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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 55 "Zuf\344llige Beispiele zur Jacobson Normalform \374ber Z[p] :
" }}{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 "" {TEXT -1 39 "Zuf\344llige Erzeugung von Matrizen \374b
er  " }{XPPEDIT 18 0 "Z_p" "6#%$Z_pG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 93 "In den weitaus meisten Versuchen gibt es nur einen nic
httrivialen invarianten Faktor(warum?)." }}{PARA 0 "" 0 "" {TEXT -1 
114 "Allerdings sind diese oft zerlegbar und f\374hren so zu einer von
 der Frobeniusform abweichenden Jacobson-Normalform." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 6 "#p:=2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "#die:=rand(0..p):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#n:=11;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "#A:=matrix(n,n,(i,j)->die());" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 " ein  geeignetes Beispiel mit p=2,
 n=11ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 400 "#A := matrix([[1, 1, \+
1, 1, 0, 1, 1, 0, 1, 1, 0], [0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1], [0, 0, \+
0, 0, 1, 0, 0, 0, 0, 0, 1], [1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0], [1, 0, \+
1, 0, 1, 1, 0, 1, 0, 1, 1], [1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0], [1, 1, \+
1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0], [1, 1, \+
0, 0, 1, 1, 1, 0, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0], [1, 1, \+
0, 1, 0, 1, 0, 1, 1, 0, 1]]);" }{TEXT -1 6 "      " }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 56 "ein weiteres Beispiel, ebenfalls mit p=2 und n=11
,  ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 397 "A:=matrix([[1, 0, 1, 1,
 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 0, 1,
 1, 1, 0, 1, 0, 1, 0], [0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1], [0, 1, 1, 1,
 0, 1, 1, 1, 1, 0, 1], [1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1], [0, 0, 0, 1,
 0, 1, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1], [1, 0, 0, 0,
 0, 1, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1], [1, 0, 1, 1,
 1, 1, 1, 0, 1, 1, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'m
atrixG6#7-7-\"\"\"\"\"!F*F*F*F+F+F*F+F+F+7-F+F+F*F+F*F*F*F+F+F+F+7-F+F
+F+F*F*F*F+F*F+F*F+7-F+F*F+F*F*F+F*F*F*F+F*7-F+F*F*F*F+F*F*F*F*F+F*7-F
*F+F+F*F+F+F*F*F+F+F*7-F+F+F+F*F+F*F+F*F+F+F+7-F+F+F*F+F+F+F+F*F*F+F*7
-F*F+F+F+F+F*F+F+F+F+F*7-F+F*F+F+F+F*F+F*F*F*F*7-F*F+F*F*F*F*F*F+F*F*F
*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Im zweiten Beispiel ergibt s
ich als Smith form der charakteristischen Matrix A:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "SxA:=Smith(diag(evalm(seq(x,k=1..n)-A)),x
) mod p;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#cp:=charpoly(A,x) mod \+
p;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "#Factor(cp) mod p;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "#F:=Frobenius(A) mod p;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$SxAG-%'matrixG6#7-7-\"\"\"\"\"!F+F+F+F+F+F+F+F+F+
7-F+F*F+F+F+F+F+F+F+F+F+7-F+F+F*F+F+F+F+F+F+F+F+7-F+F+F+F*F+F+F+F+F+F+
F+7-F+F+F+F+F*F+F+F+F+F+F+7-F+F+F+F+F+F*F+F+F+F+F+7-F+F+F+F+F+F+F*F+F+
F+F+7-F+F+F+F+F+F+F+F*F+F+F+7-F+F+F+F+F+F+F+F+F*F+F+7-F+F+F+F+F+F+F+F+
F+%\"xGF+7-F+F+F+F+F+F+F+F+F+F+,.*$)F5\"#5F*F**$)F5\"\"*F*F**$)F5\"\")
F*F**$)F5\"\"%F*F**$)F5\"\"$F*F**$)F5\"\"#F*F*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 35 "\"Die\" Liste der Elemenarteiler ist:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 211 "ET:=[]:for l to n do if not SxA[l,l]=1\nthen \+
ET:=[op(ET), seq(op(1,op(k,op(2,Factors(SxA[l,l]) mod p)))^(op(2,op(k,
op(2,Factors(SxA[l,l]) mod p)))),k=1..nops(op(2,Factors(SxA[l,l]) mod \+
p)))]; fi;od;\nET:=sort(ET);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ETG
7&*$)%\"xG\"\"#\"\"\"F(*$),(F&F*F(F*F*F*\"\"$F**$),&F(F*F*F*F)F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Die Jacobson-Normalform  von A lau
tet jetzt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "m:=nops(ET);\nfor l \+
to m do \nefs:=Factors(ET[l]) mod p;\neli:=op(op(2,efs));\nep:=op(1,el
i);\ned:=degree(ep,x);\nea:=op(2,eli);\nJJ:=diag(seq(map(modp,companio
n(ep,x),p),r=1..ea)):\nfor k to ea-1 do JJ[k*ed+1,k*ed]:=1;od:\nJ||l:=
evalm(JJ);\nod:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J(A):=diag(seq(J
||k,k=1..m));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"JG6#%\"AG-%'matrixG6#7-7-\"\"!F-F-F-F-F
-F-F-F-F-F-7-\"\"\"F-F-F-F-F-F-F-F-F-F-F,7-F-F-F-F-F/F-F-F-F-F-F-7-F-F
-F-F/F/F-F-F-F-F-F-7-F-F-F-F-F/F-F/F-F-F-F-7-F-F-F-F-F-F/F/F-F-F-F-7-F
-F-F-F-F-F-F/F-F/F-F-7-F-F-F-F-F-F-F-F/F/F-F-7-F-F-F-F-F-F-F-F-F-F/F-7
-F-F-F-F-F-F-F-F-F-F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 208 "...we
r Lust hat, kann das ja noch so umschreiben, dass die relevanten Diago
nalk\344stchen markiert sind. Ansonsten muss man halt genau hinsehen, \+
um zu \374berpr\374fen, ob tats\344chlich eine Jacobson-Noralform vorl
iegt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "6 0 0
" 51 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }