{VERSION 5 0 "IBM INTEL LINUX" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Im Folgenden sollen einige
 Maple-Kommandos zu fortgeschrittenen Fragen der" }}{PARA 0 "" 0 "" 
{TEXT -1 36 "Linearen Algebra vorgestellt werden." }}}{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 88 
"Man beachte, dass im Folgenden Matrizen stets zeilenweise einzugeben \+
sind, dass wir aber" }}{PARA 0 "" 0 "" {TEXT -1 38 "Vektoren als Spalt
en interpretieren!!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=
matrix([[1,1],[0,1]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Deren t
ransponierte Matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "tra
nspose(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "B:=matrix([[7
/6,-2/3,1/6],[-2/3,2/3,-2/3],[1/6,-2/3,7/6]]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "transpose(B);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 61 "Offenkundig ist B symmetrisch. Das kann maple auch so pr
\374fen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(B-transpo
se(B));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "C:=matrix([[1,3*
I],[-3*I,1]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Hier ist nat
\374rlich das Bilden der konjugiert transponierten Matrix angemessen:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "htranspose(C);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Und die Hermitizit\344t von C pr
\374ft man so:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(C-h
transpose(C));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Betrachten wir \+
schliesslich noch eine Drehmatrix:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "DD:=matrix([[1/2+sqrt(2)/4,-1/2+sqrt(2)/4,-1/2],[-1/
2+sqrt(2)/4,1/2+sqrt(2)/4,-1/2],[1/2,1/2,sqrt(2)/2]]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(transpose(DD)&*DD);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 92 "Damit ist DD (reell) orthogonal. Dass DD \+
in der Tat eine Drehmatrix ist, zeigt schliesslich:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "det(DD);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 67 "Auch die Determinanten der \374brigen Matrizen sind schnell ber
echnet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(B);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "det(C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
101 "Kommen wir nun zur Eigenwerttheorie. Man kann z.B. direkt die cha
rakterisitischen Polynome bestimmen:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "PA(lambda):=charpoly(A,lambda);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 32 "Deren Eigenwerte, Eigenvektoren:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "eigenvectors(B);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 94 "B ist symmetrisch, alle Eigenwerte reell und es gibt eine
 Orthonormal-Basis aus Eigenvektoren:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "GramSchmidt([vector([-1,0,1]),vector([-1,1,-1]),vecto
r([1,2,1])],normalized);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Bilde
 daraus die Transformationsmatrix:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 121 "BB:=matrix([[-1/2*sqrt(2),-1/3*sqrt(3),1/6*sqrt(6)],
[0,1/3*sqrt(3),1/3*sqrt(6)],[1/2*sqrt(2),-1/3*sqrt(3),1/6*sqrt(6)]]);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Dieser Basiswechsel diagonali
siert B:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalm(simplify(
inverse(BB)&*B&*BB));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Analog m
it C:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(C);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "GramSchmidt([vector([I,1]
),vector([-I,1])],normalized);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "CC:=matrix([[1/2*I*sqrt(2
),-1/2*I*sqrt(2)],[1/2*sqrt(2),1/2*sqrt(2)]]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "simplify(evalm(inverse(CC)&*C&*CC));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Und schliesslich mit DD:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvectors(DD);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "GramSchmidt([vector([1/2*I*
sqrt(2), 1/2*I*sqrt(2), 1]), vector([-1/2*I*sqrt(2), -1/2*I*sqrt(2), 1
]) ,vector([-1, 1, 0])],normalized);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 102 "DDD:=matrix([[1/2*I,-1/2*I,-1/2*sqrt(2) ],[1/2*I,-1/
2*I, 1/2*sqrt(2) ],[1/2*sqrt(2),1/2*sqrt(2), 0]]);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "simplify(evalm(inverse(DDD)&*DD&*DDD));" }}
}}{MARK "47" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }