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{SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 "Beispiel Vorlesung 11. Ju
ni 2003" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart:\nwith(li
nalg):\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names \+
norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 48 "Definition der Elementarmatrizen als Prozeduren
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P:=proc(n,i,j) local k;swaprow
(diag(seq(1,k=1..n)),i,j);end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "S
:=proc(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 13 "Erl\344uterung: " }{MPLTEXT 1 0 8 "evalm(M)" }{TEXT -1 23 "   w
ertet den Ausdruck " }{MPLTEXT 1 0 1 "M" }{TEXT -1 83 " als Matrix aus
, falls dies m\366glich ist. Die Matrizenmultiplikation wird dabei mit
 " }{MPLTEXT 1 0 2 "&*" }{TEXT -1 12 "  angegeben." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 59 "A:=matrix(4,5,[4,2,3,4,5,6,3,4,5,6,-2,4,5,4
,6,12,5,4,3,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#
7&7'\"\"%\"\"#\"\"$F*\"\"&7'\"\"'F,F*F-F/7'!\"#F*F-F*F/7'\"#7F-F*F,\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Ich gehe nach dem in der \+
Vorlesung vom 11. Juni angegebenen Algorithmus vor mit dem Zusatz dass
 bei der Suche nach der Zeilennummer k stets ein minimales k genommen \+
wird." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A1:=evalm(P(4,1,2)
&*Q(4,1,2,-1)&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG
6#7&7'\"\"#\"\"\"F+F+F+7'\"\"%F*\"\"$F-\"\"&7'!\"#F-F/F-\"\"'7'\"#7F/F
-F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A2:=evalm(Q(4,1
,4,-6)&*Q(4,1,3,1)&*Q(4,1,2,-2)&*A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#A2G-%'matrixG6#7&7'\"\"#\"\"\"F+F+F+7'\"\"!F-F+F*\"\"$7'F-\"\"&
\"\"'F0\"\"(7'F-!\"\"!\"#!\"$!\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "A3:=evalm(P(4,2,3)&*A2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A3G-%'matrixG6#7&7'\"\"#\"\"\"F+F+F+7'\"\"!\"\"&\"\"
'F.\"\"(7'F-F-F+F*\"\"$7'F-!\"\"!\"#!\"$!\"'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "A4:=evalm(P(4,2,4)&*A3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A4G-%'matrixG6#7&7'\"\"#\"\"\"F+F+F+7'\"\"!!\"\"!\"#
!\"$!\"'7'F-F-F+F*\"\"$7'F-\"\"&\"\"'F5\"\"(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "A5:=evalm(Q(4,2,4,5)&*A4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A5G-%'matrixG6#7&7'\"\"#\"\"\"F+F+F+7'\"\"!!\"\"!\"#
!\"$!\"'7'F-F-F+F*\"\"$7'F-F-!\"%!#5!#B" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "A6:=evalm(Q(4,3,4,4)&*A5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A6G-%'matrixG6#7&7'\"\"#\"\"\"F+F+F+7'\"\"!!\"\"!\"#
!\"$!\"'7'F-F-F+F*\"\"$7'F-F-F-F/!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 39 "Diese Matrix ist offensichtlich in ZSF." }}{PARA 0 "" 0 "" 
{TEXT -1 89 "Nun reduzieren wir die Matrix A6 um die dann eindeutig (!
) Hermiteform von A zu erhalten:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 
"A7:=evalm(Q(4,2,1,-1)&*S(4,2,-1)&*A6);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#A7G-%'matrixG6#7&7'\"\"#\"\"!!\"\"!\"#!\"&7'F+\"\"\"F*\"\"$\"
\"'7'F+F+F0F*F17'F+F+F+F-!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "A8:=evalm(Q(4,3,2,-2)&*Q(4,3,1,1)&*A7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A8G-%'matrixG6#7&7'\"\"#\"\"!F+F+!\"#7'F+\"\"\"F+!\"
\"F+7'F+F+F.F*\"\"$7'F+F+F+F,!#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "A9:=evalm(Q(4,4,3,-1)&*S(4,4,-1)&*A8);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#A9G-%'matrixG6#7&7'\"\"#\"\"!F+F+!\"#7'F+\"\"
\"F+!\"\"F+7'F+F+F.F+!\")7'F+F+F+F*\"#6" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 161 "Wenn man die Hermiteform schon hat, ist der Weg zur Smit
hform in diesem Beispiel nicht mehr weit, allerdings wird man die Smit
hform direkt i.A. anders berechnen." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A10:=ev
alm(A9&*Q(5,5,1,1)&*Q(5,4,2,1)&*Q(5,5,3,8)&*Q(5,5,4,-5));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$A10G-%'matrixG6#7&7'\"\"#\"\"!F+F+F+7'F+\"\"
\"F+F+F+7'F+F+F-F+F+7'F+F+F+F*F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "A11:=evalm(A10&*P(5,5,4));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$A11G-%'matrixG6#7&7'\"\"#\"\"!F+F+F+7'F+\"\"\"F+F+F+
7'F+F+F-F+F+7'F+F+F+F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"A12:=evalm(A11&*Q(5,5,4,-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$A
12G-%'matrixG6#7&7'\"\"#\"\"!F+F+F+7'F+\"\"\"F+F+F+7'F+F+F-F+F+7'F+F+F
+F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A13:=evalm(P(4,1,4
)&*A12&*P(5,4,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$A13G-%'matrix
G6#7&7'\"\"\"\"\"!F+F+F+7'F+F*F+F+F+7'F+F+F*F+F+7'F+F+F+\"\"#F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Dies ist die Smithform von A unte
r der Voraussetzung: alle von 0 verschiedenen Eintr\344ge sind positiv
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }