{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 35 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "examples" {TEXT -1 8 "Examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 2 "" 1 "" {TEXT -1 79 "Warning, the protected names norm and trace have bee n\nredefined and unprotected" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := matrix(3,3, [1,-3,3,3,-5,3,6,-6,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"!\"$\"\"$7%F,!\"&F,7%\"\"'! \"'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e := eigenvalue s(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG6%\"\"%!\"#F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "v := [eigenvectors(A)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7$7%!\"#\"\"#<$-%'vectorG6#7%\" \"\"F.\"\"!-F+6#7%!\"\"F/F.7%\"\"%F.<#-F+6#7%F.F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "v[1][1]; # The first eigenvalue" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "v[1][2]; # Its multiplicity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "v[1][3]; # Its ei genvectors" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7%\"\"\"F( \"\"!-F%6#7%!\"\"F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v[ 2][1]; # The second eigenvalue" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "v[2][2]; # Its multipl icity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Here is a numerical example (A is a symmetric Toeplitz ma trix) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A := toeplitz([1.0 ,2.0,3.0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%$ \"#5!\"\"$\"#?F,$\"#IF,7%F-F*F-7%F/F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$! +++++?!\"*$!+(=@c,(!#5$\"+>@c,dF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%7 %$!+$=@c,(!#5\"\"\"<#-%'vectorG6#7%$!+I$H^k$F&$\"+'*4!*o&)F&$!+M$H^k$F &7%$\"+:@c,d!\"*F'<#-F*6#7%$\"+/!G\"fgF&$\"+[8*\\:&F&$\"+,!G\"fgF&7%$! +-+++?F6F'<#-F*6#7%$\"+6y1rqF&$\"$Z\"!#7$!+4y1rqF&" }}}{PARA 0 "" 0 " " {TEXT -1 52 "Here is the same example done with exact arithmetic " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A := toeplitz([1,2,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#\"\"$ 7%F+F*F+7%F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenva lues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"#,&#\"\"&\"\"#\"\"\"*$ \"#T#F(F'F+,&F%F(F)#!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%!\"#\"\"\"<# -%'vectorG6#7%!\"\"\"\"!F%7%,&#\"\"&\"\"#F%*$\"#T#F%F1F4F%<#-F(6#7%F%, &#!\"$\"\"%F%F2#F%F " 0 "" {MPLTEXT 1 0 28 "eigenvectors(A, 'implicit'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%-%'RootOfG6#,(*$%#_ZG\"\"#\"\"\" F)!\"&!\"%F+F+<#-%'vectorG6#7%,&F$#F+\"\"%#!\"\"F5F+F+F37%!\"#F+<#-F06 #7%F7\"\"!F+" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Here is an example of a \+ symbolic matrix " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "T := toe plitz([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7 %7%%\"aG%\"bG%\"cG7%F+F*F+7%F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvalues(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,& %\"aG\"\"\"%\"cG!\"\",(F$F%F&#F%\"\"#*$,&*$F&F*F%*$%\"bGF*\"\")F)F),(F $F%F&F)F+#F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eigenvect ors(T,'implicit');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%-%'RootOfG6#,, *$%#_ZG\"\"#\"\"\"*&,&%\"aG!\"#%\"cG!\"\"F+F)F+F+*$F.F*F+*&F0F+F.F+F+* $%\"bGF*F/F+<#-%'vectorG6#7%F+,$*&,(F0F+F$F1F.F+F+F5F1F1F+7%,&F.F+F0F1 F+<#-F86#7%F+\"\"!F1" }}}{PARA 0 "" 0 "" {TEXT -1 75 "The sqrt(2) in t he following example makes the computation more difficult. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "T := toeplitz([sqrt(2),a,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7%7%*$\"\"##\"\"\"F+ %\"aG\"\"$7%F.F*F.7%F/F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%,(#\"\"$\"\" #\"\"\"*$F'#F(F'F(*$,&\"\"*F(*$%\"aGF'\"\")F*F*F(<#-%'vectorG6#7%F(,$* &,&F%F(F+#!\"\"F'F(F/F:F:F(7%,(F%F(F)F(F+F9F(<#-F36#7%F(,$*&,&F%F(F+F* F(F/F:F:F(7%,&F)F(!\"$F(F(<#-F36#7%F:\"\"!F(" }}}{PARA 0 "" 0 "" {TEXT -1 137 "This final example shows a case where the algebraic mult iplicity of the eigenvalue u is 4 but the dimension of the eigenspace \+ is only 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "B := matrix(4 ,4,[0,1,0,0,-u^2,2*u,0,0,-u*s,s,u,0,-u*t,t,0,u]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7&7&\"\"!\"\"\"F*F*7&,$*$%\"uG\"\"#! \"\",$F/F0F*F*7&,$*&F/F+%\"sGF+F1F6F/F*7&,$*&F/F+%\"tGF+F1F:F*F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%\"uG\"\"%<%-%'vectorG6#7&*$F$!\"\"\"\"\" \"\"!F.-F(6#7&F.F.F.F--F(6#7&F.F.F-F." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }