{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 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 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 0 }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 0 }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 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "helvet ica" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 279 "" 0 "" {TEXT 256 40 "Themenvorschl\344ge f \374r den 22. Januar 2003" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 2 "a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "C := matrix([[1,I ],[-I,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7$7$ \"\"\"^#F*7$^#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "de t(C - lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%'lam bdaGF&!\"\"*$)F'F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f actor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%'lambdaG\"\"\",&\"\"#! \"\"F$F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"!\"\"\"<#-%'vectorG6#7$^#! \"\"F%7%\"\"#F%<#-F(6#7$^#F%F%" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 2 "b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "B := transpose(matrix([[-I , 1],[I,1]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$ 7$^#!\"\"^#\"\"\"7$F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " inverse(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$^##\"\" \"\"\"#F)7$^##!\"\"F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " e^C = evalm(B &* diag(1, exp(2)) &* inverse(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG%\"CG-%'matrixG6#7$7$,&#\"\"\"\"\"#F.*&F-F.-%$ex pG6#F/F.F.,&^##!\"\"F/F.*&^#F-F.F1F.F.7$,&F9F.*&F5F.F1F.F.F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "e^(C*t) = evalm(B &* diag(1, exp(2*t)) &* inverse(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG* &%\"CG\"\"\"%\"tGF(-%'matrixG6#7$7$,&#F(\"\"#F(*&F0F(-%$expG6#,$*&F1F( F)F(F(F(F(,&^##!\"\"F1F(*&^#F0F(F3F(F(7$,&F=F(*&F9F(F3F(F(F/" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 2 "c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A := matrix([[0,-1],[1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"AG-%'matrixG6#7$7$\"\"!!\"\"7$\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'matrixG6#7$7$!\"\"\"\"!7$F)F(" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 101 "Damit lassen sich die Potenzen von A explizit angeben und die Exp onentialreihe auswerten mit Ergebnis" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e^(A*t) = exponential(A*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ )%\"eG*&%\"AG\"\"\"%\"tGF(-%'matrixG6#7$7$-%$cosG6#F),$-%$sinGF1!\"\"7 $F3F/" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 11 "Alternativ:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "det(A-lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%'lambdaG\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%^#\"\"\"F%<#-%'vectorG6#7$F%^#!\"\"7%F+F%<#-F(6#7$F%F$" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 76 "Wenn man mit -i beginnt, kann ma n mit derselben Matrix B arbeiten wie bei b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "e^(A*t) = evalm(B &* diag(exp(-I*t), exp(I*t)) &* inv erse(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG*&%\"AG\"\"\"%\"tG F(-%'matrixG6#7$7$,&*&#F(\"\"#F(-%$expG6#*&F)F(^#F(F(F(F(*&F1F(-F46#*& ^#!\"\"F(F)F(F(F(,&*&^##F=F2F(F9F(F(*&^#F1F(F3F(F(7$,&*&FCF(F9F(F(*&F@ F(F3F(F(F/" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 2 "d)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "M := matrix([[1,t],[1,1]]);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7$7$\"\"\"%\"tG7$F*F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "det(M - lambda);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&\"\"#F$%'lambdaGF$!\"\"*$) F'F&F$F$%\"tGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(%, lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"\"F$*$%\"tG#F$\"\"# F$,&F$F$F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvec ts(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&\"\"\"F%*$%\"tG#F%\"\"#F %F%<#-%'vectorG6#7$F&F%7%,&F%F%F&!\"\"F%<#-F,6#7$,$F&F1F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "B := transpose(matrix([[sqrt(t), 1] ,[-sqrt(t), 1]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG 6#7$7$*$%\"tG#\"\"\"\"\"#,$F*!\"\"7$F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7$7$,$*&\"\"\"F**&\"\"#F*%\"tG#F*F,!\"\"F*#F*F,7$,$*&F*F**&F,F*F -#F*F,F/F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalm(B &* \+ diag(exp(1 + sqrt(t)), exp(1 - sqrt(t))) &* inverse(B)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*&#\"\"\"\"\"#F+-%$expG6#,&F+ F+*$%\"tGF*F+F+F+*&F*F+-F.6#,&F+F+F1!\"\"F+F+,&*&F*F+*&F2F*F-F+F+F+*&# F+F,F+*&F2F*F4F+F+F77$,&*&F*F+*&F-F+F2#F7F,F+F+*&#F+F,F+*&F4F+F2FBF+F7 F(" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 137 "Ausklammern von e macht \+ das Ganze etwas \374bersichtlicher, aber nicht viel.\ne) Die Matrix i st offensichtlich ihre eigene Transponierte.\nf)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a = conjugate(1-2*I); b = conjugate(1+I); c = conjuga te(3-I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG^$\"\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG^$\"\"\"!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"cG^$\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 3 "g) " } {XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 29 " symmetrisch un d Hermitesch, " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{TEXT -1 6 " und " }{XPPEDIT 18 0 "A[3];" "6#&%\"AG6#\"\"$" }{TEXT -1 38 " symme trisch, aber nicht Hermitesch, " }{XPPEDIT 18 0 "A[4];" "6#&%\"AG6#\" \"%" }{TEXT -1 37 " Hermitesch, aber nicht symmetrisch, " }{XPPEDIT 18 0 "A[5];" "6#&%\"AG6#\"\"&" }{TEXT -1 107 " weder symmetrisch noch Hermitsch, da die Diagonaleintraege einer Hermitschen Matrix reell se in m\374ssen, " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 107 " aus demselben Grund symmetrisch, aber nicht Hermitesch. Nach de m Satz aus der Vorlesung gibt es daher zu " }{XPPEDIT 18 0 "A[1];" "6# &%\"AG6#\"\"\"" }{TEXT -1 6 " und " }{XPPEDIT 18 0 "A[4];" "6#&%\"AG6 #\"\"%" }{TEXT -1 31 " Basen aus Eigenvektoren. Bei " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 110 " sind offensichtlich bereits die Vektoren der Standardbasis Eigenvektoren, und wenn man genau hins ieht, ist " }{XPPEDIT 18 0 "A[3] = I*A[1];" "6#/&%\"AG6#\"\"$*&%\"IG \"\"\"&F%6#F*F*" }{TEXT -1 25 " , also gibt es auch zu " }{XPPEDIT 18 0 "A[3];" "6#&%\"AG6#\"\"$" }{TEXT -1 174 " eine. Bei den beiden v erbleibenden Matrizen ist die Frage nicht einfach ohne Rechnung entsch eidbar. Mit Rechnung folgt, dass es solche Basen gibt. (Das war nicht gefragt.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A2 := matrix(4,4, [1, I,1,I,I,1,I,1,1,I,1,I,I,1,I,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #A2G-%'matrixG6#7&7&\"\"\"^#F*F*F+7&F+F*F+F*F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%^$\"\"#!\"#\"\"\"<#-%'vectorG6#7&!\"\"F'F-F'7%^$F%F%F'<#-F*6#7 &F'F'F'F'7%\"\"!F%<$-F*6#7&F5F-F5F'-F*6#7&F-F5F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A5 := matrix(4,4,[I$4, -I,I,-I,I,-I,I$3, -I $3, I]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G-%'matrixG6#7&7&^#\" \"\"F*F*F*7&^#!\"\"F*F-F*7&F-F*F*F*7&F-F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&7%,&^$!\"\"\"\"\"F'*$\"\"##F'F)F'F'<#-%'vectorG6#7&F'^#F',(^$#F&F)F 3F'*&F)F&F)F*F&*&^#F3F'F$F'F',(F2F'*&F)F&F)F*F'F5F'7%,&F%F'F(F&F'<#-F- 6#7&F'F0,(F2F'*&F)F&F)F*F'*&F6F'F:F'F',(F2F'*&F)F&F)F*F&FAF'7%,&^$F'F' F'F(F'F'<#-F-6#7&,(^$F*F3F'*&F)F&F)F*F&*&^#F*F'FEF'F',(FNF'FLF'*&F)F&F )F*F'F0F'7%,&FFF'F(F&F'<#-F-6#7&,(FLF'*&F)F&F)F*F'*&FOF'FSF'F',(FZF'FL F'*&F)F&F)F*F&F0F'" }}}{EXCHG {PARA 281 "" 0 "" {TEXT -1 2 " \n" }}} {EXCHG {PARA 282 "" 0 "" {TEXT -1 5 " h) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := matrix(3,3, [-2,-3,-1,1,2,1,2,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%!\"#!\"$!\"\"7%\"\"\" \"\"#F.7%F/F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "charpol \+ := unapply(sort(det(A - lambda), lambda), lambda);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(charpolGf*6#%'lambdaG6\"6$%)operatorG%&arrowGF(,* *$)9$\"\"$\"\"\"!\"\"*$)F/\"\"#F1F1F/F1F1F2F(F(F(" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 58 "Probiere, ob Teiler des konstanten Terms Null stellen sind:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "charpol(1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "charpol(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "charpol(lambda)/((lambd a+1)*(lambda-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,**$)%'lambdaG \"\"$\"\"\"!\"\"*$)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 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$%'lambdaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "factor(charpol(lambda));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%'lambdaG\"\"\"F'F'F'),&F&F'F'!\"\"\"\"#F'F*" }}} {EXCHG {PARA 266 "" 0 "" {TEXT -1 29 "Eigenvektor zum Eigenwert -1:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalm(A + 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!\"\"!\"$F(7%\"\"\"\"\"$F+7%\"\"#F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm((A+1) &* [x, y, z]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,(%\"xG!\"\"*&\"\"$ \"\"\"%\"yGF,F)%\"zGF),(F(F,*&F+F,F-F,F,F.F,,(*&\"\"#F,F(F,F,*&F3F,F-F ,F,*&F3F,F.F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve(\{ op(convert(%, list))\}, \{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<%/%\"zGF%/%\"yG\"\"!/%\"xG,$F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "v1 := [-1, 0, 1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#v1G7%!\"\"\"\"!\"\"\"" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 30 "Eig envektoren zum Eigenwert 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval m(A - 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!\"$F(!\" \"7%\"\"\"F+F+7%\"\"#F-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm((A-1) &* [x, y, z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'vectorG6#7%,(*&\"\"$\"\"\"%\"xGF*!\"\"*&F)F*%\"yGF*F,%\"zGF,,(F+F*F.F *F/F*,&*&\"\"#F*F+F*F**&F3F*F.F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve(\{op(convert(%, list))\}, \{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yGF%/%\"zG\"\"!/%\"xG,$F%!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v2 := [-1,1,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%!\"\"\"\"\"\"\"!" }}}{EXCHG {PARA 267 " " 0 "" {TEXT -1 145 "Da der Eigenwert trotz algebraischer Vielfachheit zwei nur geometrische Vielfachheit eins hat, brauchen wir noch einen \+ Hauptvektor zweiter Stufe." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm ((A-1)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"%F( \"\"!7%F)F)F)7%!\"%F,F)" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 88 "Hier gibt es viel Auswahl; ein einfacher, von v2 linear unabh\344ngiger Lo esungsvektor ist " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v3 := [0,0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# v3G7%\"\"!F&\"\"\"" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 28 "i) Matrix des Basiswechsels:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "B := transpo se(matrix([v1, v2, v3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%' matrixG6#7%7%!\"\"F*\"\"!7%F+\"\"\"F+7%F-F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'matrixG6#7%7%!\"\"F(\"\"!7%F)\"\"\"F)7%F+F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "evalm(A &* v3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%!\"\"\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(% - v3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' vectorG6#7%!\"\"\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(A &* v3) = v2 + v3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'v ectorG6#7%!\"\"\"\"\"F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Dreiecksgestalt := matrix([[-1,0,0],[0,1,1],[0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0DreiecksgestaltG-%'matrixG6#7%7%!\"\"\"\" !F+7%F+\"\"\"F-7%F+F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " Diag := diag(-1,1,1); N := evalm(Dreiecksgestalt - Diag);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DiagG-%'matrixG6#7%7%!\"\"\"\"!F+7%F+\"\"\"F +7%F+F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7%7%\" \"!F*F*7%F*F*\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ev alm(N^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(F( F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Dreiecksgestalt^n = Diag^n + n*Diag^(n-1)*N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%0Dreie cksgestaltG%\"nG,&)%%DiagGF&\"\"\"*(F&F*)F),&F&F*F*!\"\"F*%\"NGF*F*" } }}{EXCHG {PARA 271 "" 0 "" {TEXT -1 9 "n gerade;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Dreiecksgestalt^n = evalm(1 + n*Diag &* N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%0DreiecksgestaltG%\"nG-%'matrixG6#7%7%\" \"\"\"\"!F-7%F-F,F&7%F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(B &* rhs(%) &* inverse(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%,&\"\"\"F)%\"nG!\"\",$F*F+F,7%F*,&F)F)F*F)F*7%\"\"! F0F)" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 11 "n ungerade:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Dreiecksgestalt^n = evalm(Diag + n*N);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/)%0DreiecksgestaltG%\"nG-%'matrixG6#7 %7%!\"\"\"\"!F-7%F-\"\"\"F&7%F-F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(B &* rhs(%) &* inverse(B));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7%7%,&\"\"\"!\"\"%\"nGF*,&\"\"#F*F+F*,$F+ F*7%F+,&F)F)F+F)F+7%F-F-F)" }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 2 "j) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "e^(Dreiecksgestalt*t) = e^(Diag *t) * e^(N*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"eG*&%0Dreiecksg estaltG\"\"\"%\"tGF(*&)F%*&%%DiagGF(F)F(F()F%*&%\"NGF(F)F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "exponential(Diag*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%-%$expG6#,$%\"tG!\"\" \"\"!F.7%F.-F)6#F,F.7%F.F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalm( 1 + N*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6# 7%7%\"\"\"\"\"!F)7%F)F(%\"tG7%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalm(exponential(Diag*t) * (1 + N*t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%-%$expG6#,$%\"tG!\"\"\"\"!F.7%F .-F)6#F,*&F0\"\"\"F,F37%F.F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "e_hoch_At := evalm(B &* % &* inverse(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*e_hoch_AtG-%'matrixG6#7%7%,&-%$expG6#,$%\"tG!\"\"\" \"\"*&-F,6#F/F1F/F1F0,(F+F1F3F0F2F0,$F2F07%F2,&F3F1F2F1F27%,&F+F0F3F1F :F3" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 2 "k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "allg_Loesung := evalm(e_hoch_At &* [x0, y0, z0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-allg_LoesungG-%'vectorG6#7%,(*&,&-% $expG6#,$%\"tG!\"\"\"\"\"*&-F-6#F0F2F0F2F1F2%#x0GF2F2*&,(F,F2F4F1F3F1F 2%#y0GF2F2*(F4F2F0F2%#z0GF2F1,(*(F4F2F0F2F6F2F2*&,&F4F2F3F2F2F9F2F2F:F 2,(*&,&F,F1F4F2F2F6F2F2*&FBF2F9F2F2*&F4F2F;F2F2" }}}{EXCHG {PARA 275 " " 0 "" {TEXT -1 2 "l)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalm(e_ho ch_At &* [1,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,& -%$expG6#,$%\"tG!\"\"\"\"\"*(\"\"#F.-F)6#F,F.F,F.F-,$*(F0F.F1F.F,F.F., &F(F-*&F0F.F1F.F." }}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 2 "m)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(t = t-3, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&-%$expG6#,&%\"tG!\"\"\"\"$\"\"\"F/*( \"\"#F/-F)6#,&F,F/F.F-F/F4F/F-,$*(F1F/F2F/F4F/F/,&F(F-*&F1F/F2F/F/" }} }{EXCHG {PARA 277 "" 0 "" {TEXT -1 2 "n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "x := sort(expand(allg_Loesung[1], exp(t)));\ny := so rt(expand(allg_Loesung[2], exp(t)));\nz := sort(expand(allg_Loesung[3] , exp(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,.*(-%$expG6#%\"t G\"\"\"F*F+%#x0GF+!\"\"*(F'F+F*F+%#y0GF+F-*(F'F+F*F+%#z0GF+F-*&F'F+F/F +F-*&F'F-F,F+F+*&F'F-F/F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG, **(-%$expG6#%\"tG\"\"\"F*F+%#x0GF+F+*(F'F+F*F+%#y0GF+F+*(F'F+F*F+%#z0G F+F+*&F'F+F.F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,,*&-%$expG6 #%\"tG\"\"\"%#x0GF+F+*&F'F+%#y0GF+F+*&F'F+%#z0GF+F+*&F'!\"\"F,F+F2*&F' F2F.F+F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(x, t=infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%'signumG6#,(%#x0G\"\" \"%#y0GF*%#z0GF*F*%)infinityGF*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(y, t=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'signumG6#,(%#x0G\"\"\"%#y0GF)%#z0GF)F)%)infinityGF)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(z, t=infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'signumG6#,(%#x0G\"\"\"%#y0GF)%#z0 GF)F)%)infinityGF)" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 22 "Die Bedin gung ist also" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x0 + y0 + z0 = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%#x0G\"\"\"%#y0GF&%#z0GF&\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 13 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }