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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 38 "Beispiel f
\374r die Vorlesung am 8.6.2005" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 50 "Nachweis, dass zwei Bei
spielmatrizen \344hnlich sind " }}{PARA 265 "" 0 "" {TEXT -1 55 "und B
estimmung einer entsprechenden Basiswechselmatrix." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }{TEXT -1 
64 "Es gibt in Maple neben 'linalg' auch das Paket 'LinearAlgebra'. " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Gegeben sind die folgenden Matr
izen A,B aus " }{XPPEDIT 18 0 "Z[2]^`3x3`" "6#)&%\"ZG6#\"\"#%$3x3G" }
{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "A:=matri
x([[1, 0, 0], [0, 1, 1], [1, 1, 0]]):B:=matrix([[0,1,1], [0, 1, 0], [1
, 1, 1]]):A=evalm(A),B=evalm(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
56 "(xE-A) und (xE-B) haben die gleiche normierte Smithform:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 121 "xE:=diag(x,x,x):`Smithform von xE-A`=Smi
th(evalm(xE-A),x,U,V) mod 2,\n`Smithform von xE-B`=Smith(evalm(xE-B),x
,W,X) mod 2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Die beiden Matriz
en A, B sind also nach Satz 10.5 \344hnlich." }}{PARA 0 "" 0 "" {TEXT 
-1 36 "Es gilt mit  U,V,W,X,P,Q  aus GL(3, " }{XPPEDIT 18 0 "Z[2];" "6
#&%\"ZG6#\"\"#" }{TEXT -1 33 "[x]):   U (xE-A) V = W (xE-B) X ." }}
{PARA 0 "" 0 "" {TEXT -1 75 "und mit  P=W^(-1) U  und  Q=X V^(-1)  gil
t deswegen:   P (xE-A) = (xE-B) Q." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
103 "U=evalm(U),V=evalm(V),W=evalm(W),X=evalm(X);P,Q:=map(modp,map(exp
and,evalm((Inverse(W) mod 2)&*U)),2 )," }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 54 "map(modp,map(expand,evalm(X&*(Inverse(V) mod 2))),2 );" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "map(modp,map(expand,evalm( P &* (xE-A) - (xE-B) &* Q \+
 )),2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Bestimmung einer Basis
wechselmatrix T, mit der gilt:  " }{XPPEDIT 18 0 "T^`-1`*A*T = B;" "6#
/*()%\"TG%#-1G\"\"\"%\"AGF(F&F(%\"BG" }{TEXT -1 3 "  ." }}{PARA 0 "" 
0 "" {TEXT -1 71 "Dazu werden zuerst die Matrizen P und Q als Matrixpo
lynome geschrieben." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Mcoeff:=(M,
k)->map(coeff,M,x,k):Mpoly:=(M,d)->add(evalm(Mcoeff(M,d-k))*x^(d-k),k=
0..d):P=Mpoly(P,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Q=Mp
oly(Q,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Nun ist Links- und \+
Rechts-Einsetzung m\366glich zur Gewinnung von T. Man erh\344lt bei Li
nksseinsetzung von B in P:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "T:=ma
p(modp,evalm(add(   B&^k&*Mcoeff(P,k)   ,k=0..3)),2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "oder mit gleichem Ergebnis bei Rechtseins
etzung von A in Q:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "map(modp,eval
m(add(   Mcoeff(Q,k)&*A&^k   ,k=0..3)),2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "Probe: es muss nun gelten     B T - T A = 0 ." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "map(modp,evalm(B&*T-T&*A),2);" }}}}{MARK "0
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