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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Illustration zu Satz 2 in \+
 TeilA, \2473." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 58 "Berechnung der SVD mit elementaren orthogalen Umformungen
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" 
{XPPEDIT 18 0 "A := matrix([[1, 2, 2, 1], [-6, 2, -2, 6]]);" "6#>%\"AG
-%'matrixG6#7$7&\"\"\"\"\"#F+F*7&,$\"\"'!\"\"F+,$F+F/F." }{TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "eD:=proc(i,j,a,b,n) local
 M;\nif i=j then ERROR(\"i darf nicht gleich j sein\")\nfi:\nM:=evalm(
array(1..n,1..n,identity));\nM[i,i]:=a/sqrt(a^2+b^2):M[i,j]:=b/sqrt(a^
2+b^2):M[j,i]:=-b/sqrt(a^2+b^2):M[j,j]:=a/sqrt(a^2+b^2):\nevalm(M); \n
end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "m:=3:n:=4:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A:=matrix(m,n,[1,2,2,1,-6,2,
-2,6,-3,8,4,9]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rank(A);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "orthogonale Zeilenumformungen
: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A0:=evalm(A):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "k:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "Q|
|k=eD(1,2,A||(k-1)[1,1],A||(k-1)[2,1],m);A||k:=evalm(eD(1,2,A||(k-1)[1
,1],A||(k-1)[2,1],m)&*A||(k-1));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 109 "k:=2:Q||k=eD(1,3,A||(k-1)[1,1],A||(k-1)[3,1],m);A||k
:=evalm(eD(1,3,A||(k-1)[1,1],A||(k-1)[3,1],m)&*A||(k-1));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "k:=3:Q||k=eD(2,3,A||(k-1)[2,2],A||
(k-1)[3,2],m);A||k:=evalm(eD(2,3,A||(k-1)[2,2],A||(k-1)[3,2],m)&*A||(k
-1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Statt A zu transponiere
n und wieder eine orthogonale Zeilenstufenform zu berechnen f\374hre i
ch orthogonale Spaltenumformungen durch." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "A4:=evalm(A3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 
"k:=5:Q||k=eD(1,2,A||(k-1)[1,1],-A||(k-1)[1,2],n);A||k:=evalm(A||(k-1)
&*eD(1,2,A||(k-1)[1,1],-A||(k-1)[1,2],n));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "k:=6:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Q||k:=e
D(1,3,A||(k-1)[1,1],-A||(k-1)[1,3],n);A||k:=map(simplify,evalm(A||(k-1
)&*eD(1,3,A||(k-1)[1,1],-A||(k-1)[1,3],n)));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "k:=7:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Q||k
:=eD(1,4,A||(k-1)[1,1],-A||(k-1)[1,4],n);A||k:=map(simplify,evalm(A||(
k-1)&*eD(1,4,A||(k-1)[1,1],-A||(k-1)[1,4],n)));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "k:=8:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Q|
|k:=eD(2,3,A||(k-1)[2,2],-A||(k-1)[2,3],n);A||k:=map(simplify,evalm(A|
|(k-1)&*eD(2,3,A||(k-1)[2,2],-A||(k-1)[2,3],n)));" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 5 "k:=9:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "
Q||k:=eD(2,4,A||(k-1)[2,2],-A||(k-1)[2,4],n);A||k:=map(simplify,evalm(
A||(k-1)&*eD(2,4,A||(k-1)[2,2],-A||(k-1)[2,4],n)));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "S:=submatrix(A||k,1..2,1..2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=evalm(transpose(S)&*S);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "charpoly(B,t);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "eigenvectors(B);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }