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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "L\366sung zu Aufgabe (12)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Auszu
g aus einem Maple-Arbeitsblatt" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "Vorgegangen wird wie beim Beweis von Satz
 13.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 24 "Die gegebene Matrix ist " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "A:=matrix(3,2,[-2,4,28/10,4/10,38/10,-16/10]);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "#map(evalf,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG-%'matrixG6#7%7$!\"#\"\"%7$#\"#9\"\"&#\"\"#F/7$#\"#>F/#!\")F/
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Definition elementarer nxn-Dr
ehungsmatrizen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "eD:=proc(i,j,a,
b,n) local M;\nif i=j then ERROR(\"i darf nicht gleich j sein\")\nfi:
\nM:=evalm(array(1..n,1..n,identity));\nM[i,i]:=a/sqrt(a^2+b^2):M[i,j]
:=b/sqrt(a^2+b^2):M[j,i]:=-b/sqrt(a^2+b^2):M[j,j]:=a/sqrt(a^2+b^2):\ne
valm(M); \nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "1. Schritt: B
erechnung eines invertierbaren quadratischen Blocks durch elementare o
rthogonale Umformungen." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A0:=eval
m(A):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=1:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "A||k:=evalm(eD(1,2,A||(k-1)[1,1],A||(k-1)[2,1],3)&*A
||(k-1));\nU:=eD(1,2,A||(k-1)[1,1],A||(k-1)[2,1],3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7%7$,$*(\"\"#\"\"\"\"\"&!\"\"\"#u#
F-F,F-,$*(\"#VF-\"$&=F/F0F1F/7$\"\"!,$*(\"#:F-\"#PF/F0F1F/7$#\"#>F.#!
\")F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'matrixG6#7%7%,$*(\"
\"&\"\"\"\"#u!\"\"F.#F-\"\"#F/,$*(\"\"(F-F.F/F.F0F-\"\"!7%,$*(F4F-F.F/
F.F0F/F*F57%F5F5F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=2:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "A||k:=map(simplify,evalm(eD(1,
3,A||(k-1)[1,1],A||(k-1)[3,1],3)&*A||(k-1)));\nU:=evalm(eD(1,3,A||(k-1
)[1,1],A||(k-1)[3,1],3)&*U);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G
-%'matrixG6#7%7$,$*(\"\"$\"\"\"\"\"&!\"\"\"#t#F-\"\"#F-,$*(\"$3\"F-\"$
l$F/F0F1F/7$\"\"!,$*(\"#:F-\"#PF/\"#uF1F/7$F8,$**F;F-\"%,FF/F=F1F0F1F-
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'matrixG6#7%7%,$*(\"#5\"\"
\"\"$>#!\"\"\"#t#F-\"\"#F/,$*(\"#9F-F.F/F0F1F-,$*(\"#>F-F.F/F0F1F-7%,$
*(\"\"(F-\"#uF/F=F1F/,$*(\"\"&F-F=F/F=F1F/\"\"!7%,$**\"#&*F-\"&1i\"F/F
=F1F0F1F-,$**\"$L\"F-FFF/F=F1F0F1F/,$**F2F-F.F/F=F1F0F1F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=3:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 127 "A||k:=map(simplify,evalm(eD(2,3,A||(k-1)[2,2],A||(k-1)[3,2],3
)&*A||(k-1)));\nU:=evalm(eD(2,3,A||(k-1)[2,2],A||(k-1)[3,2],3)&*U);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G-%'matrixG6#7%7$,$*(\"\"$\"\"\"
\"\"&!\"\"\"#t#F-\"\"#F-,$*(\"$3\"F-\"$l$F/F0F1F/7$\"\"!,$*(\"#IF-F0F/
F0F1F-7$F8F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'matrixG6#7%7%
,$*(\"#5\"\"\"\"$>#!\"\"\"#t#F-\"\"#F/,$*(\"#9F-F.F/F0F1F-,$*(\"#>F-F.
F/F0F1F-7%,$*(\"#AF-F.F/F0F1F-,$*(\"#8F-F.F/F0F1F-,$*(F2F-F.F/F0F1F-7%
#F/\"\"$#F2FD#!\"#FD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(A3-U&*A);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7$\"\"!F(F'F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 46 "Die folgende Teilmatrix ist jetzt inverti
erbar" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S:=submatrix(A3,1..2,1..2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'matrixG6#7$7$,$*(\"\"$\"
\"\"\"\"&!\"\"\"#t#F-\"\"#F-,$*(\"$3\"F-\"$l$F/F0F1F/7$\"\"!,$*(\"#IF-
F0F/F0F1F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Nun wird zuerst  " 
}{XPPEDIT 18 0 "B = transpose(S)*S;" "6#/%\"BG*&-%*transposeG6#%\"SG\"
\"\"F)F*" }{TEXT -1 20 "    diagonalisiert: " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "B:=evalm(transpose(S)&*S);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$#\"$d'\"#D#!$C$F,7$F-#\"$o%F," }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "das charakteristische Polynom ist
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "charpoly(B,t);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,(*$)%\"tG\"\"#\"\"\"F(*&\"#XF(F&F(!\"\"\"$C$F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&%\"tG\"\"\"\"\"*!\"\"F&,&F%F&\"#OF(F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 101 "Berechnung einer orthonormierten Eigenve
ktorbasis. Beachte: die Orthogonalit\344t gilt automatisch, da  " }
{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 19 "   symmetrisch ist." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(B):" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 97 "Q:=concat(vector([-4/3, 1])/norm(vector([-4
/3, 1]),2),vector([1, 4/3])/norm(vector([1, 4/3]),2));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"QG-%'matrixG6#7$7$#!\"%\"\"&#\"\"$F,7$F-#\"\"%
F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "evalm(transpose(Q)&*Q);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$F)F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 35 "Festlegung der orthogonalen Matrix " }{XPPEDIT 18 0 "P;" 
"6#%\"PG" }{TEXT -1 31 " , wie im Beweis von Satz 13.2:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "P:=evalm(diag(6,3)^(-1)&*transpose(Q)&*transpo
se(S));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7$7$,$*(
\"\")\"\"\"\"#t!\"\"F.#F-\"\"#F/,$*(\"\"$F-F.F/F.F0F-7$F2,$*(F,F-F.F/F
.F0F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "evalm(transpose(P)&*P);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$F)F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 42 "Die Singul\344rwertzerlegung von A ist jetzt:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalm(   (diag(P,1)&*U)   &*   A   \+
&*   Q   );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7$\"\"'\"
\"!7$F)\"\"$7$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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