{VERSION 4 0 "IBM INTEL NT" "4.0" }
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1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 
0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(linalg):\nwith(
plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names n
orm and trace have been redefined and unprotected\n" }}{PARA 7 "" 1 "
" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Einige Hilfsprozeduren:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Schreibt die Gleichung der Quadrik
 hin" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "quadrik:=proc(T,a,
r,X)\nreturn(evalm(multiply(transpose(X),T,X)+multiply(transpose(a),X)
+r)[1,1]);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Das Gram
-Schmidt-Verfahren:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "add
ition:=(v,w)->(evalm(v+w)):\nskalarm:=(lambda,v)->(evalm(lambda*v)):\n
sigma:=(v,w)->((multiply(transpose(map(conjugate,v)),w))[1,1]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "gramschmidt:=proc(basis,sig
ma,addition,skalarmult)\nlocal j,jj,vj,obasis;\nobasis:=[skalarm(1/sqr
t(sigma(basis[1],basis[1])),basis[1])];\nfor j from 2 to nops(basis) d
o\n  vj:=basis[j]:\n  for jj from 1 to nops(obasis) do\n    vj:=additi
on(vj,skalarm(-sigma(obasis[jj],basis[j]),obasis[jj])):\n  od:\n  obas
is:=[op(obasis),simplify(skalarm(1/sqrt(sigma(vj,vj)),vj))]:\nod:\nret
urn(obasis):\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Die No
rm eines Vektors" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unprote
ct(norm);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "norm:=(v)->(sq
rt(sigma(v,v))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Spalten e
iner Matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "spalten:=pr
oc(A)\nlocal L;\nL:=convert(transpose(A),listlist);\nreturn([seq(trans
pose(matrix([L[i]])),i=1..nops(L))])\nend proc:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 66 "Der Kern einer Matrix (gibt eine Liste von Spaltenve
ktoren zur\374ck)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "kern:
=proc(A)\nlocal K;\nK:=nullspace(A);\nreturn([seq(transpose(matrix([K[
j]])),j=1..nops(K))]);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
141 "Erstelle f\374r die Hauptachsentransformation eine Liste der Eige
nvektoren einer diagonalisierbaren Matrix, wobei die EV zum EW 0 am En
de stehen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "eigenvektoren
:=proc(T)\nlocal EV,lambda,n,E;\nn:=nops(convert(T,listlist));\nE:=dia
g(seq(1,j=1..n));\nEV:=[]:\nfor lambda in \{eigenvalues(T)\} do\nif la
mbda<>0 then EV:=[op(EV),op(kern(T-lambda*E))];fi\nod:\nEV:=[op(EV),op
(kern(T))];\nreturn(EV);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 50 "Erzeugt die SO(n) Hauptachsentransformationsmatrix" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "hauptachsentr:=proc(T)\nlocal Q;\n
Q:=concat(op(gramschmidt(eigenvektoren(T),sigma,addition,skalarm)));\n
if det(Q)=-1 then Q:=mulcol(Q,1,-1);fi;\nreturn(evalm(Q));\nend proc:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Transformiert eine Quadrik un
ter einer Bewegung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "sub
st:=proc(X,bewegung)\nreturn(\{seq(X[j,1]=bewegung[j,1],j=1..nops(conv
ert(X,listlist)))\});\nend proc:\n\ntransformierte:=proc(quadrik,beweg
ung,X,Y)\nlocal substitutionsliste;\nsubstitutionsliste:=subst(X,beweg
ung);\nreturn(sort(simplify(subs(substitutionsliste,quadrik))));\nend \+
proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Schrittweise Beispielrec
hnung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "T:=matrix([[1,-1
,0,0],[1,1,1,1],[0,0,0,0],[0,0,0,0]]):\nT:=multiply(transpose(T),T);\n
a:=matrix([[1],[2],[-1],[-2]]);\nr:=2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"TG-%'matrixG6#7&7&\"\"#\"\"!\"\"\"F,7&F+F*F,F,7&F,F,F,F,F." 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'matrixG6#7&7#\"\"\"7#\"\"#7
#!\"\"7#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "X:=matrix([[x1],[x2],[x3],[x
4]]):\nY:=matrix([[y1],[y2],[y3],[y4]]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "f:=quadrik(T,a,r,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"fG,4*&,(%#x1G\"\"#%#x3G\"\"\"%#x4GF+F+F(F+F+*&,(%#x2GF)F*F+F,F+
F+F/F+F+*&,*F(F+F/F+F*F+F,F+F+F*F+F+*&F1F+F,F+F+F(F+*&F)F+F/F+F+F*!\"
\"*&F)F+F,F+F4F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=4;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "m:=n-nops(kern(T));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"mG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "E:=diag(seq(1,j=1..n)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Ha
uptachsentransformation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
Q:=hauptachsentr(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'matri
xG6#7&7&,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F/#F0F/\"\"!F37&,$F+F3F3F4F37&F
4F3F6F17&F4F3F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DT:=si
mplify(multiply(transpose(Q),T,Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#DTG-%'matrixG6#7&7&\"\"#\"\"!F+F+7&F+\"\"%F+F+7&F+F+F+F+F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Bewegung:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "x=evalm(Q)*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%\"xG*&-%'matrixG6#7&7&,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F0#F1F0\"\"!F4
7&,$F,F4F4F5F47&F5F4F7F27&F5F4F+F2F1%\"yGF1" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "transformierte(f,multiply(Q,Y),X,Y);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,.*$)%#y1G\"\"#\"\"\"F'*&\"\"%F()%#y2GF'F(F(*(#F
(F'F(-%%sqrtG6#F'F(%#y3GF(F(*&\"\"$F(%#y4GF(F(*(F.F(F/F(F&F(F(F'F(" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Quadratische Erg\344nzung:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "b:=transpose(multiply(transp
ose(a),Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'matrixG6#7&7#,
$*$-%%sqrtG6#\"\"#\"\"\"#F0F/7#\"\"!F)7#\"\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "b1:=submatrix(b,1..m,1..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b1G-%'matrixG6#7$7#,$*$-%%sqrtG6#\"\"#\"\"\"#F0F/7#
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "if m<n then b1:=st
ackmatrix(b1,transpose(matrix([[seq(0,i=m+1..n)]])));fi;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#b1G-%'matrixG6#7&7#,$*$-%%sqrtG6#\"\"#\"\"\"#
F0F/7#\"\"!F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "b2:=eval
m(b-b1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G-%'matrixG6#7&7#\"\"
!F)7#,$*$-%%sqrtG6#\"\"#\"\"\"#F2F17#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "iD:=diag(inverse(submatrix(DT,1..m,1..m)),seq(0,i=m
+1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#iDG-%'matrixG6#7&7&#\"
\"\"\"\"#\"\"!F-F-7&F-#F+\"\"%F-F-7&F-F-F-F-F1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "c:=evalm(1/2*multiply(iD,b1));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"cG-%'matrixG6#7&7#,$*$-%%sqrtG6#\"\"#\"\"\"#F0
\"\")7#\"\"!F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Bewegung" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x=evalm(Q)*(y-evalm(c));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG*&-%'matrixG6#7&7&,$*$-%%sqrtG6#
\"\"#\"\"\"#!\"\"F0#F1F0\"\"!F47&,$F,F4F4F5F47&F5F4F7F27&F5F4F+F2F1,&%
\"yGF1-F'6#7&7#,$F,#F1\"\")7#F5FCFCF3F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "transformierte(f,multiply(Q,Y-c),X,Y);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,,*$)%#y1G\"\"#\"\"\"F'*&\"\"%F()%#y2GF'F(F(*(#F
(F'F(-%%sqrtG6#F'F(%#y3GF(F(*&\"\"$F(%#y4GF(F(#\"#J\"#;F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "Drehung des Ausartungsraums:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "Falls b2=0 oder m=n-1 ist dies nicht notw
endig." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Anderenfalls:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "erg\344nze" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 93 "b2n:=skalarm(1/norm(b2),b2):\nB1:=concat(op([op(
spalten(submatrix(E,1..n,1..m))),op([b2n])]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B1G-%'matrixG6#7&7%\"\"\"\"\"!F+7%F+F*F+7%F+F+,$*&-%
%sqrtG6#\"#QF*-F16#\"\"#F*#F*F37%F+F+,$*$F0F*#\"\"$\"#>" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 21 "zu einer SO(n) Matrix" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 180 "if rank(B1)<n then \n  kB1:=kern(transpose
(B1));\n  B:=concat(op(B1),op(gramschmidt(kB1,sigma,addition,skalarm))
);\nelse\n  B:=B1;\nfi:\nif det(B)=-1 then B:=mulcol(B,n,-1);fi;\nprin
t(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F)
F)7&F)F(F)F)7&F)F),$*&-%%sqrtG6#\"#QF(-F/6#\"\"#F(#F(F1,$*&-F/6#\"#>F(
F2F(#!\"$F:7&F)F),$*$F.F(#\"\"$F:,$*$F8F(#F(F:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 9 "Bewegung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "y=evalm(Q)*evalm(B)*(x-evalm(c));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%\"yG*(-%'matrixG6#7&7&,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F0#F1F0\"\"!
