{VERSION 4 0 "Mac OS X" "4.0" } {USTYLETAB {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ord ered List 5" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Lef t Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Norma l259" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 4 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal258" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "C ourier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Wid th" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 6 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Exercis e" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 -12 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal257" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 } {CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small" -1 201 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 202 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "T imes" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "T imes" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels " -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading " -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal " -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple N ame" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D O utput" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dict ionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 } {CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 4 4 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 203 28 "Eigenvalues and Eigenvect ors" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 35 "Worksheet written by Russell Blyth." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "restart: with(LinearAlgebra): with(plots): with(plottools):" } }}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 38 "Computing Eigenvalues and Eige nvectors" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 87 "Let's investigate a \+ \"random\" 3 x 3 matrix using the algorithm developed in the text. " } }{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "randomize(19):\n" }{MPLTEXT 1 0 48 "A := RandomMatrix(3,3,gene rator=rand(-5..5));\nR" }{MPLTEXT 1 0 9 "ank(A);\n" }{MPLTEXT 1 0 24 " I3 := IdentityMatrix(3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 60 "Comp ute the characteristic polynomial, which is det(A-t*I3)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "chpoly := Determinant(A - t*I3);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 28 "Set equal to zero and solve." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "AEvals := solve(chpoly = 0 ,t);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 45 "In this case we have f ound three eigenvalues." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 29 "Next compute the eigenspaces." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 50 "AEspace1 := LinearSolve(A - AEvals[1]*I3,<0, 0,0>);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 37 "And then extract the b asis vectors(s)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "AEbasis1 := [subs(_t[3]=1,AEspace1)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 26 "For the second eigenvalue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "AEspace2 := LinearSolve(A - AEvals[2]*I3,<0,0,0>);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "AEbasis2 := [subs(_t0[2]=1,AEspace2 )];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 29 "And for the third eigenva lue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "AEspace3 := LinearS olve(A - AEvals[3]*I3,<0,0,0>);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "AEbasis3 := [subs(_t1[3]=1, AEspace3)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 56 "Let's check that these vectors are in fa ct eigenvectors:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A.AEbas is1[1];\n" }{MPLTEXT 1 0 25 "AEvals[1]* AEbasis1[1];\n" }{MPLTEXT 1 0 16 "A.AEbasis2[1];\n" }{MPLTEXT 1 0 25 "AEvals[2]* AEbasis2[1];\n" } {MPLTEXT 1 0 16 "A.AEbasis3[1];\n" }{MPLTEXT 1 0 25 "AEvals[3]* AEbasi s3[1];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 101 "Now compute the cha nge of coordinate matrix Q; it has columns which are the eigenvectors \+ we computed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Q := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 115 "Compute the product Qinv * A * Q, which should give us the diago nal matrix D with the eigenvalues on the diagonal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "DD := Q^(-1).A.Q;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 204 163 "Let's see the geometric effect of these eigenvalues a nd eigenvectors. Rotate the plots if necessary to see the eigenvector \+ and its image under multiplication by A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "x1 := line([0,0,0], convert(AEbasis1[1],list),color = red, thickness = 5):\n" }{MPLTEXT 1 0 85 "lambdax1 := line([0,0,0], c onvert(A.AEbasis1[1],list),color = blue, thickness = 1):\n" }{MPLTEXT 1 0 38 "display(\{x1,lambdax1\}, axes=normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "x2 := line([0,0,0], convert(AEbasis2[1],list),c olor = red, thickness = 5):\n" }{MPLTEXT 1 0 85 "lambdax2 := line([0,0 ,0], convert(A.AEbasis2[1],list),color = blue, thickness = 1):\n" } {MPLTEXT 1 0 38 "display(\{x2,lambdax2\}, axes=normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "x3 := line([0,0,0], convert(AEbasis 3[1],list),color = red, thickness = 5):\n" }{MPLTEXT 1 0 85 "lambdax3 \+ := line([0,0,0], convert(A.AEbasis3[1],list),color = blue, thickness = 1):\n" }{MPLTEXT 1 0 38 "display(\{x3,lambdax3\}, axes=normal);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 204 20 "See them all at once" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "display(\{x1,lambdax1,x2,lambdax2,x 3,lambdax3\}, axes=normal, scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 71 "Maple has commands for computing eigenvalues and e igenvectors directly." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ei genvectors(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 213 "We get a vect or which contains the eigenvalues and a matrix which has as its column s eigenvectors corresponding to the eigenvalues. This data can be extr acted by first placing the data into a sequence, as follows:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evA := [Eigenvectors(A)];" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 12 "For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evA[1][1];\n" }{MPLTEXT 1 0 46 "evA[1][3]; Column(evA[2],1); Column(evA[2],3);" }}}{SECT 1 {PARA 209 "" 0 "" {TEXT 203 8 "Exercise" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 38 "1) Repe at the above for the matrix B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B := Matrix([-1,0,5,0,4,0,5,0,-1]);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }