) 0 (t) 9 (subspaces,) ks 74 2382 p m (with) 18 (the) 18 (imp) 1 (ortan) 0 (t) 16 (n) 0 (umerical) 18 (prop) 1 (ert) 0 (y) 16 (that) 17 (it) 18 (pro) 0 (vides) 17 (an) 18 (orthonormal) 16 (basis) 18 (for) 17 (suc) 0 (h) 17 (spaces.) ks 74 2421 p 756 2 ru cmr6.300 @sf 126 2449 p m (1) s cmr9.300 @sf 143 2464 p save [<03FFE000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C00 00001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000801C0080801C 0080801C0080C01C0180401C0100401C0100701C07007FFFFF00> 25 26 -1 0 28] 84 @pc restore 28 r m (his) 14 (w) 0 (ork) 13 (w) 0 (as) 12 (supp) 1 (orted) 15 (in) 14 (part) ks 14 r save [<10F0001B0C001E07001C03001C03801C01C01C01C01C01C01C01C01C01C01C01C01C01801C0380 1E07001F0E001CF8001C00001C00001C00001C00001C00001C00001C00001C00001C0000FC0000> 18 26 -1 0 21] 98 @pc restore 21 r m (y) 13 (the) ks 13 r save [ 26 26 -1 0 29] 65 @pc restore 29 r m (pplied) s 15 r save [ 30 26 -2 0 35] 77 @pc restore 35 r m (athematical) 16 (Sciences) 15 (su) ks save [<10F0001B0C001E07001C03001C03801C01C01C01C01C01C01C01C01C01C01C01C01C01801C0380 1E07001F0E001CF8001C00001C00001C00001C00001C00001C00001C00001C00001C0000FC0000> 18 26 -1 0 21] 98 @pc restore 21 r m (program) 15 (of) 13 (the) ks 14 r save [<007F000001C1C000070070000E0038001C001C003C001E0038000E0078000F0078000F00F00007 80F0000780F0000780F0000780F0000780F0000780F0000780F00007807000070078000F003800 0E003C001E001C001C000E0038000700700001C1C000007F0000> 25 26 -2 0 30] 79 @pc restore 30 r save [<7F8FE3FC0E0380E00E0380E00E0380E00E0380E00E0380E00E0380E00E0380E00E0380E00E0380 E00E0380E00E0380E00E0380E00E0380E00E0380E0FFFFFFE00E0380000E0380000E0380000E03 80000E0380000E038040060380E00703C06001C17040007E1F80> 30 26 0 0 32] 14 @pc restore 32 r m (ce) 13 (of) ks 13 r save [ 22 26 -2 0 26] 69 @pc restore 26 r m (nergy) s 74 2510 p save [ 25 26 -2 0 28] 82 @pc restore 28 r m (esearc) 0 (h,) ks 11 r save [<003E0000C1800180400300400700200600200E00100E00100E00100E00100E00100E00100E0010 0E00100E00100E00100E00100E00100E00100E00100E00100E00100E00100E00100E0038FFE1FF> 24 26 -2 0 29] 85 @pc restore 29 r m (.S.) s 10 r save [ 24 26 -2 0 29] 68 @pc restore 29 r m (epartmen) 0 (t) 11 (of) ks 11 r save [ 22 26 -2 0 26] 69 @pc restore 26 r m (nergy) -2 (,) 11 (under) ks 11 r save [<003F0001C0C00300300E00081C00081C0004380004780002700002700002F00000F00000F00000 F00000F00000F000007000027000027800023800061C00061C000E0E001E03002E01C0C6003F02> 23 26 -2 0 28] 67 @pc restore 28 r m (on) 0 (tract) ks 10 r save [ 24 26 -2 0 29] 68 @pc restore 29 r save [ 22 26 -2 0 26] 69 @pc restore 26 r m (-) s save [ 26 26 -1 0 29] 65 @pc restore 29 r save [<003F0001C0C00300300E00081C00081C0004380004780002700002700002F00000F00000F00000 F00000F00000F000007000027000027800023800061C00061C000E0E001E03002E01C0C6003F02> 23 26 -2 0 28] 67 @pc restore 28 r m (05-84) s save [<007F000001C1C000070070000E0038001C001C003C001E0038000E0078000F0078000F00F00007 80F0000780F0000780F0000780F0000780F0000780F0000780F00007807000070078000F003800 0E003C001E001C001C000E0038000700700001C1C000007F0000> 25 26 -2 0 30] 79 @pc restore 30 r save [ 25 26 -2 0 28] 82 @pc restore 28 r m (21400,) 12 (and) 11 (in) 12 (part) ks 11 r save [<10F0001B0C001E07001C03001C03801C01C01C01C01C01C01C01C01C01C01C01C01C01801C0380 1E07001F0E001CF8001C00001C00001C00001C00001C00001C00001C00001C00001C0000FC0000> 18 26 -1 0 21] 98 @pc restore 21 r m (y) 11 (the) 11 (Science) ks 12 r save [ 26 26 -1 0 29] 65 @pc restore 29 r m (lliance,) s 74 2555 p m (a) 13 (state-supp) 1 (orted) 15 (program) 14 (at) 12 (the) ks 14 r save [<003E0000C1800180400300400700200600200E00100E00100E00100E00100E00100E00100E0010 0E00100E00100E00100E00100E00100E00100E00100E00100E00100E00100E00100E0038FFE1FF> 24 26 -2 0 29] 85 @pc restore 29 r m (niv) 0 (ersit) 0 (y) 12 (of) ks 13 r save [<03FFE000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C00 00001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000801C0080801C 0080801C0080C01C0180401C0100401C0100701C07007FFFFF00> 25 26 -1 0 28] 84 @pc restore 26 r m (ennessee.) s cmr6.300 @sf 126 2586 p m (2) s cmr9.300 @sf 143 2601 p save [<0020004000003000C000007000E000007000E000007801E000007801E00000E801D00000E40390 0000E403900001C403880001C403880001C20708000382070400038207040003810E040007010E 020007010E020007009C020007009C02000E009C01000E007801000E007801001C007800801C00 7800803C007801C0FF83FF0FF0> 36 26 -1 0 39] 87 @pc restore 37 r m (ork) 16 (on) 17 (this) 18 (pap) 1 (er) 17 (w) 0 (as) 16 (started) 17 (as) 16 (a) ks 17 r save [<7E00E300E18041C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001 C001C001C00FC0000000000000000000000000018003C003C00180> 10 33 2 7 12] 106 @pc restore 12 r m (oin) 0 (t) 16 (e) ks save [<7F0FF00E07000E07000E07000E07000E07000E07000E07000E07000E07000E07000E07000E0700 0E07000E0700FFFFF00E07000E07000E07000E07000E07000E070006070407078E0183C600FC7C> 23 26 0 0 22] 11 @pc restore 22 r m (ort) 17 (with) 17 (James) ks 17 r save [ 24 26 -2 0 29] 72 @pc restore 29 r m (.) s 16 r save [<0020004000003000C000007000E000007000E000007801E000007801E00000E801D00000E40390 0000E403900001C403880001C403880001C20708000382070400038207040003810E040007010E 020007010E020007009C020007009C02000E009C01000E007801000E007801001C007800801C00 7800803C007801C0FF83FF0FF0> 36 26 -1 0 39] 87 @pc restore 39 r m (ilkin) 1 (son) 19 (in) 17 (1983.) ks 28 r save [ 26 26 -1 0 29] 65 @pc restore 29 r m (fter) 16 (Jim's) 17 (un) 0 (timely) ks 74 2647 p m (death,) 11 (the) 11 (w) 0 (ork) 9 (la) 0 (y) 11 (un) ks save [<7F0FE00E07000E07000E07000E07000E07000E07000E07000E07000E07000E07000E07000E0700 0E07000E0700FFFF000E00000E00000E00000E00000E00000E020006070007030001820000FC00> 19 26 0 0 21] 12 @pc restore 21 r m (nished) 13 (for) 10 (a) 11 (n) 0 (um) ks save [<10F0001B0C001E07001C03001C03801C01C01C01C01C01C01C01C01C01C01C01C01C01801C0380 1E07001F0E001CF8001C00001C00001C00001C00001C00001C00001C00001C00001C0000FC0000> 18 26 -1 0 21] 98 @pc restore 22 r m (er) 9 (of) 10 (y) 0 (ears.) ks 16 r save [<03FFE000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C00 00001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000801C0080801C 0080801C0080C01C0180401C0100401C0100701C07007FFFFF00> 25 26 -1 0 28] 84 @pc restore 28 r m (he) 10 (authors) 11 (recen) 0 (tly) 11 (came) 11 (across) 11 (parts) 11 (of) 10 (Jim's) 10 (handwritten) ks 74 2692 p m (man) 0 (uscript) 14 (and) 14 (completed) 14 (the) 13 (w) 0 (ork.) ks cmr10.329 @sf 1008 2850 p m (1) s @eop @end .