{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 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 "" -1 256 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "Geneva" 1 18 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 258 "Geneva" 1 12 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 266 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }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 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 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 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Monaco" 1 9 255 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE " " -1 -1 "Geneva" 1 18 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 256 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 256 2 "\011\011" }{TEXT 257 32 "\011Quadratic Forms and Eigenvalues" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 258 368 "This worksheet explores the relationship between symmet ric matrices and\nquadratic forms. It shows how the eigenvalues of suc h a matrix relate to\nthe geometric character of the graph of the quad ratic form. It also discusses\nin the context of an example how the ei genvectors of the symmetric matrix\ndetermine a rotation of coordinate s making the quadratic form diagonal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 " Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warni ng, new definition for trace" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 259 57 "The symmetric matrix corresponding to 2 x^2 + 8 x y + y^2" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := matrix(2,2,[2,4,4,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7$7$\"\"#\"\"%7$F+\" \"\"" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 260 24 "A general vector in R^ 2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v := vector(2,[x,y]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7$%\"xG%\"yG" }}} {EXCHG {PARA 256 "" 0 "" {TEXT 261 58 "Verifying that A corresponds to this quadratic expression." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "expn 1 := expand(dotprod(v,evalm(A &* v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expn1G,(*$%\"xG\"\"#F(*&F'\"\"\"%\"yGF*\"\")*$F+F(F*" }}} {EXCHG {PARA 256 "" 0 "" {TEXT 262 199 "Compute the eigenvalues (retur ned as a list below) and eigenvectors\n(returned in the columns of A_ evects) of the matrix A.\nRecall a nonzero v is eigenvector with eigen value lambda if A v = lambda v." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " evalf(Eigenvals(A,A_evects));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VE CTORG6#7$$!+w)G6`#!\"*$\"+t)G6`&F)" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 263 52 "The columns here are the corresponding eigenvectors." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "print(A_evects);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7$$\"+Jc-=m!#5$!+g " 0 "" {MPLTEXT 1 0 70 "for j from 1 to 2 do\n \011print(sqrt(sum((A_evects[i,j])^2,i=1..2)));\n\011od;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+++++5!\"*" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 265 93 "(printf(`Len gth of column %d is %g\\n`,j,sqrt(sum((A_evects[i,j])^2,i=1..2))) woul d be nicer.)" }}{PARA 256 "" 0 "" {TEXT 266 206 "The eigenvectors of a symmetric matrix can always be chosen to be an orthonormal\nbasis for R^n, and this reflects itself in the identity below.\nThis identity a lso means A_evects is the matrix of a rotation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "print(evalm(inverse(A_evects) - transpose(A_evects))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7$\"\"!F(F'" }}} {EXCHG {PARA 256 "" 0 "" {TEXT 267 308 "In general, if an n x n matri x A has n independent eigenvectors in the columns of\nA, then inverse (A_evects) &* A &* A_evects will be a diagonal matrix.\n\nIf we trans form coordinates by [x_new y_new] = A_evects &* [x y], then B will\nbe the symmetric matrix corresponding to expn1 in coordinates x_new,y_ne w." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B := evalm(transpose(A_evects ) &* A &* A_evects);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRI XG6#7$7$$!+v)G6`#!\"*$\"\"$F,7$$\"\"#F,$\"+u)G6`&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v_new := vector(2,[x_new,y_new]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&v_newG-%'VECTORG6#7$%&x_newG%&y_new G" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 268 43 "The same expression in th e new coordinates." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "expn2 := expa nd(dotprod(v_new,evalm(B &* v_new)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expn2G,(*$%&x_newG\"\"#$!+v)G6`#!\"**&F'\"\"\"%&y_newGF-$\"\"&F+ *$F.F($\"+u)G6`&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot3 d(expn1,x=-5..5,y=-5..5,axes=boxed,shading=zhue,view=-10..10);" }} {PARA 13 "" 1 "" {GLPLOT3D 308 308 308 {PLOTDATA 3 "6'-%%GRIDG6%;$!\"& \"\"!$\"\"&F)F&7;7;$\"$v#F)$\"1yxxxFSVD!#8$\"1xxxxxFSBF2$\"1++++]iS@F2 $\"1WWWWWWW>F2$\"16666ht^\" F2$\"1++++]i:5F2$\"1xxxxxx-%)!#9$\"1xxxxx-%o'FE$\"1**************\\FE$ \"1WWWWWp]LFE$\"1566666OF2$\"1AAAAAZ`j\"FE$!1AAAAAZ.IFE$!1xxxxxFSVFE$!156666OUcFE$!1 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