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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Poly_Smith_neu.mws        \+
     12. Juni 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "Beispiel zur Berechnung der Smith-Form  von Polynom-Matri
zen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Definition der Elementarm
atrizen als Prozeduren, n gibt die Zeilenanzahl (=Spaltenanzahl) an:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P:=proc(n,i,j) local k;swaprow(di
ag(seq(1,k=1..n)),i,j);end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "S:=p
roc(n,i,l) local k; evalm(mulrow(diag(seq(1,k=1..n)),i,l));end:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Q:=proc(n,i,j,l) local k; evalm(add
row(diag(seq(1,k=1..n)),i,j,l));end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 
"Festlegung der Matrix:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=3:" }
{TEXT -1 38 " (Festlegung des Koeffizientenk\366rpers)" }{MPLTEXT 1 0 
1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 205 "M := map(modp,matrix([[x^2-1,
 x^3+3*x^2-x-3, x^3-3*x^2-x+3], [x^3+3*x^2-x-3, -x^4-x^3-4*x^2-5*x-1, \+
-x^3-x^2-x-1], [x^3+x^2-x-1, x^6+8*x^5+23*x^4+21*x^3-7*x^2-23*x-11, x^
5+9*x^4+32*x^3+52*x^2+41*x+13]]),p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"MG-%'matrixG6#7%7%,&*$)%\"xG\"\"#\"\"\"F/F.F/,&*$)F-\"\"$F/F/*&F
.F/F-F/F/F07%F0,,*&F.F/)F-\"\"%F/F/*&F.F/F2F/F/*&F.F/F,F/F/F-F/F.F/,**
&F.F/F2F/F/*&F.F/F,F/F/*&F.F/F-F/F/F.F/7%,*F1F/F+F/*&F.F/F-F/F/F.F/,.*
$)F-\"\"'F/F/*&F.F/)F-\"\"&F/F/*&F.F/F8F/F/*&F.F/F,F/F/F-F/F/F/,,*$FHF
/F/*&F.F/F2F/F/F+F/*&F.F/F-F/F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
100 "Abk\374rzung der stets notwendigen Operationen \"Reduzieren modul
o p\" und  \"entwickeln nach Potenzen \" :" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "RE:=X->map(sort,map(modp,map(expand,evalm(X)),p));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REGf*6#%\"XG6\"6$%)operatorG%&arro
wGF(-%$mapG6$%%sortG-F-6%%%modpG-F-6$%'expandG-%&evalmG6#9$%\"pGF(F(F(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Im " }{TEXT 256 15 "ersten Schritt \+
" }{TEXT -1 89 "werden durch elementare Zeilen/Spalten-Umformungen die
 Eintr\344ge modulo M[1,1] reduziert: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "M1:=RE(Q(3,1,2,-quo(M[2,1],M[1,1],x)) &* M);\n\nU:=Q(
3,1,2,-quo(M[2,1],M[1,1],x)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1
G-%'matrixG6#7%7%,&*$)%\"xG\"\"#\"\"\"F/F.F/,&*$)F-\"\"$F/F/*&F.F/F-F/
F/F07%\"\"!,**$)F-\"\"%F/F/*&F.F/F2F/F/F-F/F.F/,**&F.F/F9F/F/*&F.F/F2F
/F/*&F.F/F-F/F/F.F/7%,*F1F/F+F/*&F.F/F-F/F/F.F/,.*$)F-\"\"'F/F/*&F.F/)
F-\"\"&F/F/*&F.F/F9F/F/*&F.F/F,F/F/F-F/F/F/,,*$FHF/F/*&F.F/F2F/F/F+F/*
&F.F/F-F/F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "M2:=RE(Q
(3,1,3,-quo(M1[3,1],M1[1,1],x)) &* M1);\n\nU:=RE(Q(3,1,3,-quo(M1[3,1],
M1[1,1],x)) &* U):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'matrixG
6#7%7%,&*$)%\"xG\"\"#\"\"\"F/F.F/,&*$)F-\"\"$F/F/*&F.F/F-F/F/F07%\"\"!
,**$)F-\"\"%F/F/*&F.F/F2F/F/F-F/F.F/,**&F.F/F9F/F/*&F.F/F2F/F/*&F.F/F-
F/F/F.F/7%F6,.*$)F-\"\"'F/F/*&F.F/)F-\"\"&F/F/F8F/*&F.F/F2F/F/*&F.F/F-
F/F/F/F/,,*$FFF/F/*&F.F/F9F/F/F1F/*&F.F/F,F/F/F/F/" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 85 "M3:=RE(M2 &* Q(3,2,1,-quo(M2[1,2],M2[1,1],x
)));\n\nV:=Q(3,2,1,-quo(M2[1,2],M2[1,1],x)):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M3G-%'matrixG6#7%7%,&*$)%\"xG\"\"#\"\"\"F/F.F/\"\"!,
&*$)F-\"\"$F/F/*&F.F/F-F/F/7%F0,**$)F-\"\"%F/F/*&F.F/F3F/F/F-F/F.F/,**
&F.F/F9F/F/*&F.F/F3F/F/*&F.F/F-F/F/F.F/7%F0,.*$)F-\"\"'F/F/*&F.F/)F-\"
\"&F/F/F8F/*&F.F/F3F/F/*&F.F/F-F/F/F/F/,,*$FFF/F/*&F.F/F9F/F/F2F/*&F.F
/F,F/F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "M4:=RE(M3 &*
 Q(3,3,1,-quo(M3[1,3],M3[1,1],x)));\n\nV:=RE(V&*Q(3,3,1,-quo(M3[1,3],M
3[1,1],x))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M4G-%'matrixG6#7%7%
,&*$)%\"xG\"\"#\"\"\"F/F.F/\"\"!F07%F0,**$)F-\"\"%F/F/*&F.F/)F-\"\"$F/
F/F-F/F.F/,**&F.F/F4F/F/*&F.F/F7F/F/*&F.F/F-F/F/F.F/7%F0,.*$)F-\"\"'F/
F/*&F.F/)F-\"\"&F/F/F3F/*&F.F/F7F/F/*&F.F/F-F/F/F/F/,,*$FCF/F/*&F.F/F4
F/F/*$F7F/F/*&F.F/F,F/F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "E
s lagen (untypischerweise) in der ersten Zeile und in der ersten Spalt
e nurVielfache von " }{XPPEDIT 18 0 "x^2+2" "6#,&*$%\"xG\"\"#\"\"\"F&F
'" }{TEXT -1 6 "  vor." }}{PARA 0 "" 0 "" {TEXT -1 76 "Sch\366n, aber \+
teilt nun der (1,1)-Eintrag auch alle \374brigen Eintr\344ge von M4 ??
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Ich b
erechne die Divisionsreste:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "RE(map(u->rem(u,M4[1,1],x),M4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'matrixG6#7%7%\"\"!F(F(7%F(F(,&%\"xG\"\"\"F,F,7%F(F(,&*&\"\"#F,F+F
,F,F0F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 125 "Entsprechend addiere ich  bei M4 die zweite Zeile zur er
sten. Dadurch wandert der (2,3)-Eintrag von M4 in die (1,3)-Position:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M5:=RE(Q(3,2,1,1)&*M4);
\n\nU:=RE(Q(3,2,1,1)&*U):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M5G-%'
matrixG6#7%7%,&*$)%\"xG\"\"#\"\"\"F/F.F/,**$)F-\"\"%F/F/*&F.F/)F-\"\"$
F/F/F-F/F.F/,**&F.F/F2F/F/*&F.F/F5F/F/*&F.F/F-F/F/F.F/7%\"\"!F0F77%F<,
.*$)F-\"\"'F/F/*&F.F/)F-\"\"&F/F/F1F/*&F.F/F5F/F/*&F.F/F-F/F/F/F/,,*$F
CF/F/*&F.F/F2F/F/*$F5F/F/*&F.F/F,F/F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Nun wird wie im ersten Sc
hritt  weiterverfahren:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 
"M6:=RE(M5&*Q(3,2,1,-quo(M5[1,2],M5[1,1],x))&*Q(3,3,1,-quo(M5[1,3],M5[
1,1],x)));\n\nV:=RE(V&*Q(3,2,1,-quo(M5[1,2],M5[1,1],x))&*Q(3,3,1,-quo(
M5[1,3],M5[1,1],x))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M6G-%'matr
ixG6#7%7%,&*$)%\"xG\"\"#\"\"\"F/F.F/\"\"!,&F-F/F/F/7%F0,**$)F-\"\"%F/F
/*&F.F/)F-\"\"$F/F/F-F/F.F/,**&F.F/F5F/F/*&F.F/F8F/F/*&F.F/F-F/F/F.F/7
%F0,.*$)F-\"\"'F/F/*&F.F/)F-\"\"&F/F/F4F/*&F.F/F8F/F/*&F.F/F-F/F/F/F/,
,*$FDF/F/*&F.F/F5F/F/*$F8F/F/*&F.F/F,F/F/F/F/" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Nun m\374ssen 1. und 3
. Spalte vertauscht werden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "M7:=RE(M6&*P(3,1,3));\n\nV:=RE(V&*P(3,1,3)):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M7G-%'matrixG6#7%7%,&%\"xG\"\"\"F,F,\"\"!,&*$)F+\"\"
#F,F,F1F,7%,**&F1F,)F+\"\"%F,F,*&F1F,)F+\"\"$F,F,*&F1F,F+F,F,F1F,,**$F
5F,F,*&F1F,F8F,F,F+F,F1F,F-7%,,*$)F+\"\"&F,F,*&F1F,F5F,F,*$F8F,F,*&F1F
,F0F,F,F,F,,.*$)F+\"\"'F,F,*&F1F,FAF,F,F<F,*&F1F,F8F,F,*&F1F,F+F,F,F,F
,F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 33 ".. weiter, wie im ersten Schritt:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 158 "M8:=RE(Q(3,1,3,-quo(M7[3,1],M7[1,1],x))&*Q(3,1,2,-qu
o(M7[2,1],M7[1,1],x))&*M7);\n\nU:=RE(Q(3,1,3,-quo(M7[3,1],M7[1,1],x))&
*Q(3,1,2,-quo(M7[2,1],M7[1,1],x))&*U):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#M8G-%'matrixG6#7%7%,&%\"xG\"\"\"F,F,\"\"!,&*$)F+\"\"#F,F,F1F,
7%F-,**$)F+\"\"%F,F,*&F1F,)F+\"\"$F,F,F+F,F1F,,**$)F+\"\"&F,F,*&F1F,F8
F,F,F/F,F1F,7%F-,.*$)F+\"\"'F,F,*&F1F,F<F,F,F4F,*&F1F,F8F,F,*&F1F,F+F,
F,F,F,,0*&F1F,FBF,F,*&F1F,F<F,F,F4F,*&F1F,F8F,F,*&F1F,F0F,F,*&F1F,F+F,
F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "M9:=RE(M8&*Q(3,3,
1,-quo(M8[1,3],M8[1,1],x)));\n\nV:=RE(V&*Q(3,3,1,-quo(M8[1,3],M8[1,1],
x))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M9G-%'matrixG6#7%7%,&%\"xG
\"\"\"F,F,\"\"!F-7%F-,**$)F+\"\"%F,F,*&\"\"#F,)F+\"\"$F,F,F+F,F4F,,**$
)F+\"\"&F,F,*&F4F,F5F,F,*$)F+F4F,F,F4F,7%F-,.*$)F+\"\"'F,F,*&F4F,F9F,F
,F0F,*&F4F,F5F,F,*&F4F,F+F,F,F,F,,0*&F4F,FAF,F,*&F4F,F9F,F,F0F,*&F4F,F
5F,F,*&F4F,F=F,F,*&F4F,F+F,F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Teilt jetzt der (1,1)-Eintrag von \+
M9 auch alle \374brigen Eintr\344ge von M9 ??" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Ich berechne die Division
sreste:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RE(map(u->rem(u,M9[1,1],
x),M9));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7%7%\"\"!F(F(F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 4 "ja !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 61 "Nun kann mit der folgenden Teilmatix weitergearbeitet werden:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "B:=submatrix(M9,2..3,2..3
);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$,**$)%\"
xG\"\"%\"\"\"F/*&\"\"#F/)F-\"\"$F/F/F-F/F1F/,**$)F-\"\"&F/F/*&F1F/F2F/
F/*$)F-F1F/F/F1F/7$,.*$)F-\"\"'F/F/*&F1F/F6F/F/F+F/*&F1F/F2F/F/*&F1F/F
-F/F/F/F/,0*&F1F/F>F/F/*&F1F/F6F/F/F+F/*&F1F/F2F/F/*&F1F/F:F/F/*&F1F/F
-F/F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Mit B wird genauso v
erfahren:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "B1:=RE(Q(2,1,2
,-quo(B[2,1],B[1,1],x))&*B);\n\nU:=RE(diag(1,Q(2,1,2,-quo(B[2,1],B[1,1
],x)))&*U):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1G-%'matrixG6#7$7$,
**$)%\"xG\"\"%\"\"\"F/*&\"\"#F/)F-\"\"$F/F/F-F/F1F/,**$)F-\"\"&F/F/*&F
1F/F2F/F/*$)F-F1F/F/F1F/7$,**&F1F/F2F/F/F9F/F-F/F1F/,.*&F1F/)F-\"\"(F/
F/*&F1F/)F-\"\"'F/F/*&F1F/F6F/F/*&F1F/F:F/F/*&F1F/F-F/F/F1F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "B2:=RE(P(2,1,2)&*B1);\n\nU:=
RE(diag(1,P(2,1,2))&*U):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B2G-%'m
atrixG6#7$7$,**&\"\"#\"\"\")%\"xG\"\"$F-F-*$)F/F,F-F-F/F-F,F-,.*&F,F-)
F/\"\"(F-F-*&F,F-)F/\"\"'F-F-*&F,F-)F/\"\"&F-F-*&F,F-F2F-F-*&F,F-F/F-F
-F,F-7$,**$)F/\"\"%F-F-*&F,F-F.F-F-F/F-F,F-,**$F;F-F-*&F,F-F.F-F-F1F-F
,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "B3:=RE(Q(2,1,2,-quo(
B2[2,1],B2[1,1],x))&*B2);\n\nU:=RE(diag(1,Q(2,1,2,-quo(B2[2,1],B2[1,1]
,x)))&*U):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B3G-%'matrixG6#7$7$,*
*&\"\"#\"\"\")%\"xG\"\"$F-F-*$)F/F,F-F-F/F-F,F-,.*&F,F-)F/\"\"(F-F-*&F
,F-)F/\"\"'F-F-*&F,F-)F/\"\"&F-F-*&F,F-F2F-F-*&F,F-F/F-F-F,F-7$,&F1F-F
,F-,0*&F,F-)F/\"\")F-F-*&F,F-F5F-F-*&F,F-F8F-F-*$F;F-F-*$F.F-F-*&F,F-F
/F-F-F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "B4:=RE(P(2,1,2
)&*B3);\n\nU:=RE(diag(1,P(2,1,2))&*U):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#B4G-%'matrixG6#7$7$,&*$)%\"xG\"\"#\"\"\"F/F.F/,0*&F.F/)F-\"\"
)F/F/*&F.F/)F-\"\"(F/F/*&F.F/)F-\"\"'F/F/*$)F-\"\"&F/F/*$)F-\"\"$F/F/*
&F.F/F-F/F/F.F/7$,**&F.F/F>F/F/F+F/F-F/F.F/,.*&F.F/F5F/F/*&F.F/F8F/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 98 "B5:=RE(Q(2,1,2,-quo(B4[2,1],B4[1,1],x))&*B4);\n\nU:=R
E(diag(1,Q(2,1,2,-quo(B4[2,1],B4[1,1],x)))&*U):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B5G-%'matrixG6#7$7$,&*$)%\"xG\"\"#\"\"\"F/F.F/,0*&F.
F/)F-\"\")F/F/*&F.F/)F-\"\"(F/F/*&F.F/)F-\"\"'F/F/*$)F-\"\"&F/F/*$)F-
\"\"$F/F/*&F.F/F-F/F/F.F/7$\"\"!,2*&F.F/)F-\"\"*F/F/*&F.F/F5F/F/*$F8F/
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 98 "B6:=RE(B5&*Q(2,2,1,-quo(B5[1,2],B5[1,1],x)));
\n\nV:=RE(V&*diag(1,Q(2,2,1,-quo(B5[1,2],B5[1,1],x)))):" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#B6G-%'matrixG6#7$7$,&*$)%\"xG\"\"#\"\"\"F/F.F
/\"\"!7$F0,2*&F.F/)F-\"\"*F/F/*&F.F/)F-\"\"(F/F/*$)F-\"\"'F/F/*$)F-\"
\"&F/F/*$)F-\"\"%F/F/*&F.F/)F-\"\"$F/F/F+F/*&F.F/F-F/F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Wir habe
n Gl\374ck und sind bereits so gut wie am Ziel. Nur noch eine Normieru
ng ist vorzunehmen, um Eindeutigkeit zu erreichen:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 50 "B7:=RE(S(2,2,2)&*B6);\n\nU:=RE(diag(1,S(2,2
,2))&*U):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B7G-%'matrixG6#7$7$,&*
$)%\"xG\"\"#\"\"\"F/F.F/\"\"!7$F0,2*$)F-\"\"*F/F/*$)F-\"\"(F/F/*&F.F/)
F-\"\"'F/F/*&F.F/)F-\"\"&F/F/*&F.F/)F-\"\"%F/F/*$)F-\"\"$F/F/*&F.F/F,F
/F/F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "Die Smithform von A latet nun:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "`Smithform von M`= diag(M9[1,1],B7);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%0Smithform~von~MG-%'matrixG6#7%7%,&%\"xG\"\"\"
F,F,\"\"!F-7%F-,&*$)F+\"\"#F,F,F2F,F-7%F-F-,2*$)F+\"\"*F,F,*$)F+\"\"(F
,F,*&F2F,)F+\"\"'F,F,*&F2F,)F+\"\"&F,F,*&F2F,)F+\"\"%F,F,*$)F+\"\"$F,F
,*&F2F,F1F,F,F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 37 "Zum Vergleich das Ergebnis von Maple:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 17 "Smith(M,x) mod p;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%,&%\"xG\"\"\"F*F*\"\"!F+7%F+,&*$)F)\"\"
#F*F*F0F*F+7%F+F+,2*$)F)\"\"*F*F**$)F)\"\"(F*F**&F0F*)F)\"\"'F*F**&F0F
*)F)\"\"&F*F**&F0F*)F)\"\"%F*F**$)F)\"\"$F*F**&F0F*F/F*F*F)F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 40 "In Anwendungen ist es h\344ufig notwendi
g, " }{TEXT 257 23 "nicht nur die Smithform" }{TEXT -1 10 ", sondern \+
" }{TEXT 258 32 "auch die Transformationsmatrizen" }{TEXT -1 104 " zu \+
kennen. In unserem Falle sind dies die unterwegs neben der Rechnung he
r erzeugten Matrizen U und V. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 25 "Die Probe best\344tigt dies:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "RE(U&*M&*V);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%,&%\"xG\"\"\"F*F*\"\"!F+7%F+,&*$)F)\"\"
#F*F*F0F*F+7%F+F+,2*$)F)\"\"*F*F**$)F)\"\"(F*F**&F0F*)F)\"\"'F*F**&F0F
*)F)\"\"&F*F**&F0F*)F)\"\"%F*F**$)F)\"\"$F*F**&F0F*F/F*F*F)F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "37 0 0" 0 }{VIEWOPTS 1 0 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }