1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
|
/*
* 4006.maxima
*
* (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
*/
A: matrix([ a+b*%i, 1, 0, 0 ],
[ 0, a+b*%i, 0, 0 ],
[ 0, 0, a-b*%i, 1 ],
[ 0, 0, 0, a-b*%i ]);
expand(charpoly(A, x));
S: (1/sqrt(2)) * matrix([ 1, -%i, 0, 0 ],
[ 0, 0, 1, -%i ],
[ 1, %i, 0, 0 ],
[ 0, 0, 1, %i ]);
B: expand(invert(S).A.S);
C: subst(2, a, B);
C: subst(3, b, C);
A: subst(2, a, A);
A: subst(3, b, A);
U: matrix([ 1, 0, 1, 0 ],
[ 0, 1, 1, 2 ],
[ 0, 0, 1, 0 ],
[ 0, 0, 0, 1 ]);
V: matrix([ 1, 0, 0, 0 ],
[ 0, 1, 0, 0 ],
[ 0, 1, 1, 0 ],
[ 1, 0, 0, 1 ]);
T: U.V;
invert(T);
D: T.C.invert(T);
p: expand(charpoly(D, x));
factor(p);
lambda: 2+3*%i;
Dlambda: ratsimp(expand(D - lambda * identfor(D)));
rank(Dlambda);
/* D2: expand(Dlambda.Dlambda); */
/* rank(D2); */
load(functs);
/*
E: Dlambda;
E[1]: (rational(1/E[1,1]))*E[1]$
E[2]: E[2] - E[2,1] * E[1]$
E[3]: E[3] - E[3,1] * E[1]$
E[4]: E[4] - E[4,1] * E[1]$
E: ratsimp(E)$
E[2]: (rational(1/E[2,2])) * E[2]$
E[3]: E[3] - E[3,2] * E[2]$
E[4]: E[4] - E[4,2] * E[2]$
E: ratsimp(E)$
E[3]: (rational(1/E[3,3])) * E[3]$
E[4]: E[4] - E[4,3] * E[3]$
E: ratsimp(E)$
E[2]: E[2] - E[2,3] * E[3]$
E[1]: E[1] - E[1,3] * E[3]$
E: ratsimp(E)$
E[1]: E[1] - E[1,2] * E[2]$
E: ratsimp(E)$
E;
*/
b1: matrix([1+%i],[2+2*%i],[%i],[1]);
ratsimp(D.b1 - lambda*b1);
G: Dlambda;
G: addcol(G, b1);
G[1]: (rational(1/G[1,1]))*G[1]$
G[2]: G[2] - G[2,1] * G[1]$
G[3]: G[3] - G[3,1] * G[1]$
G[4]: G[4] - G[4,1] * G[1]$
G: ratsimp(G)$
G[2]: (rational(1/G[2,2])) * G[2]$
G[3]: G[3] - G[3,2] * G[2]$
G[4]: G[4] - G[4,2] * G[2]$
G: ratsimp(G)$
G[3]: (rational(1/G[3,3])) * G[3]$
G[4]: G[4] - G[4,3] * G[3]$
G: ratsimp(G)$
G[2]: G[2] - G[2,3] * G[3]$
G[1]: G[1] - G[1,3] * G[3]$
G: ratsimp(G)$
G[1]: G[1] - G[1,2] * G[2]$
G: ratsimp(G)$
G;
b2: matrix([ G[1,5] ], [ G[2,5] ], [ G[3,5] ], [ G[4,5] ]);
expand(D.b2 - lambda * b2 - b1);
c1: 2 * realpart(b1);
d1: 2 * imagpart(b1);
c2: 2 * realpart(b2);
d2: 2 * imagpart(b2);
D.c1 - 2 * c1 + 3 * d1;
D.d1 - 3 * c1 - 2 * d1;
D.c2 - 2 * c2 + 3 * d2 - c1;
D.d2 - 3 * c2 - 2 * d2 - d1;
|
