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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "A:=matrix([[1,2,2],[2,1,2],[2,2,1]]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#F+7%F+F*F+7%F+F+F
*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Das charakteristische Polyno
m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "det(t*diag(1,1,1)-A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"tG\"\"$\"\"\"F(*&F'F()F&\"
\"#F(!\"\"*&\"\"*F(F&F(F,\"\"&F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"
tG\"\"\"\"\"&!\"\"F&),&F%F&F&F&\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "nullspace(A+diag(1,1,1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$-%'vectorG6#7%\"\"\"\"\"!!\"\"-F%6#7%F)F(F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "nullspace(A-5*diag(1,1,1));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7%\"\"\"F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "also ist A diagonalisierbar." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "iT:=matrix([[1,0,1],[1,-1,-1
],[1,1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#iTG-%'matrixG6#7%7%
\"\"\"\"\"!F*7%F*!\"\"F-7%F*F*F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "T:=inverse(iT);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"TG-%'matrixG6#7%7%#\"\"\"\"\"$F*F*7%#!\"\"F,F.#\"\"#F,7%F0F.F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "multiply(T,A,iT);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"&\"\"!F)7%F)!\"\"F)7%F)F
)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "# konstruiere ortho
gonales Q:\n# wende dazu Gram-Schmidt auf die Eigenr\344ume an:\niQ:=m
ulcol(iT,1,1/sqrt(3));\niQ:=mulcol(iQ,2,1/sqrt(2));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#iQG-%'matrixG6#7%7%,$*$-%%sqrtG6#\"\"$\"\"\"#F0F/
\"\"!F07%F*!\"\"F47%F*F0F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#iQG-%
'matrixG6#7%7%,$*$-%%sqrtG6#\"\"$\"\"\"#F0F/\"\"!F07%F*,$*$-F-6#\"\"#F
0#!\"\"F8F:7%F*,$F5#F0F8F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "iQ:=addcol(iQ,2,3,-1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#iQG-%'matrixG6#7%7%,$*$-%%sqrtG6#\"\"$\"\"\"#F0F/\"\"!F07%F*,$*$-F-
6#\"\"#F0#!\"\"F8F97%F*,$F5#F0F8F9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "iQ:=mulcol(iQ,3,1/sqrt(3/2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#iQG-%'matrixG6#7%7%,$*$-%%sqrtG6#\"\"$\"\"\"#F0F/\"
\"!,$*$-F-6#\"\"'F0F17%F*,$*$-F-6#\"\"#F0#!\"\"F=,$F4#F?F77%F*,$F:#F0F
=F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "In den Spalten von iQ steh
t dann die Orthonormalbasis aus Eigenvektoren." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "Q:=inverse(iQ);\nmultiply(Q,A,iQ);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'matrixG6#7%7%,$*$-%%sqrtG6#\"\"$\"
\"\"#F0F/F*F*7%\"\"!,$*$-F-6#\"\"#F0#!\"\"F8,$F5#F0F87%,$*$-F-6#\"\"'F
0F1,$F?#F:FBFC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"
&\"\"!F)7%F)!\"\"F)7%F)F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Q \+
ist tats\344chlich orthogonal:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "multiply(Q,transpose(Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "B:=matrix([[2,1,1],[2,2,1],[-2,1,2]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%\"\"#\"\"\"F+7%F*F*F+7
%!\"#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "det(t*diag(1,1
,1)-B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"tG\"\"\"\"\"#!\"\"F&
,(*$)F%F'F&F&*&\"\"%F&F%F&F(\"\"$F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"
tG\"\"\"\"\"#!\"\"F&,&F%F&F&F(F&,&F%F&\"\"$F(F&" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 28 "also ist B diagonalisierbar." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 24 "Bestimme die Eigenr\344ume:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "nullspace(B-diag(1,1,1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#-%'vectorG6#7%\"\"!!\"\"\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "nullspace(B-2*diag(1,1,1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#<#-%'vectorG6#7%\"\"\"\"\"#!\"#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "nullspace(B-3*diag(1,1,1));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#<#-%'vectorG6#7%!\"#!\"$\"\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "iT:=matrix([[0,1,2],[-1,2,3],[1,-2,-1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#iTG-%'matrixG6#7%7%\"\"!\"\"\"\"\"#
7%!\"\"F,\"\"$7%F+!\"#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"T:=inverse(iT);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#
7%7%\"\"##!\"$F*#!\"\"F*7%\"\"\"F.F.7%\"\"!#F0F*F3" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 17 "multiply(T,B,iT);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)\"\"#F)7%F)F)\"\"$" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "C:=matrix([[6,3,1],[-6,-2,-
1],[6,4,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7%7
%\"\"'\"\"$\"\"\"7%!\"'!\"#!\"\"7%F*\"\"%F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "det(t*diag(1,1,1)-C);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**$)%\"tG\"\"$\"\"\"F(*&\"\"(F()F&\"\"#F(!\"\"*&\"#;F(
F&F(F(\"#7F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"tG\"\"\"\"\"$!\"\"F&),&F%F&
\"\"#F(F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "nullspace(C-
3*diag(1,1,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7%\"
\"\"#!\"$\"\"##\"\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "n
ullspace(C-2*diag(1,1,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vec
torG6#7%\"\"\"!\"#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "also i
st C nicht diagonalisierbar." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "
da das charakteristische Polynom in Linearfaktoren zerf\344llt exist e
ine Jordansche Normalform.\nBis auf Vertauschen der Jordanbl\366cke ha
t diese die Form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "J:=dia
g(JordanBlock(2,2),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'mat
rixG6#7%7%\"\"#\"\"\"\"\"!7%F,F*F,7%F,F,\"\"$" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 34 "mit der eingebauten Maple-Routine:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "jordan(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'matrixG6#7%7%\"\"$\"\"!F)7%F)\"\"#\"\"\"7%F)F)F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 6 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }