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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):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Zuf\344llige Er
zeugung von Matrizen \374ber  " }{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 nichttrivialen 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 5 "p:=2;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "#die:=rand(0..p):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 6 "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 Beis
piel, 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]]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 79 "Im zweiten Beispiel ergibt sich als Smith
 form der charakteristischen Matrix A:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "#map(modp,evalm(diag(seq(x,k=1..n))-A),p);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "SxA:=Smith(evalm(diag(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;" }}}{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\n
then 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);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
44 "Die Jacobson-Normalform  von A lautet jetzt:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 235 "m:=nops(ET);\nfor l to m do \nefs:=Factors(ET[l]) mo
d p;\neli:=op(op(2,efs));\nep:=op(1,eli);\ned:=degree(ep,x);\nea:=op(2
,eli);\nJJ:=diag(seq(map(modp,companion(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));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 208 "...wer Lust hat, kann das ja noch so umschreib
en, dass die relevanten Diagonalk\344stchen markiert sind. Ansonsten m
uss man halt genau hinsehen, um zu \374berpr\374fen, ob tats\344chlich
 eine Jacobson-Noralform vorliegt." }}}{EXCHG {PARA 0 "> " 0 "" 
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