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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 62 "Ein Beispiel zur Rechts- und Linksdivision bei Polynommatrizen
" }}{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):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected \+
names norm and trace have been redefined and unprotected\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "M:=matrix(2,2,[x+1,x^4+x+1,x^3+x,x^
5+x^4+1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7$7$,&
%\"xG\"\"\"F,F,,(*$)F+\"\"%F,F,F+F,F,F,7$,&*$)F+\"\"$F,F,F+F,,(*$)F+\"
\"&F,F,F.F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Gradberechnung
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Mdeg:=M->max(seq(seq(degree(M[
i,j],x),j=1..2),i=1..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MdegGf
*6#%\"MG6\"6$%)operatorG%&arrowGF(-%$maxG6#-%$seqG6$-F06$-%'degreeG6$&
9$6$%\"iG%\"jG%\"xG/F;;\"\"\"\"\"#/F:F>F(F(F(" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 24 "Koeffizientenberechnung:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "Mcoeff:=(M,k)->map(coeff,M,x,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'McoeffGf*6$%\"MG%\"kG6\"6$%)operatorG%&arrowGF)-%$ma
pG6&%&coeffG9$%\"xG9%F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Da
rstellung von M als Matrixpolynom:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "Mpoly:=(M,d)->add(evalm(Mcoeff(M,k))*x^k,k=0..d);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&MpolyGf*6$%\"MG%\"dG6\"6$%)operatorG%&arrowGF)-
%$addG6$*&-%&evalmG6#-%'McoeffG6$9$%\"kG\"\"\")%\"xGF8F9/F8;\"\"!9%F)F
)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Im Beispiel ist:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "`Grad von M`=Mdeg(M),`   und  M`=Mpoly(M,
5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%+Grad~von~MG\"\"&/%*~~~und~~M
G,.-%'matrixG6#7$7$\"\"\"F.7$\"\"!F.F.*&-F*6#7$F-7$F.F0F.%\"xGF.F.*&-F
*6#7$7$F0F0F;F.)F6\"\"#F.F.*&-F*6#7$F;F5F.)F6\"\"$F.F.*&-F*6#7$F/F/F.)
F6\"\"%F.F.*&-F*6#7$F;F/F.)F6F%F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 41 "Division der Matrix M von rechts durch L:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "L:=matrix(2,2,[x^2,1,1,x^2]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"LG-%'matrixG6#7$7$*$)%\"xG\"\"#\"\"\"F.7$F.F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "L=Mpoly(L,2);dL:=Mdeg(L
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"LG,(-%'matrixG6#7$7$\"\"!\"
\"\"7$F,F+F,*&-F'6#7$7$F+F+F2F,%\"xGF,F,*&-F'6#7$F-F*F,)F3\"\"#F,F," }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dLG\"\"#" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 11 "1. Schritt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "M0:=evalm(M):Q:=matrix(2,2,0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "k:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "dM:=Mdeg(M||(k-1)):\nM
||k:=M||(k-1)-x^(dM-dL)*Mcoeff(M||(k-1),dM)&*L;\nM||k:=map(modp,evalm(
M||k),2)\n;M||k=Mpoly(M||k,dM);\nQ:=evalm(x^(dM-dL)*Mcoeff(M||(k-1),dM
)+Q):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G,&%#M0G\"\"\"-%#&*G6$*&
)%\"xG\"\"$F'-%'matrixG6#7$7$\"\"!F47$F4F'F'%\"LG!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7$7$,&%\"xG\"\"\"F,F,,(*$)F+\"\"
%F,F,F+F,F,F,7$F+,&F.F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M1G,
.-%'matrixG6#7$7$\"\"\"F+7$\"\"!F+F+*&-F'6#7$F*7$F+F-F+%\"xGF+F+*&-F'6
#7$7$F-F-F8F+)F3\"\"#F+F+*&F5F+)F3\"\"$F+F+*&-F'6#7$F,F,F+)F3\"\"%F+F+
*&F5F+)F3\"\"&F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "2. Schritt:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "k:=2:\ndM:=Mdeg(M||(k-1)):\nM|
|k:=M||(k-1)-x^(dM-dL)*Mcoeff(M||(k-1),dM)&*L;\nM||k:=map(modp,evalm(M
||k),2);\nM||k=Mpoly(M||k,dM);\nQ:=evalm(x^(dM-dL)*Mcoeff(M||(k-1),dM)
+Q):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G,&%#M1G\"\"\"-%#&*G6$*&)
%\"xG\"\"#F'-%'matrixG6#7$7$\"\"!F'F3F'%\"LG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M2G-%'matrixG6#7$7$,(%\"xG\"\"\"F,F,*$)F+\"\"#F,F,,&
F+F,F,F,7$,&F+F,F-F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M2G,,-%'m
atrixG6#7$7$\"\"\"F+7$\"\"!F+F+*&-F'6#7$F*7$F+F-F+%\"xGF+F+*&-F'6#7$F2
F2F+)F3\"\"#F+F+*&-F'6#7$7$F-F-F>F+)F3\"\"$F+F+*&F;F+)F3\"\"%F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "3. Schritt:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 169 "k:=3:\ndM:=Mdeg(M||(k-1)):\nM||k:=M||(k-1)-x^(dM-dL)
*Mcoeff(M||(k-1),dM)&*L;\nM||k:=map(modp,evalm(M||k),2);\nM||k=Mpoly(M
||k,dM);\nQ:=evalm(x^(dM-dL)*Mcoeff(M||(k-1),dM)+Q):" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#M3G,&%#M2G\"\"\"-%#&*G6$-%'matrixG6#7$7$F'\"\"!F/
%\"LG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'matrixG6#7$7$,&
%\"xG\"\"\"F,F,F+7$F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M3G,(
-%'matrixG6#7$7$\"\"\"\"\"!7$F,F,F+*&-F'6#7$7$F+F+F*F+%\"xGF+F+*&-F'6#
7$F-F-F+)F3\"\"#F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Nun kann \+
aus Gradgr\374nden nicht mehr weiter reduziert werden." }}{PARA 0 "" 
0 "" {TEXT -1 41 "Das Ergebnis der Division von rechts ist:" }}{PARA 
259 "" 0 "" {TEXT -1 7 "M=Q*L+R" }}{PARA 0 "" 0 "" {TEXT -1 3 "mit" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Q=evalm(Q), R=evalm(M||k);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"QG-%'matrixG6#7$7$\"\"\"*$)%\"
xG\"\"#F*7$F*,&F+F**$)F-\"\"$F*F*/%\"RG-F&6#7$7$,&F-F*F*F*F-7$F-\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "map(modp,map(expand,evalm(M-Q&*L-M||k)),2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!F(F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Was ergibt die \+
Division von links in diesem Beispiel?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 11 "1. Schritt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"M0:=evalm(M):Q:=matrix(2,2,0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k
:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "dM:=Mdeg(M||(k-1)):\nM||k:
=M||(k-1)-x^(dM-dL)*L&*Mcoeff(M||(k-1),dM);\nM||k:=map(modp,evalm(M||k
),2);\nM||k=Mpoly(M||k,dM);\nQ:=evalm(x^(dM-dL)*Mcoeff(M||(k-1),dM)+Q)
:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G,&%#M0G\"\"\"-%#&*G6$*&)%\"
xG\"\"$F'%\"LGF'-%'matrixG6#7$7$\"\"!F57$F5F'!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7$7$,&%\"xG\"\"\"F,F,,**$)F+\"\"%F,
F,F+F,F,F,*$)F+\"\"$F,F,7$,&F1F,F+F,,&F.F,F,F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#M1G,.-%'matrixG6#7$7$\"\"\"F+7$\"\"!F+F+*&-F'6#7$F*7
$F+F-F+%\"xGF+F+*&-F'6#7$7$F-F-F8F+)F3\"\"#F+F+*&-F'6#7$F,F2F+)F3\"\"$
F+F+*&-F'6#7$F,F,F+)F3\"\"%F+F+*&F5F+)F3\"\"&F+F+" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 11 "2. Schritt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 181 
"k:=2:\ndM:=Mdeg(M||(k-1)):\nM||k:=M||(k-1)-x^(dM-dL)*L&*Mcoeff(M||(k-
1),dM);\nM||k:=map(modp,map(expand,evalm(M||k)),2);\nM||k=Mpoly(M||k,d
M);\nQ:=evalm(x^(dM-dL)*Mcoeff(M||(k-1),dM)+Q):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M2G,&%#M1G\"\"\"-%#&*G6$*&)%\"xG\"\"#F'%\"LGF'-%'mat
rixG6#7$7$\"\"!F'F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'m
atrixG6#7$7$,&%\"xG\"\"\"F,F,,*F+F,F,F,*$)F+\"\"$F,F,*$)F+\"\"#F,F,7$,
&F.F,F+F,,&F1F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M2G,,-%'matr
ixG6#7$7$\"\"\"F+7$\"\"!F+F+*&-F'6#7$F*7$F+F-F+%\"xGF+F+*&-F'6#7$F,F,F
+)F3\"\"#F+F+*&-F'6#7$F,F2F+)F3\"\"$F+F+*&-F'6#7$7$F-F-FDF+)F3\"\"%F+F
+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "3. Schritt:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 181 "k:=3:\ndM:=Mdeg(M||(k-1)):\nM||k:=M||(k-1)-x^(d
M-dL)*L&*Mcoeff(M||(k-1),dM);\nM||k:=map(modp,map(expand,evalm(M||k)),
2);\nM||k=Mpoly(M||k,dM);\nQ:=evalm(x^(dM-dL)*Mcoeff(M||(k-1),dM)+Q):
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G,&%#M2G\"\"\"-%#&*G6$*&%\"xG
F'%\"LGF'-%'matrixG6#7$7$\"\"!F'7$F'F3!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M3G-%'matrixG6#7$7$\"\"\",(%\"xGF*F*F**$)F,\"\"#F*F*
7$F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#M3G,*-%'matrixG6#7$7$\"\"
\"F+7$\"\"!F+F+*&-F'6#7$F,F*F+%\"xGF+F+*&-F'6#7$F,F,F+)F2\"\"#F+F+*&-F
'6#7$7$F-F-F=F+)F2\"\"$F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "4.
 Schritt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "k:=4:\ndM:=Mdeg(M||(k
-1)):\nM||k:=M||(k-1)-x^(dM-dL)*L&*Mcoeff(M||(k-1),dM);\nM||k:=map(mod
p,map(expand,evalm(M||k)),2);\nM||k=Mpoly(M||k,dM);\nQ:=evalm(x^(dM-dL
)*Mcoeff(M||(k-1),dM)+Q):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M4G,&%
#M3G\"\"\"-%#&*G6$%\"LG-%'matrixG6#7$7$\"\"!F'F0!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#M4G-%'matrixG6#7$7$\"\"\"%\"xG7$F+F+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%#M4G,(-%'matrixG6#7$7$\"\"\"\"\"!7$F,F,F+*
&-F'6#7$7$F,F+7$F+F+F+%\"xGF+F+*&-F'6#7$F-F-F+)F4\"\"#F+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "Nun kann aus Gradgr\374nden nicht mehr we
iter reduziert werden." }}{PARA 0 "" 0 "" {TEXT -1 40 "Das Ergebnis de
r Division von links ist:" }}{PARA 259 "" 0 "" {TEXT -1 7 "M=L*Q+R" }}
{PARA 0 "" 0 "" {TEXT -1 3 "mit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "Q=evalm(Q), R=evalm(M||k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
/%\"QG-%'matrixG6#7$7$\"\"!,(%\"xG\"\"\"F-F-*$)F,\"\"#F-F-7$F,,(F-F-F.
F-*$)F,\"\"$F-F-/%\"RG-F&6#7$7$F-F,7$F,F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "Probe:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "map(m
odp,map(expand,evalm(M-L&*Q-M||k)),2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$\"\"!F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }