{VERSION 4 0 "HP RISC UNIX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 24 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 "" 0 258 1 {CSTYLE " " -1 -1 "" 1 14 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 "" 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 264 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 33 "Die Jordan-Zerlegung ein er Matrix" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "unprotect(D): # In dieser Sitzung kann D nicht als Differentiati on verwendet werden" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "D := diag(2$2,1$3):\nN := matrix(5,5,[0,1,0$11, 2, 3, 0$4, 4, 0$5]):\nA := evalm(D+N);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 108 "Diese Matrix A kann zerlegt werden in eine Diagonalmatrix und eine damit kommutieren de obere Dreiecksmatrix:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(D);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalm(D & * N) = evalm(N &*D);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "e_h och_Dt := diag(exp(2*t)$2, exp(t)$3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(N &* N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(N^3);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 77 "Die drit te und damit auch alle h\366heren Potenzen von N verschwinden also, d. h." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "ex p(N*t) = Sum(N^i/i!*t^i, i=0..2);\ne_hoch_Nt := evalm(1 + N*t + N^2/2* t^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "e^A = e^D * e^N; e valm(e_hoch_Dt &* e_hoch_Nt);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 78 "Zur Probe noch einmal mit der eingebauten Matrixexponentialfunktio n von Maple:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exponential(A*t);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 58 "Ge nauso lassen sich auch Potenzen von A leicht ausrechnen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A^10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A^10 = Sum(binomial(10, i)*D ^(N-i)*N^i, i=0..10);" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 86 "Da all e Potenzen von N ab der dritten verschwinden, gen\374gen die ersten dr ei Summanden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalm(D^10 + 10*D^9&*N + 45*D^8&*N^2);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 105 "WARNUNG: Das alles funktioniert nur, weil D und N kommutieren!\nI m allgemeinen kann man nicht so rechnen! " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "D := diag(1,2); N := matrix( 2,2,[0,3,0,0]); A := evalm(D+N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalm(D&*N) <> evalm(N&*D);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "exponential(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalm(exponential(D) &* exponential(N));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalm(exponential(N) &* exponential (D));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A^2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(D^2 + 2*D&*N + N^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(D^2 + D&*N + N&*D + N^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A^10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(D^10 + 10*D^9&*N);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(D^10 + 10*N&*D^9);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalm(D^10 + D^9&*N + N&*D ^9 + sum(evalm(D^j&*N&*D^(9-j)), j=1..8));" }}}{EXCHG {PARA 258 "" 0 " " {TEXT 256 47 "Anwendung auf ein Differentialgleichungssystem:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "diff(x (t), t) = 2*x(t) - y(t) - z(t);\ndiff(y(t), t) = x(t) + 5*y(t) + 2*z(t );\ndiff(z(t), t) = -x(t) - 2*y(t) + z(t);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "A := matrix(3,3,[2,-1,-1,1,5,2,-1,-2,1]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "det(A-lambda);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 102 "Zum Eigenwert 3 mit algebraischer Vielfachheit zwei gibt es also nur einen eindimensionalen Eigenraum." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v1 := vector([1, -1, 1]); # \+ Eigenvektor zu 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v2 := v ector([0, -1, 1]); # Eigenvektor zu 3" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 78 "Hauptvektoren zweiter Stufe zum Eigenwert drei werden ann ulliert von (A-3E)^2:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm((A-3)^2);" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 84 "Der Eigenvektor v2 ist eine L\366sung; eine davon linear unabh \344ngige weitere L\366sung ist" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "v3 := vector([1,-1,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalm((A-3)^2 &* v3);" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 108 "Um A bez\374glich der Basis v1, v2, v3 ausdr\374 cken zu k\366nnen, m\374ssen wir wissen, wohin A den Vektor v3 abbilde t:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ev alm(A &* v3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalm(% - \+ 3*v3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalm(3*v3 + v2); " }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 70 "Also ist A v3 = v2 + 3 v3 u nd A hat bez\374glich der neuen Basis die Form" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "M := transpose(matrix([[2,0,0],[0,3,0],[0,1,3] ]));" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 67 "Zerlegung in Diagonalma trix und damit kommutierende Dreiecksmatrix:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "D := diag(2,3,3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "N := matrix(3,3, [0$5, 1, 0$ 3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "e_hoch_Dt := diag(e xp(2*t), exp(3*t)$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ev alm(N^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "e_hoch_Nt := e valm(1 + N*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalm(e_h och_Dt &* e_hoch_Nt);" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 105 "Das m u\337 noch zur\374cktransformiert werden in die alte Basis.\nDie Matri x des Basiswechsels zu v1, v2, v3 ist" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "B := transpose(matrix(map(convert, \+ [v1, v2, v3], list)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "i nverse(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "e_hoch_At := \+ evalm(B &* e_hoch_Dt &* e_hoch_Nt &* inverse(B));" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 14 "Zum Vergleich:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exponential(A*t);" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 72 "Der allgemeine L\366sungsvektor des Different ialgleichungssystems ist somit" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 58 "Loesung := expand(evalm(e_hoch_At &* [x[0], \+ y[0], z[0]]));" }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 71 "Die drei L \366sungsfunktionen sind, in Abh\344ngigkeit von den Anfangswerten " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x(t) := Loesung[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y(t) := Loe sung[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "z(t) := Loesung [3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }