{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 "Normal260" -1 203 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 "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 "Ord ered List 3" -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 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 "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 "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 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 205 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 Pr inted Output" -1 6 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 "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" -1 206 1 {CSTYLE "" -1 -1 "Ti mes" 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 "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 "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 Width" -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 "Exercise" -1 207 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" -1 208 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 "Order ed List 2" -1 209 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 "Tim es" 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 Inp ut" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Mat h Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help F ixed" -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 Ti tle" -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 "Times" 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 "Times" 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 Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary 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 "Ti mes" 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 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Tim es" 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 "Tim es" 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 Ital ic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 210 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 }{PSTYLE "" -1 211 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 }{PSTYLE "" -1 212 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 41 "Diagonalizability and Inv ariant Subspaces" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 17 "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 17 "Diagonalizability" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 289 "If an nxn matrix has n distinct eigenva lues, then we know that it is diagonalizable because each eigenvector \+ has a corresponding eigenspace of dimension at least one (actually, of dimension exactly one in this case), and a collection of basis vector s for these eigenspaces is independent. " }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A := Matrix([[-3, 4, 0, -5], [0, 1, 0, -5], [-5, 4, 2, -5], [0, 0, 0, -4]]);\n" }{MPLTEXT 1 0 32 "CharacteristicPolynomial(A,t);\n" }{MPLTEXT 1 0 23 "evA := Eig envectors(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 232 "What if some e igenvalues have algebraic multiplicity greater than one? Then whether \+ or not the matrix is diagonalizable depends on the existence of enough independent eigenvectors in each eigenspace. Consider the following e xamples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "B := Matrix([[- 4, 6, 0, -6], [0, 2, 0, -6], [-6, 6, 2, -6], [0, 0, 0, -4]]);\n" } {MPLTEXT 1 0 32 "CharacteristicPolynomial(B,t);\n" }{MPLTEXT 1 0 16 "E igenvectors(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 25 "Thus B is dia gonalizable." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "C := Matrix ([[-4, 6, 0, -6], [0, 2, 0, -6], [-6, 7, 2, -7], [0, 0, 0, -4]]);\n" } {MPLTEXT 1 0 33 "CharacteristicPolynomial(C,t);\nE" }{MPLTEXT 1 0 17 " igenvectors(C);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 92 "But C is no t diagonalizable, since the eigenspace for the eigenvalue 2 has dimens ion only 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "E := Matrix( [[-4, 6, 0, -5], [0, 2, 0, -6], [-6, 7, 2, -6], [0, 0, 0, -4]]);\n" } {MPLTEXT 1 0 33 "CharacteristicPolynomial(E,t);\nE" }{MPLTEXT 1 0 15 " igenvectors(E);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 89 "E has deficit s of eigenvectors for both eigenvalues, and thus is also not diagonali zable." }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 203 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT 200 68 "1. What do you conclude about the diagonalizabil ity of the matrix G?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "G := Matrix([[2, 1, 0, 0], [0, 2, 0, 0], [1, 1, 2, -1], [0, 0, 0, 2]]);" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 252 "If a matrix is diagonalizable then it is possible to easily compute (high) powers of that matrix. A pply this process to the matrix A from above. First we construct the d iagonalized matrix Da and the change of coordinate matrix Qa s.t. Da = inv(Qa)*A*Qa" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Qa := evA[ 2];\n" }{MPLTEXT 1 0 21 "Da := Qa^(-1).A.Qa;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 86 "Now to compute A to the power n, we can compute usin g the diagonal matrix, using A = " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mrow(Typesetting :-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\" , lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \+ \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1 \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0 ,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting: -mi(\"Qa\"), Typesetting:-mo(\"⁢\", form = \"infix\", f ence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \" 0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting :-mi(\"Da\"), Typesetting:-mo(\"⁢\", form = \"infix\", \+ fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \+ \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"inv\"), Typesetting:-mo(\"⁡ \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Com ment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\" , fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mrow(Typesetting:-mi(\"Qa\")), Typesetting:-mo (\")\", form = \"postfix\", fence = \"true\", separator = \"false\", l space = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \+ \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1 \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0 ,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Types etting:-mi(\"\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \+ \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \+ \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \")), Typesetting:-mi(\"n\"), superscriptshift = \"0\"), Typesetting:- mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6#Q!F'-I%msupGF$6%-F#6%-I#moGF$63Q\"(F'/%%formGQ'prefixF'/%&fenc eGQ%trueF'/%*separatorGQ&falseF'/%'lspaceGQ.thinmathspaceF'/%'rspaceGF C/%)stretchyGF=/%*symmetricGF@/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F' /%(largeopGF@/%.movablelimitsGF@/%'accentGF@/%0font_style_nameGQ+2D~Co mmentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255 ,255]F'-F#6(-F,6#Q#QaF'-F563Q1⁢F'/F9Q&infixF'/F/FB Q$0emF'/FEFho/FGF@FHFJFMFPFRFTFVFYFfnFin-F,6#Q#DaF'Fao-F#6&-F,6#Q$invF '-F563Q0⁡F'FdoFfoF>FgoFioFjoFHFJFMFPFRFTFVFYFfnFin-F#6%F 4-F#6#F^o-F563Q\")F'/F9Q(postfixF'F;F>FA/FEQ2verythinmathspaceF'FFFHFJ FMFPFRFTFVFYFfnFinF+F+Fjp-F,6#Q\"nF'/%1superscriptshiftGQ\"0F'F+" } {TEXT 204 4 " =\n" }{TEXT 204 7 "Qa * (D" }{XPPEDIT 18 0 "Typesetting: -mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mrow(Types etting:-mi(\"a\"), Typesetting:-mo(\")\", form = \"postfix\", fence = \+ \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \+ \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \")), Typesetting:-mi(\"n\"), superscriptshift = \"0\"), Typesetting:- mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6#Q!F'-I%msupGF$6%-F#6$-F,6#Q\"aF'-I#moGF$63Q\")F'/%%formGQ(post fixF'/%&fenceGQ%trueF'/%*separatorGQ&falseF'/%'lspaceGQ.thinmathspaceF '/%'rspaceGQ2verythinmathspaceF'/%)stretchyGF@/%*symmetricGFC/%(maxsiz eGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGFC/%.movablelimitsGFC/%'acce ntGFC/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0 ,0,0]F'/%+backgroundGQ.[255,255,255]F'-F,6#Q\"nF'/%1superscriptshiftGQ \"0F'F+" }{TEXT 204 10 " * inv(Qa)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Dan := Matrix([[Da[1,1]^n,0,0,0],\n[" }{MPLTEXT 1 0 38 "0,Da[2,2]^n,0,0],[0,0,Da[3,3]^n,0],\n[" }{MPLTEXT 1 0 21 "0,0,0,Da [4,4]^n]]);\n" }{MPLTEXT 1 0 15 "Qa.Dan.Qa^(-1);" }}}{SECT 1 {PARA 211 "" 0 "" {TEXT 203 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT 200 52 "2. Fin d a formula for the nth power of the matrix H." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "H := Matrix([[-1,-1,1], [2,4,2], [3,1,1]]);" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT 205 19 "Invariant subspaces" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 84 "Consider the linear operator T with matrix E above. Compute so me T-cyclic subspaces." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "E; \n" }{MPLTEXT 1 0 23 "evE := Eigenvectors(E);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 30 "Start with the vector v1 = e1:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "v1 := <1,0,0,0>;\n" }{MPLTEXT 1 0 12 "Tv1 := E .v1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 63 "\{v1, T(v1)\} is indepen dent - how about adding another vector?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "T2v1 := E.Tv1;" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"T\"), Typesetting: -mn(\"2\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typeset ting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6%-F,6#Q\"TF'-I#mnGF$6#Q\"2F'/%1sup erscriptshiftGQ\"0F'F+F+" }{TEXT 204 89 "(v1) is in span(\{v1, T(v1)\} ), so the T-cyclic subspace generated by v1 has dimension 2." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 115 "Next we choo se another vector to start with, and we pick something not in the firs t T-cyclic subspace, say v2 = e2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v2 := <0,1,0,0>;\n" }{MPLTEXT 1 0 12 "Tv2 := E.v2;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 62 "\{v2, T(v2)\} is independent, h ow about adding another vector?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "T2v2 := E.Tv2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 16 "It app ears that " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), T ypesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting: -mi(\"T\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typeset ting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6%-F,6#Q\"T F'-I#mnGF$6#Q\"2F'/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 204 59 "(v2) \+ can be added without losing independence; let's check." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "; Rank(); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 17 "Thus \{v2, T(v2)," } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"T\"), T ypesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesetting:-mi(\"\" )), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6%-F,6#Q\"TF'-I#mnGF$6#Q \"2F'/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 204 57 "(v2)\} is independ ent, and we try adding one more vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "T3v2 := E.T2v2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 17 "It is clear that " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:- mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(T ypesetting:-mi(\"T\"), Typesetting:-mn(\"3\"), superscriptshift = \"0 \"), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF $6%-F,6#Q\"TF'-I#mnGF$6#Q\"3F'/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 204 32 "(v2) is in the span(\{v2, T(v2)," }{XPPEDIT 18 0 "Typesetting: -mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-msup(Typesetting:-mi(\"T\"), Typesetting:-mn(\"2\"), supe rscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q! F'-F#6%F+-I%msupGF$6%-F,6#Q\"TF'-I#mnGF$6#Q\"2F'/%1superscriptshiftGQ \"0F'F+F+" }{TEXT 204 60 "(v2)\}) since the final coordinate is zero, \+ and \{v2, T(v2)," }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typese tting:-mi(\"T\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), T ypesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6%-F, 6#Q\"TF'-I#mnGF$6#Q\"2F'/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 204 54 "(v2)\} already spans the subspace of all such vectors." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 203 8 "Exercise " }}{PARA 0 "" 0 "" {TEXT 204 161 "3. Find a basis for the T-cyclic su bspace generated by the vector e4, where E represents the linear opera tor T. What can you say about this T-cyclic subspace of " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesett ing:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"R\"), Typesetting:- mn(\"4\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-F#6%F+-I%msupGF$6%-F,6#Q\"RF'-I#mnGF$6#Q\"4F'/%1supe rscriptshiftGQ\"0F'F+F+" }{TEXT 204 1 "?" }{TEXT 200 0 "" }}{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 }