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
|
.TH kroneck 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
kroneck - Kronecker form of matrix pencil
.SH CALLING SEQUENCE
.nf
[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F)
[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(E,A)
.fi
.SH PARAMETERS
.TP
F
: real matrix pencil \fVF=s*E-A\fR
.TP
E,A
: two real matrices of same dimensions
.TP 10
Q,Z
:
two square orthogonal matrices
.TP
Qd,Zd
:
two vectors of integers
.TP 15
numbeps,numeta
:
two vectors of integers
.SH DESCRIPTION
Kronecker form of matrix pencil: \fVkroneck\fR computes two
orthogonal matrices \fVQ, Z\fR which put the pencil \fVF=s*E -A\fR into
upper-triangular form:
.nf
| sE(eps)-A(eps) | X | X | X |
|----------------|----------------|------------|---------------|
| O | sE(inf)-A(inf) | X | X |
Q(sE-A)Z = |---------------------------------|----------------------------|
| | | | |
| 0 | 0 | sE(f)-A(f) | X |
|--------------------------------------------------------------|
| | | | |
| 0 | 0 | 0 | sE(eta)-A(eta)|
.fi
The dimensions of the four blocks are given by:
.PP
\fVeps=Qd(1) x Zd(1)\fR, \fVinf=Qd(2) x Zd(2)\fR,
\fVf = Qd(3) x Zd(3)\fR, \fVeta=Qd(4)xZd(4)\fR
.LP
The \fVinf\fR block contains the infinite modes of
the pencil.
.LP
The \fVf\fR block contains the finite modes of
the pencil
.LP
The structure of epsilon and eta blocks are given by:
.PP
\fVnumbeps(1)\fR = # of eps blocks of size 0 x 1
.PP
\fVnumbeps(2)\fR = # of eps blocks of size 1 x 2
.PP
\fVnumbeps(3)\fR = # of eps blocks of size 2 x 3 etc...
.PP
\fVnumbeta(1)\fR = # of eta blocks of size 1 x 0
.PP
\fVnumbeta(2)\fR = # of eta blocks of size 2 x 1
.PP
\fVnumbeta(3)\fR = # of eta blocks of size 3 x 2 etc...
.LP
The code is taken from T. Beelen (Slicot-WGS group).
.SH EXAMPLE
.nf
F=randpencil([1,1,2],[2,3],[-1,3,1],[0,3]);
Q=rand(17,17);Z=rand(18,18);F=Q*F*Z;
//random pencil with eps1=1,eps2=1,eps3=1; 2 J-blocks @ infty
//with dimensions 2 and 3
//3 finite eigenvalues at -1,3,1 and eta1=0,eta2=3
[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F);
[Qd(1),Zd(1)] //eps. part is sum(epsi) x (sum(epsi) + number of epsi)
[Qd(2),Zd(2)] //infinity part
[Qd(3),Zd(3)] //finite part
[Qd(4),Zd(4)] //eta part is (sum(etai) + number(eta1)) x sum(etai)
numbeps
numbeta
.fi
.SH SEE ALSO
gschur, gspec, systmat, pencan, randpencil, trzeros
|
