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{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT 257 16 "Chapter 18\n\nThe " } {TEXT 316 13 "LinearAlgebra" }{TEXT 317 8 " Package" }{TEXT 314 1 "\n " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 31 " \251 Copyright 2003 by Andr\351 Heck." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 262 2 "1." }{TEXT -1 80 " Consider the following matrices.\n \+ " }{XPPEDIT 18 0 "A=MATRIX([ [1, 0, 2], [2, -1, 3], [4, 1, 8]])" "6#/%\"AG-%'MATRIXG6#7%7%\"\"\"\" \"!\"\"#7%F,,$F*!\"\"\"\"$7%\"\"%F*\"\")" }{TEXT -1 5 " , " } {XPPEDIT 18 0 "B=MATRIX([[-3, 2], [0, 1], [7, 4]])" "6#/%\"BG-%'MATRIX G6#7%7$,$\"\"$!\"\"\"\"#7$\"\"!\"\"\"7$\"\"(\"\"%" }{TEXT -1 9 "\nComp ute\n" }{TEXT 263 3 "(a)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "A^(-1)" "6#) %\"AG,$\"\"\"!\"\"" }{TEXT -1 1 "\n" }{TEXT 264 3 "(b)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "A*A^t" "6#*&%\"AG\"\"\")F$%\"tGF%" }{TEXT -1 1 "\n" }{TEXT 265 3 "(c)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "B^t*A*B" "6#*()%\"B G%\"tG\"\"\"%\"AGF'F%F'" }{TEXT -1 1 "\n" }{TEXT 266 3 "(d)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(2*A+B*B^t)*A^t" "6#*&,&*&\"\"#\"\"\"%\"AGF'F '*&%\"BGF')F*%\"tGF'F'F')F(F,F'" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 289 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := Matrix([[1,0,2], [2,-1,3], [4,1,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")Og\\:-%'MATRIXG6#7%7%\" \"\"\"\"!\"\"#7%F0!\"\"\"\"$7%\"\"%F.\"\")%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MatrixInverse(A), A^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")3*pl\"-%'MATRIXG6#7%7%!#6\"\"#F-7%! \"%\"\"!\"\"\"7%\"\"'!\"\"F4%'MatrixG-F$6%\")[#)f;F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 290 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := Matrix([[1,0,2], [2,-1, 3], [4,1,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\" ))Q4t\"-%'MATRIXG6#7%7%\"\"\"\"\"!\"\"#7%F0!\"\"\"\"$7%\"\"%F.\"\")%'M atrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A.Transpose(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")!=Px\"-%'MATRIXG6#7%7 %\"\"&\"\")\"#?7%F-\"#9\"#J7%F.F1\"#\")%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 291 3 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := Matrix([[1,0,2], [2,-1,3], [4, 1,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")OI4:-% 'MATRIXG6#7%7%\"\"\"\"\"!\"\"#7%F0!\"\"\"\"$7%\"\"%F.\"\")%'MatrixG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B := Matrix([[-3,2], [0,1] , [7,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"(G$y B-%'MATRIXG6#7%7$!\"$\"\"#7$\"\"!\"\"\"7$\"\"(\"\"%%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Transpose(B).A.B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(wrp#-%'MATRIXG6#7$7$\"$v#\"$d #7$\"$8#\"$*>%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 292 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := Matrix([[1,0,2], [2,-1,3], [4,1,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"(%=$G$-%'MATRIXG6#7%7%\"\"\"\"\"!\" \"#7%F0!\"\"\"\"$7%\"\"%F.\"\")%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B := Matrix([[-3,2], [0,1], [7,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")[n&Q\"-%'MATRIXG6#7%7$!\"$\" \"#7$\"\"!\"\"\"7$\"\"(\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "(2*A + B.Transpose(B)).Transpose(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")SYC9-%'MATRIXG6#7%7%!\"$\"\"\"!#57% \"#E\"#V\"$.\"7%\"$d\"\"$F#\"$M'%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 267 2 "2." } {TEXT -1 60 " Compute the Wronskian of the following list of functions : [" }{XPPEDIT 18 0 "cos(x),sin(x),e^x" "6%-%$cosG6#%\"xG-%$sinG6#F&)% \"eGF&" }{TEXT -1 81 "]. Compute the determinant of this matrix. Do th e same computation for the list [" }{XPPEDIT 18 0 "cosh(x),sinh(x),e^x " "6%-%%coshG6#%\"xG-%%sinhG6#F&)%\"eGF&" }{TEXT -1 3 "].\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "with(VectorCalculus):" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, the assigned names <,> and <|> now have a global binding \n" }}{PARA 7 "" 1 "" {TEXT -1 110 "Warning, these protected names hav e been redefined and unprotected: *, +, ., Vector, diff, int, limit, s eries\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Wronskian([cos(x ), sin(x), exp(x)], x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6 %\");'y^\"-%'MATRIXG6#7%7%-%$cosG6#%\"xG-%$sinGF.-%$expGF.7%,$F0!\"\"F ,F27%,$F,F6F5F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LinearAlgebra:-Determinant(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* (\"\"#\"\"\")-%$cosG6#%\"xGF%F&-%$expGF*F&F&*(F%F&)-%$sinGF*F%F&F,F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$expG6#%\"xGF&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Wronskian([cosh(x), sinh(x), exp(x)], x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")+Os:-% 'MATRIXG6#7%7%-%%coshG6#%\"xG-%%sinhGF.-%$expGF.7%F0F,F2F+%'MatrixG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LinearAlgebra:-Determinant (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 268 2 "3." } {TEXT -1 502 " Create a 5x5 matrix with entries randomly chosen as uni variate polynomials with integral coefficients, 3 terms, and degree le ss than 5. Compute the characteristic polynomial and verify the Cayley -Hamilton theorem by substitution of the matrix in the characteristic \+ polynomial when it is collected in the main variable. Also compute the Horner form of the characteristic polynomial and verify the Cayley-Ha milton theorem by substitution of the matrix in the polynomial in Horn er form. Compare timings.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearA lgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "poly := proc() \+ randpoly(x, terms=3, degree=5) end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A := RandomMatrix(5, 5, generator=poly);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")o\"yh\"-%'MATRIXG6#7'7',( \"#z\"\"\"*&\"#(*F0)%\"xG\"\"$F0F0*&\"#]F0)F4\"\"#F0F0,(*&\"#XF0)F4\" \"%F0F0*&\"\")F0F3F0!\"\"*&\"#$*F0F8F0FA,(*&\"#xF0)F4\"\"&F0F0*&\"#mF0 F=F0F0*&\"#aF0F8F0F0,(*&\"#7F0FGF0FA*&\"#=F0F=F0FA*&\"#JF0F8F0F0,(\"#h FA*&\"#\"*F0F3F0FA*&\"#ZF0F8F0FA7',(\"#%*F0*&\"#`F0FGF0F0*$F3F0FA,(*& \"#%)F0F=F0FA*&\"#>F0F3F0F0*&F7F0F4F0FA,(\"#F0F4F0F07',(FfqF0 *&FhpF0F=F0F0*&\"\"*F0F3F0F0,(*&\"#iF0F=F0F0*&\"#6F0F3F0F0*&\"#))F0F8F 0F0,(\"#dF0*&FerF0F8F0FA*&\"#5F0F4F0F0,(*&\"\"(F0F=F0FA*&\"#eF0F3F0F0* &FfnF0F4F0FA,(*&\"#9F0FGF0FA*&F`rF0F3F0FA*&\"#^F0F8F0FA7',(\"#VF0*&FHF 0FGF0F0*&\"#')F0F4F0FA,(*&F]qF0F3F0FA*&F@F0F8F0F0*&FcoF0F4F0FA,(F " 0 "" {MPLTEXT 1 0 71 "characteristicPolynomial := collect(CharacteristicPolynomial(A , t), t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%9characteristicPolynomi alG,fn*&\"+g#[dW\"\"\"\")%\"xG\"#>F(!\"\"*&\"+EgUKBF()F*\"#?F(F,*&\")+ Nx9F()F*\"#CF(F(*&\")bjShF()F*\"#BF(F,*&\"+VyMJ7F()F*\"#@F(F,*&\"))z- \\*F()F*\"#AF(F(*$)%\"tG\"\"&F(F(*&,,\"#DF,*&\"$b\"F()F*\"\"%F(F(*&\"# ()F()F*\"\"#F(F,*&\"$u\"F()F*\"\"$F(F,*&\"#mF(F*F(F(F()FCFKF(F(*&,6\"% ClF,*&\"&us#F()F*\"\"(F(F,*&\"%zpF()F*\"\"'F(F,*&\"%y@F()F*\"\")F(F,*& \"&Q.\"F(F*F(F,*&\"%VsF(FJF(F,*&\"&JG#F()F*FDF(F(*&\"%EPF(FRF(F,*&\"%h ()F(FNF(F(*&\"%5LF()F*\"\"*F(F,F()FCFSF(F(*&,B\"&qD*F,*&\"'w;mF(FfnF(F (*&\"(u2n\"F(FjnF(F,*&\"'AxrF(F]pF(F,*&\"(O.A\"F(F^oF(F,*&\"'))[]F(F*F (F,*&\"'K4aF()F*\"#6F(F,*&\"'lG5F()F*\"#:F(F,*&\"(:dW\"F()F*\"#5F(F(*& \"'T#>\"F()F*\"#9F(F,*&\"'J^9F()F*\"#7F(F,*&\"&,`$F()F*\"#8F(F(*&\"'^@ ^F(FJF(F,*&\"(R%)[\"F(FfoF(F,*&\"'eLJF(FRF(F,*&\"'#=q&F(FNF(F,F()FCFOF (F(*&,L\")U*pD#F(*&\"(g2L#F(F)F(F,*&\"'+IpF(F/F(F(*&\")%yk]\"F(FfnF(F( *&\"*$)H(fAF(FjnF(F(*&\"*D\\^*=F(F]pF(F(*&\"(80p%F(F^oF(F,*&\")?(>^%F( F*F(F(*&\"(C%HlF(F_qF(F,*&\")dk%)RF(FcqF(F,*&\"*$[;+AF(FgqF(F,*&\")yyc nF(F[rF(F,*&\")6S=$)F(F_rF(F(*&\")Uc``F(FcrF(F,*&\"*`/Pz\"F(FJF(F,*&\" *+LS]\"F(FfoF(F,*&\")%eJ]#F(FRF(F(*&\")?'z!HF(FNF(F,*&\"(zTW*F()F*\"#= F(F(*&\")6$RH$F()F*\"#;F(F(*&\")!RKc\"F()F*\"#(F(FcqF(F,*&\",7+'p_? F(FgqF(F,*&\",9?Lq4\"F(F[rF(F,*&\",!*>K#Q6F(F_rF(F,*&\",TW-ea\"F(FcrF( F,*&\",(y/qP;F(FJF(F(*&\",Ekd=P$F(FfoF(F(*&\",]*yj;;F(FRF(F(*&\"+q@KSm F(FNF(F(*&\"+f[]E9F(FeuF(F(*&\"+OZPV?F(FiuF(F,*&\"+SE33eF(F]vF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "settime := time():" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(t=A, characteristicPoly nomial);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,fn*&\"+g#[dW\"\"\"\")%\"x G\"#>F&!\"\"*&\"+EgUKBF&)F(\"#?F&F**&\")+Nx9F&)F(\"#CF&F&*&\")bjShF&)F (\"#BF&F**&\"+VyMJ7F&)F(\"#@F&F**&\"))z-\\*F&)F(\"#AF&F&*&,L\")U*pD#F& *&\"(g2L#F&F'F&F**&\"'+IpF&F-F&F&*&\")%yk]\"F&)F(\"\"(F&F&*&\"*$)H(fAF &)F(\"\"'F&F&*&\"*D\\^*=F&)F(\"\"*F&F&*&\"(80p%F&)F(\"\")F&F**&\")?(>^ %F&F(F&F&*&\"(C%HlF&)F(\"#6F&F**&\")dk%)RF&)F(\"#:F&F**&\"*$[;+AF&)F( \"#5F&F**&\")yycnF&)F(\"#9F&F**&\")6S=$)F&)F(\"#7F&F&*&\")Uc``F&)F(\"# 8F&F**&\"*`/Pz\"F&)F(\"\"%F&F**&\"*+LS]\"F&)F(\"\"&F&F**&\")%eJ]#F&)F( \"\"$F&F&*&\")?'z!HF&)F(\"\"#F&F**&\"(zTW*F&)F(\"#=F&F&*&\")6$RH$F&)F( \"#;F&F&*&\")!RKc\"F&)F(\"#F&*& \"#OF&F\\pF&F**&\"#SF&FhpF&F&,(*&\"#tF&F\\pF&F**&F_vF&FhpF&F&*&F]pF&F( F&F&7',(F\\wF&*&F_vF&F\\pF&F&*&FQF&FdpF&F&,(*&\"#iF&F\\pF&F&*&FenF&Fdp F&F&*&\"#))F&FhpF&F&,(\"#dF&*&FenF&FhpF&F**&F]oF&F(F&F&,(*&FIF&F\\pF&F **&\"#eF&FdpF&F&*&F_tF&F(F&F*,(*&FaoF&F`pF&F**&FQF&FdpF&F**&\"#^F&FhpF &F*7',(\"#VF&*&FapF&F`pF&F&*&\"#')F&F(F&F*,(*&FdvF&FdpF&F**&FUF&FhpF&F &*&FjtF&F(F&F*,(FgrF&*&\"#nF&FdpF&F**&\"#oF&FhpF&F&,(\"#fF**&FavF&FdpF &F**&FhsF&FhpF&F*,(FjrF**&FfvF&F\\pF&F**&\"#PF&FhpF&F&%'MatrixGF&F&*$) FfqFapF&F&*&,,F_vF**&\"$b\"F&F\\pF&F&*&FeuF&FhpF&F**&\"$u\"F&FdpF&F**& F_sF&F(F&F&F&)FfqF]pF&F&*&,6\"%ClF**&\"&us#F&FHF&F**&\"%zpF&FLF&F**&\" %y@F&FTF&F**&\"&Q.\"F&F(F&F**&\"%VsF&F\\pF&F**&\"&JG#F&F`pF&F&*&\"%EPF &FdpF&F**&\"%h()F&FhpF&F&*&\"%5LF&FPF&F*F&)FfqFepF&F&*&,B\"&qD*F**&\"' w;mF&FHF&F&*&\"(u2n\"F&FLF&F**&\"'AxrF&FPF&F**&\"(O.A\"F&FTF&F**&\"')) []F&F(F&F**&\"'K4aF&FZF&F**&\"'lG5F&FhnF&F**&\"(:dW\"F&F\\oF&F&*&\"'T# >\"F&F`oF&F**&\"'J^9F&FdoF&F**&\"&,`$F&FhoF&F&*&\"'^@^F&F\\pF&F**&\"(R %)[\"F&F`pF&F**&\"'eLJF&FdpF&F**&\"'#=q&F&FhpF&F*F&)FfqFipF&F&*&\",piy qh$F&FHF&F&*&\",*[RJ4hF&FLF&F&*&\",&3-j*4#F&FPF&F**&\"+6n.`'*F&FTF&F&* &\"*y*G1OF&F(F&F&*&\",ekT/n\"F&FZF&F**&\"+T:[%>(F&FhnF&F**&\",7+'p_?F& F\\oF&F**&\",9?Lq4\"F&F`oF&F**&\",!*>K#Q6F&FdoF&F**&\",TW-ea\"F&FhoF&F **&\",(y/qP;F&F\\pF&F&*&\",Ekd=P$F&F`pF&F&*&\",]*yj;;F&FdpF&F&*&\"+q@K SmF&FhpF&F&*&\"+f[]E9F&F\\qF&F&*&\"+OZPV?F&F`qF&F**&\"+SE33eF&FdqF&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(%) assuming x: :scalar;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"('*H#G-%'MAT RIXG6#7'7'\"\"!F,F,F,F,F+F+F+F+%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "cpu_time := (time()-settime)*seconds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)cpu_timeG,$*&$\"%_;!\"$\"\"\"%(secondsGF*F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "characteristicPolynomial := convert(characteristicPolynomial, horner, t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%9characteristicPolynomialG,T*&\"+g#[dW\"\"\"\")%\"xG \"#>F(!\"\"*&\"+EgUKBF()F*\"#?F(F,*&\")+Nx9F()F*\"#CF(F(*&\")bjShF()F* \"#BF(F,*&\"+VyMJ7F()F*\"#@F(F,*&\"))z-\\*F()F*\"#AF(F(*&,N\")U*pD#F(* &\"(g2L#F(F)F(F,*&\"'+IpF(F/F(F(*&\")%yk]\"F()F*\"\"(F(F(*&\"*$)H(fAF( )F*\"\"'F(F(*&\"*D\\^*=F()F*\"\"*F(F(*&\"(80p%F()F*\"\")F(F,*&\")?(>^% F(F*F(F(*&\"(C%HlF()F*\"#6F(F,*&\")dk%)RF()F*\"#:F(F,*&\"*$[;+AF()F*\" #5F(F,*&\")yycnF()F*\"#9F(F,*&\")6S=$)F()F*\"#7F(F(*&\")Uc``F()F*\"#8F (F,*&\"*`/Pz\"F()F*\"\"%F(F,*&\"*+LS]\"F()F*\"\"&F(F,*&\")%eJ]#F()F*\" \"$F(F(*&\")?'z!HF()F*\"\"#F(F,*&\"(zTW*F()F*\"#=F(F(*&\")6$RH$F()F*\" #;F(F(*&\")!RKc\"F()F*\"#\"F(FboF(F,*&\"'J^9F(F foF(F,*&\"&,`$F(FjoF(F(*&\"'^@^F(F^pF(F,*&\"(R%)[\"F(FbpF(F,*&\"'eLJF( FfpF(F,*&\"'#=q&F(FjpF(F,*&,8\"%ClF,*&\"&us#F(FJF(F,*&\"%zpF(FNF(F,*& \"%y@F(FVF(F,*&\"&Q.\"F(F*F(F,*&\"%VsF(F^pF(F,*&\"&JG#F(FbpF(F(*&\"%EP F(FfpF(F,*&\"%h()F(FjpF(F(*&\"%5LF(FRF(F,*&,.\"#DF,*&\"$b\"F(F^pF(F(*& \"#()F(FjpF(F,*&\"$u\"F(FfpF(F,*&\"#mF(F*F(F(%\"tGF(F(FiuF(F(F(FiuF(F( F(FiuF(F(F(FiuF(F(*&\",piyqh$F(FJF(F(*&\",*[RJ4hF(FNF(F(*&\",&3-j*4#F( FRF(F,*&\"+6n.`'*F(FVF(F(*&\"*y*G1OF(F*F(F(*&\",ekT/n\"F(FfnF(F,*&\"+T :[%>(F(FjnF(F,*&\",7+'p_?F(F^oF(F,*&\",9?Lq4\"F(FboF(F,*&\",!*>K#Q6F(F foF(F,*&\",TW-ea\"F(FjoF(F,*&\",(y/qP;F(F^pF(F(*&\",Ekd=P$F(FbpF(F(*& \",]*yj;;F(FfpF(F(*&\"+q@KSmF(FjpF(F(*&\"+f[]E9F(F^qF(F(*&\"+OZPV?F(Fb qF(F,*&\"+SE33eF(FfqF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "settime := time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs (t=A, characteristicPolynomial):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(%) assuming x::scalar;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Kq89-%'MATRIXG6#7'7'\"\"!F,F,F,F,F+F+F+F +%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "cpu_time := ( time()-settime)*seconds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)cpu_tim eG,$*&$\"%LC!\"$\"\"\"%(secondsGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 103 "The computation in \+ Horner form takes more time because of the intermediate manipulation o f polynomials." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 269 2 "4." }{TEXT -1 90 " Consider the matrices\n \+ " }{XPPEDIT 18 0 "A=MATRIX([[-4, -7, 0], [0, \+ 4, 2], [-5, -7, 1]])" "6#/%\"AG-%'MATRIXG6#7%7%,$\"\"%!\"\",$\"\"(F,\" \"!7%F/F+\"\"#7%,$\"\"&F,,$F.F,\"\"\"" }{TEXT -1 72 "\nand\n \+ " }{XPPEDIT 18 0 "B=MATRIX([[-2, -1, -2], [2, 2, -2], [0, 0, 1]])" "6#/%\"BG-%'MAT RIXG6#7%7%,$\"\"#!\"\",$\"\"\"F,,$F+F,7%F+F+,$F+F,7%\"\"!F3F." }{TEXT -1 11 "\nshow that " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 66 " are similar and compute th e corresponding transformation matrix.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A \+ := Matrix([[-4,-7,0], [0,4,2], [-5,-7,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"(s\"zF-%'MATRIXG6#7%7%!\"%!\"(\"\"! 7%F0\"\"%\"\"#7%!\"&F/\"\"\"%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "B := Matrix([[-2,-1,-2], [2,2,-2], [0,0,1]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"(sm&H-%'MATRIXG6#7 %7%!\"#!\"\"F.7%\"\"#F1F.7%\"\"!F3\"\"\"%'MatrixG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 46 "q, P := IsSimilar(A, B, output=['query','C'] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"qG%\"PG6$%%trueG-%'RTABLEG 6%\")c!oU\"-%'MATRIXG6#7%7%\"\"\"\"\"!F37%!\"&\"\"(F67%#F6\"\"#F3#!\"( F9%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Equal(P^(-1) .B.P, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 270 2 "5." }{TEXT -1 5 " Let " }{TEXT 258 4 "a, b" }{TEXT -1 1 " " }{TEXT 296 1 "\316" }{TEXT -1 10 " R with 0 " }{TEXT 298 1 "\243" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 1 " " }{TEXT 299 1 "\243 " }{TEXT -1 5 " 1 , " }{XPPEDIT 18 0 "b^2=2*a*(1-a)" "6#/*$%\"bG\"\"#* (F&\"\"\"%\"aGF(,&F(F(F)!\"\"F(" }{TEXT -1 65 ", and\n \+ " }{XPPEDIT 18 0 "A=MATRIX( [[a, a-1, b], [a-1, a, b], [-b, -b, 2*a-1]])" "6#/%\"AG-%'MATRIXG6#7%7 %%\"aG,&F*\"\"\"F,!\"\"%\"bG7%,&F*F,F,F-F*F.7%,$F.F-,$F.F-,&*&\"\"#F,F *F,F,F,F-" }{TEXT -1 2 ".\n" }{TEXT 271 3 "(a)" }{TEXT -1 23 " Check w ith Maple that " }{TEXT 259 1 "A" }{TEXT -1 56 " is an orthogonal matr ix with determinant equal to one.\n" }{TEXT 272 3 "(b)" }{TEXT -1 23 " From (a) follows that " }{TEXT 260 1 "A" }{TEXT -1 64 " is a matrix t hat describes a rotation in the standard basis of " }{XPPEDIT 18 0 "R^ 3" "6#*$%\"RG\"\"$" }{TEXT -1 31 ". Determine the rotation axis.\n" }} {SECT 1 {PARA 5 "" 0 "" {TEXT 303 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A \+ := Matrix([[a,a-1,b], [a-1,a,b], [-b,-b,2*a-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"(7\"*>$-%'MATRIXG6#7%7%%\"aG,&F.\" \"\"F0!\"\"%\"bG7%F/F.F27%,$F2F1F5,&*&\"\"#F0F.F0F0F0F1%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "map(expand, A.Transpose(A)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Gn*Q\"-%'MATRIXG6#7 %7%,**&\"\"#\"\"\")%\"aGF.F/F/*&F.F/F1F/!\"\"F/F/*$)%\"bGF.F/F/,(*&F.F /F0F/F/*&F.F/F1F/F3F4F/\"\"!7%F7F,F:7%F:F:,**&F.F/F5F/F/*&\"\"%F/F0F/F /*&F@F/F1F/F3F/F/%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(simplify, %, \{b^2=2*a*(1-a)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")[G,9-%'MATRIXG6#7%7%\"\"\"\"\"!F-7%F-F,F-7%F-F-F, %'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 304 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 99 "The r otation axis is described by the eigenvector of eigenvalue 1. Let us c ompute this eigenvector." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r estart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgeb ra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := Matrix([[a,a-1 ,b], [a-1,a,b], [-b,-b,2*a-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AG-%'RTABLEG6%\")O!ye\"-%'MATRIXG6#7%7%%\"aG,&F.\"\"\"F0!\"\"%\"bG7% F/F.F27%,$F2F1F5,&*&\"\"#F0F.F0F0F0F1%'MatrixG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "(v, e) := Eigenvectors(A);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>6$%\"vG%\"eG6$-%'RTABLEG6%\")#*zS:-%'MATRIXG6#7%7#\" \"\"7#,(F1!\"\"*&\"\"#F1%\"aGF1F1*(%\"bGF1F6#F1F6^#F1F1F17#,(F1F4*&F6F 1F7F1F1*(^#F4F1F9F1F6F:F1&%'VectorG6#%'columnG-F)6%\"(s&yK-F-6#7%7%F4* &^##F4F6F1F6F:*&^#F:F1F6F:7%F1FLFO7%\"\"!F1F1%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "e[1..-1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")knk:-%'MATRIXG6#7%7#!\"\"7#\"\"\"7#\"\"! &%'VectorG6#%'columnG" }}}{PARA 0 "" 0 "" {TEXT -1 56 "So, the rotatio n axis is the intersection of the planes " }{XPPEDIT 18 0 "x+y=0" "6#/ ,&%\"xG\"\"\"%\"yGF&\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z=0" "6 #/%\"zG\"\"!" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 273 2 "6." }{TEXT -1 4 " Let" }{TEXT 261 4 " a,b" }{TEXT -1 1 " " }{TEXT 297 1 "\316" }{TEXT -1 67 " R and\n \+ " }{XPPEDIT 18 0 "A=MATRIX([[0,a,1,0,b],[1,0, 0,b,0],[0,1,b,0,1],[b,0,0,1,0],[0,b,1,0,b]])" "6#/%\"AG-%'MATRIXG6#7'7 '\"\"!%\"aG\"\"\"F*%\"bG7'F,F*F*F-F*7'F*F,F-F*F,7'F-F*F*F,F*7'F*F-F,F* F-" }{TEXT -1 2 ".\n" }{TEXT 274 3 "(a)" }{TEXT -1 54 " For what value s of a and b is the matrix A singular?\n" }{TEXT 275 3 "(b)" }{TEXT -1 84 " Determine the inverse of A (for those values of a and b for wh ich A is inversible)\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 301 3 "(a)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 "A := Matrix([[0,a,1,0,b], [1,0,0,b,0], [0,1,b, 0,1], [b,0,0,1,0], [0,b,1,0,b]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"AG-%'RTABLEG6%\")w)zr\"-%'MATRIXG6#7'7'\"\"!%\"aG\"\"\"F.%\"bG7'F0F .F.F1F.7'F.F0F1F.F07'F1F.F.F0F.7'F.F1F0F.F1%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(\"\"#\"\"\"%\"aGF&)%\"bGF%F&!\"\"F'F&*&F%F&)F)\"\"$ F&F&F)F**&F'F&)F)\"\"%F&F&*$)F)\"\"&F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(),&% \"bG\"\"\"F'!\"\"\"\"#F'),&F&F'F'F'F)F',&F&F(%\"aGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Th e matrix is singular if " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b=-1" "6#/%\"bG,$\"\"\"!\"\"" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "a=b" "6#/%\"aG%\"bG" }{TEXT -1 1 "." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 302 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "A \+ := Matrix([[0,a,1,0,b], [1,0,0,b,0], [0,1,b,0,1], [b,0,0,1,0], [0,b,1, 0,b]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"(s#*)G- %'MATRIXG6#7'7'\"\"!%\"aG\"\"\"F.%\"bG7'F0F.F.F1F.7'F.F0F1F.F07'F1F.F. F0F.7'F.F1F0F.F1%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MatrixInverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6% \")%GwQ\"-%'MATRIXG6#7'7'\"\"!,$*&\"\"\"F/,&*$)%\"bG\"\"#F/F/F/!\"\"F5 F5F,*&F3F/F0F5F,7'*&F/F/,&%\"aGF/F3F5F5F,F,F,,$F8F57'F,F,F6F,F-7'F,F6F ,F-F,7'F;F,F-F,*&,**&F2F/F:F/F/F3F/*$)F3\"\"$F/F5F:F5F5,&*&F3F/F:F/F/F /F5F/%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(facto r, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")C6%R\"-%'MATR IXG6#7'7'\"\"!,$*&\"\"\"F/*&,&%\"bGF/F/!\"\"F/,&F2F/F/F/F/F3F3F,*(F2F/ F1F3F4F3F,7'*&F/F/,&%\"aGF/F2F3F3F,F,F,,$F7F37'F,F,F5F,F-7'F,F5F,F-F,7 'F:F,F-F,**,&*&F2F/F9F/F/F/F3F/F1F3F4F3F8F3%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 276 2 "7." }{TEXT -1 2 " \n" }{TEXT 277 3 "(a)" }{TEXT -1 13 " Compute det " }{XPPEDIT 18 0 "MATRIX([[x^2+1,x,0,0],[x,x^2+1,x,0],[0,x,x^2+1,x],[ 0,0,x,x^2+1]])" "6#-%'MATRIXG6#7&7&,&*$%\"xG\"\"#\"\"\"F,F,F*\"\"!F-7& F*,&*$F*F+F,F,F,F*F-7&F-F*,&*$F*F+F,F,F,F*7&F-F-F*,&*$F*F+F,F,F," } {TEXT -1 3 ".\n\n" }{TEXT 278 3 "(b)" }{TEXT -1 14 " Compute det " } {XPPEDIT 18 0 "MATRIX([[x^2+1,x,0,0,0],[x,x^2+1,x,0,0],[0,x,x^2+1,x,0] ,[0,0,x,x^2+1,x],[0,0,0,x,x^2+1]])" "6#-%'MATRIXG6#7'7',&*$%\"xG\"\"# \"\"\"F,F,F*\"\"!F-F-7'F*,&*$F*F+F,F,F,F*F-F-7'F-F*,&*$F*F+F,F,F,F*F-7 'F-F-F*,&*$F*F+F,F,F,F*7'F-F-F-F*,&*$F*F+F,F,F," }{TEXT -1 3 ".\n\n" } {TEXT 279 3 "(c)" }{TEXT -1 256 " Looking at the results of (a) and (b ), do you have any idea what the determinant of a general matrix of th e above form is? If so, check your conjecture for a 8 x 8 matrix. If n ot, compute the determinants for matrices of dimension 6 and 7 to get \+ an idea.\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 306 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "BandMatrix([x, x^2+1, x], 1, 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")O0l9-%'MATRIXG6#7&7&,&*$)%\"xG\"\"#\"\" \"F1F1F1F/\"\"!F27&F/F,F/F27&F2F/F,F/7&F2F2F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,,*$)%\"xG\"\")\"\"\"F(*$)F&\"\"'F(F(*$)F&\"\"%F(F(*$ )F&\"\"#F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 307 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Ba ndMatrix([x, x^2+1, x], 1, 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'R TABLEG6%\")?Wu:-%'MATRIXG6#7'7',&*$)%\"xG\"\"#\"\"\"F1F1F1F/\"\"!F2F27 'F/F,F/F2F27'F2F/F,F/F27'F2F2F/F,F/7'F2F2F2F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,.*$)%\"xG\"#5\"\"\"F(*$)F&\"\")F(F(*$)F&\"\"'F(F(*$) F&\"\"%F(F(*$)F&\"\"#F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 305 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "BandMatrix([x, x^2+1, x], 1, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")_%pY\"-%'MATRIXG6#7(7(,&*$)%\"xG\"\"#\" \"\"F1F1F1F/\"\"!F2F2F27(F/F,F/F2F2F27(F2F/F,F/F2F27(F2F2F/F,F/F27(F2F 2F2F/F,F/7(F2F2F2F2F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0 \"\"\"F$*$)%\"xG\"\"#F$F$*$)F'\"\"%F$F$*$)F'\"\"'F$F$*$)F'\"\")F$F$*$) F'\"#5F$F$*$)F'\"#7F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " BandMatrix([x, x^2+1, x], 1, 7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\")k5)\\\"-%'MATRIXG6#7)7),&*$)%\"xG\"\"#\"\"\"F1F1F1F/\"\"! F2F2F2F27)F/F,F/F2F2F2F27)F2F/F,F/F2F2F27)F2F2F/F,F/F2F27)F2F2F2F/F,F/ F27)F2F2F2F2F/F,F/7)F2F2F2F2F2F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,2\"\"\"F$*$)%\"xG\"\"#F$F$*$)F'\"\"%F$F$*$)F'\"\"'F$F$*$)F'\"\")F$F$ *$)F'\"#5F$F$*$)F'\"#7F$F$*$)F'\"#9F$F$" }}}{PARA 0 "" 0 "" {TEXT 308 10 "Conjecture" }{TEXT -1 59 ": the determinant such a square banded m atrix of dimension " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 13 " is eq ual to " }{XPPEDIT 18 0 "Sum(x^(2*i), i=0..n)=1+x^2+x^4+x^6+`...`+x^(2 *(n-1))+x^(2*n)" "6#/-%$SumG6$)%\"xG*&\"\"#\"\"\"%\"iGF+/F,;\"\"!%\"nG ,0F+F+*$F(F*F+*$F(\"\"%F+*$F(\"\"'F+%$...GF+)F(*&F*F+,&F0F+F+!\"\"F+F+ )F(*&F*F+F0F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "We ve rify it for " }{XPPEDIT 18 0 "n=8" "6#/%\"nG\"\")" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "BandMatrix([x, x^2+1, x], 1, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")OhD:-%'MATRIXG6 #7*7*,&*$)%\"xG\"\"#\"\"\"F1F1F1F/\"\"!F2F2F2F2F27*F/F,F/F2F2F2F2F27*F 2F/F,F/F2F2F2F27*F2F2F/F,F/F2F2F27*F2F2F2F/F,F/F2F27*F2F2F2F2F/F,F/F27 *F2F2F2F2F2F/F,F/7*F2F2F2F2F2F2F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4\"\"\"F$*$)%\"xG\"\"#F$F$*$)F'\"\"%F$F$*$)F'\"\"'F$F$*$)F'\"\")F$ F$*$)F'\"#5F$F$*$)F'\"#7F$F$*$)F'\"#9F$F$*$)F'\"#;F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 280 2 "8." }{TEXT -1 45 " For each natural number n, the n x n m atrix " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 31 " is defi ned as\n " }{XPPEDIT 18 0 "A[n](i,j)=PIECEWISE([0, i = \+ j],[1, i <> j])" "6#/-&%\"AG6#%\"nG6$%\"iG%\"jG-%*PIECEWISEG6$7$\"\"!/ F*F+7$\"\"\"0F*F+" }{TEXT -1 59 "\n Carry out the following computatio ns for n = 3, 4 and 5.\n" }{TEXT 281 3 "(a)" }{TEXT -1 27 " Compute th e determinat of " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 1 "\n" }{TEXT 282 3 "(b)" }{TEXT -1 42 " Compute the characteristic poly nomial of " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 2 ".\n" }{TEXT 283 3 "(c)" }{TEXT -1 30 " Determine all eigenvalues of " } {XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 76 " and determine fo r each eigenvalue a basis of the corresponding eigenspace.\n" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 294 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A \+ := n -> \nMatrix(n, n, (i,j) -> if i=j then 0 else 1 end if):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Gm0;-%'MATRIXG6#7%7%\"\"!\"\"\"F-7%F-F,F -7%F-F-F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Deter minant(A(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Determinant(A(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Determinant(A(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 13 "In general, " }{XPPEDIT 18 0 "det(A[n])=(-1)^(n-1)*(n-1)" "6#/ -%$detG6#&%\"AG6#%\"nG*&),$\"\"\"!\"\",&F*F.F.F/F.,&F*F.F.F/F." } {TEXT -1 1 "." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 293 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A := n -> \nMatrix(n, n, (i,j) -> if i=j then 0 else \+ 1 end if):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Characteristi cPolynomial(A(3), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\" \"$\"\"\"F(*&F'F(F&F(!\"\"\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\" xG\"\"\"\"\"#!\"\"F&),&F%F&F&F&F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "CharacteristicPolynomial(A(4), x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**$)%\"xG\"\"%\"\"\"F(*&\"\"'F()F&\"\"#F(!\"\"*&\"\" )F(F&F(F-\"\"$F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"$!\"\"F&),&F %F&F&F&F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Characterist icPolynomial(A(5), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\" \"&\"\"\"F(*&\"#5F()F&\"\"$F(!\"\"*&\"#?F()F&\"\"#F(F-*&\"#:F(F&F(F-\" \"%F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"%!\"\"F&),&F%F&F&F& F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 40 "In general, the character polynomial of " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 13 " is equal to " }{XPPEDIT 18 0 "(x-n+1)*(x+1)^(n-1)" "6#*&,(%\"xG\"\"\"%\"nG!\"\"F&F&F&),&F%F&F&F&, &F'F&F&F(F&" }{TEXT -1 1 "." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 295 3 "( c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A := n -> \nMatrix(n, n, (i,j) -> i f i=j then 0 else 1 end if):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Eigenvectors(A(3), output=list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%!\"\"\"\"#<$-%'RTABLEG6%\")cZ8:-%'MATRIXG6#7%7#F%7#\"\"\"7# \"\"!&%'VectorG6#%'columnG-F)6%\")gO,:-F-6#7%F0F3F1F57%F&F2<#-F)6%\")W $p]\"-F-6#7%F1F1F1F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Eig envectors(A(4), output=list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7% \"\"$\"\"\"<#-%'RTABLEG6%\")k![^\"-%'MATRIXG6#7&7#F&F0F0F0&%'VectorG6# %'columnG7%!\"\"F%<%-F)6%\")W([^\"-F-6#7&7#F67#\"\"!F?F0F1-F)6%\")!GUa \"-F-6#7&F>F0F?F?F1-F)6%\")/([^\"-F-6#7&F>F?F0F?F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Eigenvectors(A(5), output=list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%!\"\"\"\"%<&-%'RTABLEG6%\")%)*\\^\"-%'MA TRIXG6#7'7#F%7#\"\"!F1F17#\"\"\"&%'VectorG6#%'columnG-F)6%\")k)\\^\"-F -6#7'F0F3F1F1F1F5-F)6%\")/*\\^\"-F-6#7'F0F1F3F1F1F5-F)6%\")W*\\^\"-F-6 #7'F0F1F1F3F1F57%F&F4<#-F)6%\")%Q]^\"-F-6#7'F3F3F3F3F3F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "In general, the eigenvalues of " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG " }{TEXT -1 5 " are " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "n-1" "6#,&%\"nG \"\"\"F%!\"\"" }{TEXT -1 37 " with multiplicity 1 and eigenvector " } {XPPEDIT 18 0 "MATRIX([[1], [1], [`...`], [1]])" "6#-%'MATRIXG6#7&7#\" \"\"7#F(7#%$...G7#F(" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 20 "1 with multiplicity " }{XPPEDIT 18 0 " n-1" "6#,&%\"nG\"\"\"F%!\"\"" } {TEXT -1 27 " and eigenspace with basis " }{XPPEDIT 18 0 "MATRIX([[-1] ,[1],[0],[0],[`...`],[`...`],[0],[0]]), MATRIX([[-1],[0],[1],[0],[`... `],[`...`],[0],[0]]),MATRIX([[-1],[0],[0],[1],[`...`],[`...`],[0],[0]] ),` ..... `,MATRIX([[-1],[0],[0],[0],[`...`],[`...`],[1],[0]]),MATRIX ([[-1],[0],[0],[0],[`...`],[`...`],[0],[1]])" "6(-%'MATRIXG6#7*7#,$\" \"\"!\"\"7#F)7#\"\"!7#F-7#%$...G7#F07#F-7#F--F$6#7*7#,$F)F*7#F-7#F)7#F -7#F07#F07#F-7#F--F$6#7*7#,$F)F*7#F-7#F-7#F)7#F07#F07#F-7#F-%)~.....~~ G-F$6#7*7#,$F)F*7#F-7#F-7#F-7#F07#F07#F)7#F--F$6#7*7#,$F)F*7#F-7#F-7#F -7#F07#F07#F-7#F)" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 284 2 "9." } {TEXT -1 25 " For each natural number " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT -1 6 ", the " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 312 3 " x " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 8 " matrix " }{XPPEDIT 18 0 "A[n ]" "6#&%\"AG6#%\"nG" }{TEXT -1 84 " is defined as\n \+ " }{XPPEDIT 18 0 "A[ n](i,j)=gcd(i,j)" "6#/-&%\"AG6#%\"nG6$%\"iG%\"jG-%$gcdG6$F*F+" }{TEXT -1 2 ".\n" }{TEXT 285 3 "(a)" }{TEXT -1 28 " Compute the determinant o f " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "n=1,2" "6$/%\"nG\"\"\"\"\"#" }{TEXT -1 12 ", ... , 15. \n" }{TEXT 286 3 "(b)" }{TEXT -1 86 " (for the mathematicians among us ) Try to find a closed formula for the general case.\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 309 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(Lin earAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A := n -> M atrix(n, n, igcd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGf*6#%\"nG6 \"6$%)operatorG%&arrowGF(-%'MatrixG6%9$F/%%igcdGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A(4); # an example" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(?IZ#-%'MATRIXG6#7&7&\"\"\"F,F,F,7&F ,\"\"#F,F.7&F,F,\"\"$F,7&F,F.F,\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(Determinant(A(n)), n=1..15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "61\"\"\"F#\"\"#\"\"%\"#;\"#K\"$#>\"$o(\"%3Y\"&K%=\"' ?V=\"'!GP(\"(gt%))\")gT3`\"*!GtYU" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 310 3 "(b)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "A := n -> Matrix(n, n, igcd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%'Matrix G6%9$F/%%igcdGF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 48 "We first try to \+ find a relationship between the " }{XPPEDIT 18 0 "det(A[n])" "6#-%$det G6#&%\"AG6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "det(A[n-1])" "6# -%$detG6#&%\"AG6#,&%\"nG\"\"\"F+!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(Determinant(A(n))/Determinant(A(n-1)) , n=2..30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6?\"\"\"\"\"#F$\"\"%F$\" \"'F%F&F%\"#5F%\"#7F&\"\")F)\"#;F&\"#=F)F(F'\"#AF)\"#?F(F+F(\"#GF)" }} }{PARA 0 "" 0 "" {TEXT -1 21 "The totient function " }{XPPEDIT 18 0 "p hi" "6#%$phiG" }{TEXT -1 25 " is defined as follows:\n " }{XPPEDIT 18 0 "phi(n)" "6#-%$phiG6#%\"nG" }{TEXT -1 50 " is the number of positive integers not exceeding " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 25 " \+ and relatively prime to " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 80 "Compare the above list of numbers wit h the first values of the totient function." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "with(numtheory, phi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seq(phi(n), n=2..30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6?\"\"\"\"\"#F$\"\"%F$\"\"'F%F&F%\"#5F%\"#7F&\"\")F)\"#;F&\"#=F)F( F'\"#AF)\"#?F(F+F(\"#GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 67 "The sequences are the same. This sup ports the following conjecture." }}{PARA 0 "" 0 "" {TEXT 311 10 "Conje cture" }{TEXT -1 2 ". " }{XPPEDIT 18 0 "det(A[n]) = product(phi(i), i= 1..n)" "6#/-%$detG6#&%\"AG6#%\"nG-%(productG6$-%$phiG6#%\"iG/F1;\"\"\" F*" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 287 3 "10." }{TEXT -1 138 " Comp ute the LU decomposition of the following matrix A, taken from [50].\n " } {XPPEDIT 18 0 "A=MATRIX([[6, 2, 1,-1], [2, 4, 1,0], [1, 1, 4,-1],[-1,0 ,-1,3]])" "6#/%\"AG-%'MATRIXG6#7&7&\"\"'\"\"#\"\"\",$F,!\"\"7&F+\"\"%F ,\"\"!7&F,F,F0,$F,F.7&,$F,F.F1,$F,F.\"\"$" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A \+ := Matrix([[6,2,1,-1], [2,4,1,0], [1,1,4,-1], [-1,0,-1,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\");?:9-%'MATRIXG6#7&7&\" \"'\"\"#\"\"\"!\"\"7&F/\"\"%F0\"\"!7&F0F0F3F17&F1F4F1\"\"$%'MatrixG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "(P, L, U) := LUDecompositi on(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"PG%\"LG%\"UG6%-%'RTABL EG6%\")OjS9-%'MATRIXG6#7&7&\"\"\"\"\"!F3F37&F3F2F3F37&F3F3F2F37&F3F3F3 F2%'MatrixG-F*6%\")![/V\"-F.6#7&F17&#F2\"\"$F2F3F37&#F2\"\"'#F2\"\"&F2 F37&#!\"\"FC#F2\"#5#!\"*\"#PF2F7-F*6%\")'\\)H9-F.6#7&7&FC\"\"#F2FH7&F3 #FJF@#FUF@F?7&F3F3#FMFJ#FLFJ7&F3F3F3#\"$\">\"#uF7" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "Equal( P.L.U, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 288 3 "11." }{TEXT -1 5 " Let " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 46 " be the following 5 x 5 matr ix over the field " }{XPPEDIT 18 0 "F[2]" "6#&%\"FG6#\"\"#" }{TEXT -1 85 " with two elements.\n \+ " }{XPPEDIT 18 0 "A=MATRIX([[0,0,0,1,0],[0,0,1,0,1 ],[0,0,0,0,1],[1,0,0,1,0],[0,1,0,0,0]])" "6#/%\"AG-%'MATRIXG6#7'7'\"\" !F*F*\"\"\"F*7'F*F*F+F*F+7'F*F*F*F*F+7'F+F*F*F+F*7'F*F+F*F*F*" }{TEXT -1 35 "\nDetermine a transformation matrix " }{XPPEDIT 18 0 "T" "6#%\" TG" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "F[2]" "6#&%\"FG6#\"\"#" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "T^(-1)*A*T" "6#*()%\"TG,$\" \"\"!\"\"F'%\"AGF'F%F'" }{TEXT -1 90 " is of the form\n \+ " } {XPPEDIT 18 0 "A=MATRIX([[B,0],[0,C]])" "6#/%\"AG-%'MATRIXG6#7$7$%\"BG \"\"!7$F+%\"CG" }{TEXT -1 8 ",\nwhere " }{XPPEDIT 18 0 "B" "6#%\"BG" } {TEXT -1 24 " is a 2 x 2 matrix over " }{XPPEDIT 18 0 "F[2]" "6#&%\"FG 6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 24 " is a 3 x 3 matrix over " }{XPPEDIT 18 0 "F[2]" "6#&%\"FG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "A := Matrix([[0,0,0,1,0], [0,0,1,0,1], [0,0,0,0,1], [1,0,0,1,0], [ 0,1,0,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\") c#Qi\"-%'MATRIXG6#7'7'\"\"!F.F.\"\"\"F.7'F.F.F/F.F/7'F.F.F.F.F/7'F/F.F .F/F.7'F.F/F.F.F.%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 133 "Simply by looking at the matrix we readily notice that the 1st and 4th basis ve ctors form an invariant subspace. So, we can take for " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 32 " the following permution matrix." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "T := Matrix([[1,0,0,0,0], [0 ,0,0,1,0], [0,0,1,0,0], [0,1,0,0,0], [0,0,0,0,1]]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"TG-%'RTABLEG6%\")_kE;-%'MATRIXG6#7'7'\"\"\"\"\"!F /F/F/7'F/F/F/F.F/7'F/F/F.F/F/7'F/F.F/F/F/7'F/F/F/F/F.%'MatrixG" }}} {PARA 0 "" 0 "" {TEXT -1 51 "Let us verify that this matrix indeed doe s the job." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T^(-1).A.T;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")w'[m\"-%'MATRIXG6#7'7' \"\"!\"\"\"F,F,F,7'F-F-F,F,F,7'F,F,F,F,F-7'F,F,F-F,F-7'F,F,F,F-F,%'Mat rixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 16238256 16266452 16648676 }{RTABLE M7R0 I5RTABLE_SAVE/16238256X,%)anythingG6"6"[gl!"%!!!#:"&"&""!F'F'"""F'F'F'F'F'F(F'F (F'F'F'F(F'F'F(F'F'F(F(F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/16266452X,%)anythingG6"6"[gl!"%!!!#:"&"&"""""!F(F(F(F(F(F(F'F(F(F (F'F(F(F(F'F(F(F(F(F(F(F(F'F& } {RTABLE M7R0 I5RTABLE_SAVE/16648676X,%)anythingG6"6"[gl!"%!!!#:"&"&""!"""F'F'F'F(F(F'F'F'F'F 'F'F(F'F'F'F'F'F(F'F'F(F(F'F& }