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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 "" 
{TEXT -1 37 "Diagonalisierung eines Endomorphismus" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 78 "A:=matrix([[4, 3, 1, -1], [-3, -2, -1, 1], \+
[-5, -4, -1, 1], [-2, -1, -1, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"AG-%'matrixG6#7&7&\"\"%\"\"$\"\"\"!\"\"7&!\"$!\"#F-F,7&!\"&!\"%F-
F,7&F0F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "det(evalm(A
-t*diag(1,1,1,1)));factor(%);solve(%%,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*$)%\"tG\"\"#\"\"\"F(*&F'F()F&\"\"$F(!\"\"*$)F&\"\"%F
(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"tG\"\"#\"\"\"),&!\"\"F'F%
F'F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#\"\"\"F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"
\"\"F#\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "nullspace(
evalm(A-diag(1,1,1,1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vecto
rG6#7&\"\"!!\"\"\"\"\"!\"#-F%6#7&F*F+F(!\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "nullspace(A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7&\"\"!F(\"\"\"F)-F
%6#7&!\"\"F)F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ev:=[ei
genvectors(A)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#evG7$7%\"\"\"\"
\"#<$-%'vectorG6#7&\"\"!!\"\"F'!\"#-F+6#7&F'F0F.!\"$7%F.F(<$-F+6#7&F.F
.F'F'-F+6#7&F/F'F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "con
vertmatrix:=proc(L)\nlocal j;\nreturn([seq(convert(L[j],matrix),j=1..n
ops(L))]);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Basis aus Eigenvektoren" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "iT:=concat(seq(op(convertmat
rix(ev[j][3])),j=1..nops(ev)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
iTG-%'matrixG6#7&7&\"\"\"\"\"!F+!\"\"7&!\"#F,F+F*7&F+F*F*F*7&!\"$F.F*F
+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "T:=inverse(iT);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7&7&\"\"%\"\"$\"\"\"
!\"\"7&!\"&!\"%F-F,7&\"\"#F,F,\"\"!7&F+F+F,F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "multiply(T,A,iT);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F)F(F)F)7&F)F)F)F)F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Mit der Maple-Funktion" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "J:=jordan(A,'U');print(U);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7&7&\"\"!F*F*F*7&F*
\"\"\"F*F*F)7&F*F*F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#
7&7&!\"$#\"#6\"\"$\"\"!#!\"\"F+7&F+F(F,F,7&#\"#;F+#!#8F+#\"\"\"F+#\"\"
#F+7&#\"\"(F+#!\"(F+F5F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 
"multiply(U,J,inverse(U))=evalm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%'matrixG6#7&7&\"\"%\"\"$\"\"\"!\"\"7&!\"$!\"#F,F+7&!\"&!\"%F,F+7&F
/F,F,F+F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 22 "Jordansche Normalform:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "A:=matrix([[-2,1,-1,1],[1,-2,2,-1],[1,-1,
1,-1],[0,0,1,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrix
G6#7&7&!\"#\"\"\"!\"\"F+7&F+F*\"\"#F,7&F+F,F+F,7&\"\"!F1F+F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "chi:=det(evalm(A-t*diag(1,1,1,1)));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$chiG,,*$)%\"tG\"\"#\"\"\"\"\"'*&\"\"%F*)F(\"\"$
F*F*F*F**&F-F*F(F*F**$)F(F-F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "solve(chi,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&!\"\"F#F#F#" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor(chi);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#*$),&\"\"\"F&%\"tGF&\"\"%F&" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "N:=evalm(A-(-1)*diag(1,1,1,1));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7&7&!\"\"\"\"\"F*F+7&F+F*
\"\"#F*F,7&\"\"!F/F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "V1:=convertmatrix(nullspa
ce(N));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G7$-%'matrixG6#7&7#\"
\"\"F*7#\"\"!F,-F'6#7&F*F,F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "V2:=convertmatrix(nullspace(N^2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#V2G7%-%'matrixG6#7&7#\"\"\"7#\"\"!F,F*-F'6#7&F*F*F,F
,-F'6#7&7#!\"#F,F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "V3:
=convertmatrix(nullspace(N^3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
V3G7&-%'matrixG6#7&7#\"\"!F*7#\"\"\"F*-F'6#7&F*F*F*F,-F'6#7&F*F,F*F*-F
'6#7&F,F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Somit hat A eine
n 3x3 Jordanblock und einen 1x1 Jordanblock  und die Jordansche Normal
form ist" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "diag(JordanBloc
k(-1,3),JordanBlock(-1,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr
ixG6#7&7&!\"\"\"\"\"\"\"!F*7&F*F(F)F*7&F*F*F(F*7&F*F*F*F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "Bestimme nun den Basiswechsel" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "Eine Basis von V3/V2 ist gegeben durch di
e Klasse von" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "w1 := matri
x([[1], [0], [0], [0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w1G-%'m
atrixG6#7&7#\"\"\"7#\"\"!F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "d
enn" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rank(concat(w1,op(V2
)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 45 "Somit gibt es 1 Jordanblock der Dimension 3x3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "w2:=multiply(N,w1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w2G-%'matrixG6#7&7#!\"\"7#\"\"\"F+7
#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Nun m\374ssen wir in V2
/V1 die Klasse von w2 zu einer Basis erg\344nzen" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "rank(concat(w2,op(V1)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Somit ist
 w2 schon eine Basis und es gibt keinen Jordanblock der Gr\366\337e 2x
2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "w3:=multiply(N,w2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w3G-%'matrixG6#7&7#\"\"\"7#\"\"!F+
F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Nun m\374ssen wir in V1 den
 Vektor w3 zu einer Basis erg\344nzen" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "V1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7
&7#\"\"\"F(7#\"\"!F*-F%6#7&F(F*F*F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "u:=matrix([[1], [1], [0], [0]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"uG-%'matrixG6#7&7#\"\"\"F)7#\"\"!F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "u und w3 sind eine Basis von V1, denn" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rank(concat(u,w3));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "Die Tranformationsmatrix lautet also" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "iT:=concat(w3,w2,w1,u);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#iTG-%'matrixG6#7&7&\"\"\"!\"\"F*F*7&\"\"!F*F-F*7&F-F
*F-F-7&F*F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "multiply
(inverse(iT),A,iT);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7
&!\"\"\"\"\"\"\"!F*7&F*F(F)F*7&F*F*F(F*7&F*F*F*F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 22 "Mit der Maple-Funktion" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "J:=jordan(A,'U');print(U);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"JG-%'matrixG6#7&7&!\"\"\"\"\"\"\"!F,7&F,F*F+F,7&F,F
,F*F,7&F,F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"
\"\"!\"\"\"\"!F)7&F*F(F)F)7&F*F(F*F*7&F(F*F*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "multiply(inverse(U),A,U);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7&!\"\"\"\"\"\"\"!F*7&F*F(F)F*7&F*F*F(F*7
&F*F*F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "19
 0 0" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 
}