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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Beispiel zur Trigonalisier
ung beliebiger komplexer n x n-Matrizen" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Dazu vgl. man das Buch, Satz
 7.4.3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 87 
"Die folgende Matrix ist nicht normal, also nicht durch eine Drehung o
rthogonalisierbar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "A:=m
atrix([[I*sqrt(2)/6,sqrt(2)/3,-I-I*sqrt(2)/6],[1/6*sqrt(2),I-1/3*I*sqr
t(2),-1/6*sqrt(2)],[-I-1/6*I*sqrt(2),-1/3*sqrt(2),1/6*I*sqrt(2)]]);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Der erste Arbeitsschritt besteht
 immer in der Bestimmung der Eigenwerte:" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvalues(A);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Die zugeh\366rigen Eigenvektoren f
indet man mit Hilfe des Gau\337schen Algorithmus" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "oder direkt mit dem Kommando:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "eigenvectors(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 88 "Die folgende Matrix entsteht so: Man nehme einen Eigenvektor, e
twa den zum Eigenwert -i," }}{PARA 0 "" 0 "" {TEXT -1 100 "erg\344anze
 diesen zu einer Basis und f\374hre anschliessend das Schmidtsche Orth
onormalisierungsverfahren" }}{PARA 0 "" 0 "" {TEXT -1 5 "durch" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "GramSchmidt([vector([1,0,1])
,vector([1,0,0]),vector([0,1,0])],normalized);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 83 "Das Ergebnis tragen wir in die erste, vorl\344ufige \+
unit\344re Transformationsmatrix ein:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "B1:=matrix([[1/sqrt(2),1/sqrt(2),0],[0,0,1],[1/sqrt(2
),-1/sqrt(2),0]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Transformat
ion von A mit B_1 liefert die erste Vereinfachung von A:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A1:=evalm(inverse(B1)&*A&*B1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(A1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 67 "Der 2 x 2-Block rechts unten muss nun gan
z analog behandelt werden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "Astrich:=matrix([[I+1/3*I*sqrt(2),2/3],[1/3,I-1/3*I*sqrt(2)]]);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Bestimmung der Eigenwerte:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eigenvalues(Astrich);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Zugeh\366riger Eigenvektor:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eigenvectors(Astrich);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Zu Basis erg\344nzen und orthogona
lisieren:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "GramSchmidt([v
ector([I*sqrt(2),1]),vector([1,0])],normalized);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 32 "Eintragen in eine unit\344re Matrix" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Bstrich:=matrix([[I*sqrt(6)/3,sqrt(
3)/3],[sqrt(3)/3,I*sqrt(6)/3]]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "inverse(Bstrich);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "A2:=simplify(evalm(inverse(Bstrich)&*Astrich&*Bstrich
));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Einf\374gen des 2 x 2-Bloc
ks in eine unit\344re 3 x 3-Matrix: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "B2:=matrix([[1,0,0],[0,I*sqrt(6)/3,sqrt(3)/3],[0,sqrt
(3)/3,I*sqrt(6)/3]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Zusammen
fassen der beiden Koordinatenwechsel:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "B:=simplify(evalm(B1&*B2));" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 61 "Da wir in drei Dimensionen arbeiten, erhalten wir berei
ts die" }}{PARA 0 "" 0 "" {TEXT -1 26 "endg\374ltige Dreiecksmatrix:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(evalm(inverse(B)
&*A&*B));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "In h\366heren Dimens
ionen h\344tte man nun dieses Verfahren entsprechend weiter" }}{PARA 
0 "" 0 "" {TEXT -1 17 "iterieren m\374ssen." }}}}{MARK "38" 0 }
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