F47&,$F,F4F4F5F47&F5F4F7F27&F5F4F+F2F1-F'6#7&7&F1F5F5F57&F5F1F5F57&F5F
5,$*&-F.6#\"#QF1F-F1#F1FD,$*&-F.6#\"#>F1F-F1#!\"$FJ7&F5F5,$*$FBF1#\"\"
$FJ,$*$FHF1#F1FJF1,&%\"xGF1-F'6#7&7#,$F,#F1\"\")7#F5FhnFhnF3F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "gl:=transformierte(f,multipl
y(Q,B,Y-c),X,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#glG,**$)%#y1G\"
\"#\"\"\"F)*&\"\"%F*)%#y2GF)F*F***#F*F)F*-%%sqrtG6#\"#>F*-F26#F)F*%#y3
GF*F*#\"#J\"#;F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Translation i
m Ausartungsraum:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Falls b2=0 n
icht notwendig" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Anderenfalls:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r1:=subs(\{y1=0,y2=0,y3=0
,y4=0\},gl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G#\"#J\"#;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nb2:=diff(gl,Y[m+1,1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nb2G,$*&-%%sqrtG6#\"#>\"\"\"-F(6#\"
\"#F+#F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "trlar:=evalm(
r1/nb2*submatrix(E,1..n,m+1..m+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&trlarG-%'matrixG6#7&7#\"\"!F)7#,$*&-%%sqrtG6#\"#>\"\"\"-F/6#\"\"#F
2#\"#J\"$/$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "transformi
erte(f,multiply(Q,B,Y-c-trlar),X,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,(*$)%#y1G\"\"#\"\"\"F'*&\"\"%F()%#y2GF'F(F(**#F(F'F(-%%sqrtG6#\"#>F
(-F06#F'F(%#y3GF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Die Gesamt
bewegung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x=evalm(Q)*eva
lm(B)*(y-evalm(c)-evalm(trlar));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
\"xG*(-%'matrixG6#7&7&,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F0#F1F0\"\"!F47&,
$F,F4F4F5F47&F5F4F7F27&F5F4F+F2F1-F'6#7&7&F1F5F5F57&F5F1F5F57&F5F5,$*&
-F.6#\"#QF1F-F1#F1FD,$*&-F.6#\"#>F1F-F1#!\"$FJ7&F5F5,$*$FBF1#\"\"$FJ,$
*$FHF1#F1FJF1,(%\"yGF1-F'6#7&7#,$F,#F1\"\")7#F5FhnFhnF3-F'6#7&FhnFhn7#
,$FG#\"#J\"$/$FhnF3F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "d.h." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "x=evalm(multiply(Q,B))*y-eva
lm(multiply(Q,B,evalm(c)+evalm(trlar)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%\"xG,&*&-%'matrixG6#7&7&,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F1#F2F1
,$*$-F/6#\"#QF2#\"\"$F:,$*$-F/6#\"#>F2#F2F:7&,$F-F5F5F6F=7&\"\"!F5,$F7
#F4FA,$F>#!\"(F:7&FFF5,$F7#!\"#FA,$F>#\"\"&F:F2%\"yGF2F2-F(6#7&7#,&#F4
\"\")F2**#\"#$*\"&_:\"F2F8F2F.F2F?F2F27#,&#F2FZF2**FfnF2F8F2F.F2F?F2F2
7#,$*(F8F2F.F2F?F2#!#J\"%wd7#,$F_o#Fao\"%))GF4" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "8 0 0" 16 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }