{VERSION 5 0 "HP RISC UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "" 0 1 0 0 0 0 0 1 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 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 O utput" 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 "" 0 256 1 {CSTYLE "" -1 -1 "He lvetica" 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 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 256 "" 0 "" {TEXT -1 40 "Themenvorschl\344ge f \374r den 15. Januar 2003" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "restart: with(linalg):\nA := diag(1,2,1); e^A = \+ exponential(diag(1,2,1));\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, t he protected names norm and trace have been redefined and unprotected \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\" !F+7%F+\"\"#F+7%F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG%\"AG -%'matrixG6#7%7%-%$expG6#\"\"\"\"\"!F07%F0-F-6#\"\"#F07%F0F0F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "A := matrix(3,3,[0,1,2,0,0,3 ,0,0,0]);\nA^2 = evalm(A^2);\nA^3 = evalm(A^3);\ne^A = exponential(A); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"!\"\" \"\"\"#7%F*F*\"\"$7%F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"A G\"\"#\"\"\"-%'matrixG6#7%7%\"\"!F.\"\"$7%F.F.F.F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*$)%\"AG\"\"$\"\"\"-%'matrixG6#7%7%\"\"!F.F.F-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG%\"AG-%'matrixG6#7%7%\"\"\"F,# \"\"(\"\"#7%\"\"!F,\"\"$7%F1F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "A := matrix([[1,0],[1,1]]);\nA^2 = evalm(A^2);\nA^3 = evalm(A^3);\ne^A = exponential(A);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"\"\"\"!7$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"#\"\"\"-%'matrixG6#7$7$F(\"\"!7$F'F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"AG\"\"$\"\"\"-%'matrixG6#7$7$F( \"\"!7$F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG%\"AG-%'matrixG6 #7$7$-%$expG6#\"\"\"\"\"!7$F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "P := diag(2*Pi*I, 2*Pi*I, 2*Pi*I);\ne^P = exponential(P);\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7%7%*&^#\"\"#\"\"\" %#PiGF-\"\"!F/7%F/F*F/7%F/F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)% \"eG%\"PG-%'matrixG6#7%7%\"\"\"\"\"!F-7%F-F,F-7%F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Q := diag(2*Pi, 2*Pi, 2*Pi);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'matrixG6#7%7%,$*&\"\"#\"\"\"% #PiGF-F-\"\"!F/7%F/F*F/7%F/F/F*" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 17 "d) Richtig, denn " }{XPPEDIT 18 0 "det(A-lambda*E);" "6#-%$detG6#, &%\"AG\"\"\"*&%'lambdaGF(%\"EGF(!\"\"" }{TEXT -1 32 " verschwindet gen au dann, wenn " }{XPPEDIT 18 0 "det(2*A-2*lambda*E);" "6#-%$detG6#,&* &\"\"#\"\"\"%\"AGF)F)*(F(F)%'lambdaGF)%\"EGF)!\"\"" }{TEXT -1 17 " ver schwindet.\ne)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A := matr ix([[1, -1, 2], [-2, 2, -3], [4, -4, 11]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"!\"\"\"\"#7%!\"#F,!\"$7%\" \"%!\"%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "det(A - lamb da);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#8\"\"\"%'lambdaGF&!\" \"*&\"#9F&)F'\"\"#F&F&*$)F'\"\"$F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(% 'lambdaG\"\"\",&F%F&F&!\"\"F&,&F%F&\"#8F(F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%\"#8\"\"\"<#-%'vectorG6#7%F%#!\")\"\"&#\"#EF-7%\"\"!F%<#-F(6#7 %F%F%F17%F%F%<#-F(6#7%F%!\"%!\"#" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 66 "(Beim Eigenwert 13 sieht der f\374nffache Eigenvektor besser au s.)\nf)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "\011B := matrix([[5,-6,-6],[3,-4,-5],[-2,4,5]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%\"\"&!\"'F+7%\"\"$!\"%!\"&7% !\"#\"\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sort(det(B - lambda));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%'lambdaG\"\"$\" \"\"!\"\"*&\"\"'F()F&\"\"#F(F(*&\"#6F(F&F(F)F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%'lambdaG\"\"\"F'!\"\"F',&F&F'\"\"#F(F',&\"\"$F(F&F'F'F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(B);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%\"\"\"F$<#-%'vectorG6#7%\"\"!!\"\"F$7%\"\" #F$<#-F'6#7%F-F$F*7%\"\"$F$<#-F'6#7%!\"$!\"#F$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "e^B = exponential(B);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG%\"BG-%'matrixG6#7%7%,&*&\"\"#\"\"\"-%$expG6#F.F /!\"\"*&\"\"$F/-F16#F5F/F/,&*&\"\"'F/F6F/F3*&F:F/F0F/F/F87%,(F0F3-F16# F/F3*&F.F/F6F/F/,(*&F.F/F>F/F/*&\"\"%F/F6F/F3*&F5F/F0F/F/,(*&FDF/F6F/F 3*&F5F/F0F/F/F>F/7%,&F>F/F6F3,&*&F.F/F6F/F/*&F.F/F>F/F3,&F>F3*&F.F/F6F /F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "e^(B*t) = exponentia l(B*t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG*&%\"BG\"\"\"%\"tG F(-%'matrixG6#7%7%,&*&\"\"$F(-%$expG6#,$*&F1F(F)F(F(F(F(*&\"\"#F(-F36# ,$*&F8F(F)F(F(F(!\"\",&*&\"\"'F(F9F(F(*&F@F(F2F(F=F>7%,(*&F8F(F2F(F(-F 36#F)F=F9F=,(*&F8F(FEF(F(*&F1F(F9F(F(*&\"\"%F(F2F(F=,(*&F1F(F9F(F(*&FK F(F2F(F=FEF(7%,&F2F=FEF(,&*&F8F(F2F(F(*&F8F(FEF(F=,&FEF=*&F8F(F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "[x(t), y(t), z(t)] = eval m(exponential(B*t) &* [0,0,7]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ 7%-%\"xG6#%\"tG-%\"yGF'-%\"zGF'-%'vectorG6#7%,&*&\"#U\"\"\"-%$expG6#,$ *&\"\"#F4F(F4F4F4F4*&F3F4-F66#,$*&\"\"$F4F(F4F4F4!\"\",(*&\"#@F4F5F4F4 *&\"#GF4F " 0 "" {MPLTEXT 1 0 29 "C := matrix([[1,I],[-I,1]]); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7$7$\"\"\"^#F* 7$^#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "det(C-lambda );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%'lambdaGF&!\" \"*$)F'F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects (C);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"!\"\"\"<#-%'vectorG6#7 $^#!\"\"F%7%\"\"#F%<#-F(6#7$^#F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ifactor(-30);\nfactor(x^3 - 12*x^2 + 41*x - 30); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%!G6#\"\"#\"\"\"-F&6#\"\"$F)-F& 6#\"\"&F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"F&!\" \"F&,&F%F&\"\"&F'F&,&F%F&\"\"'F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ifactor(33);\nfactor(x^4-8*x^3-34*x^2+8*x+33);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%!G6#\"\"$\"\"\"-F%6#\"#6F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"\"#6!\"\"F&,&F%F&F&F(F& ,&F%F&\"\"$F&F&,&F%F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ifactor(-70);\nfactor(x^4+17*x^3+69*x^2-17*x-70);\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*(-%!G6#\"\"#\"\"\"-F&6#\"\"&F)-F&6#\"\"(F)!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"\"#5F&F&,&F%F&F&! \"\"F&,&F%F&\"\"(F&F&,&F%F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ifactor(4);\nfactor(x^5-3*x^4-x^3+11*x^2-12*x+4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%!G6#\"\"#F(\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"\"\"#!\"\"F&,&F%F&F'F&F&),&F%F&F& F(\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "ifactor(12/3) ; # Das Polynom mu\337 zuerst durch den f\374hrenden Koeffizenten divi diert werden\nfactor(3*x^6+6*x^5-12*x^4-30*x^3-3*x^2+24*x+12);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%!G6#\"\"#F(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"$\"\"\",&%\"xGF&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 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 396 "p) Eigenvektoren sind P olynome, deren zweite Ableitung ein Vielfaches des Polynoms selbst ist . Somit sind alle h\366chstens linearen Polynome (au\337er dem Nullpol ynom) Eigenvektoren zum Eigenwert null, aber sonst gibt es keine Eigen werte und Eigenvektoren, denn die zweite Ableitung eines Polynoms (ung leich dem Nullpolynom) hat kleineren Grad als das Polynom selbst.\nq) \+ Ein Eigenvektor zum Eigenwert " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG " }{TEXT -1 78 " ist eine beliebig oft stetig differenzierbare Funktio n, deren Ableitung das " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" } {TEXT -1 63 "-fache der Funktion selbst ist. Also sind genau die Funkt ionen " }{XPPEDIT 18 0 "x(t) = C*exp(lambda(t));" "6#/-%\"xG6#%\"tG*&% \"CG\"\"\"-%$expG6#-%'lambdaG6#F'F*" }{TEXT -1 6 " mit " }{XPPEDIT 18 0 "C <> 0;" "6#0%\"CG\"\"!" }{TEXT -1 18 " Eigenvektoren zu " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 24 ", und insbesonder e ist " }{TEXT 256 4 "jede" }{TEXT -1 28 " reelle Zahl ein Eigenwert. \+ " }}}}{MARK "1 0 0" 23 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }