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kroneck

Kronecker form of matrix pencil

Syntax

```[Q, Z, Qd, Zd, numbeps, numbeta] = kroneck(F)
[Q, Z, Qd, Zd, numbeps, numbeta] = kroneck(E,A)```

Arguments

F

real matrix pencil `F=s*E-A`

E,A

two real matrices of same dimensions

Q,Z

two square orthogonal matrices

Qd,Zd

two vectors of integers

numbeps,numeta

two vectors of integers

Description

Kronecker form of matrix pencil: `kroneck` computes two orthogonal matrices `Q, Z` which put the pencil `F=s*E -A` into upper-triangular form:

```           | 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)|
```

The dimensions of the four blocks are given by:

`eps=Qd(1) x Zd(1)`, `inf=Qd(2) x Zd(2)`, `f = Qd(3) x Zd(3)`, `eta=Qd(4)xZd(4)`

The `inf` block contains the infinite modes of the pencil.

The `f` block contains the finite modes of the pencil

The structure of epsilon and eta blocks are given by:

`numbeps(1)` = `#` of eps blocks of size 0 x 1

`numbeps(2)` = `#` of eps blocks of size 1 x 2

`numbeps(3)` = `#` of eps blocks of size 2 x 3 etc...

`numbeta(1)` = `#` of eta blocks of size 1 x 0

`numbeta(2)` = `#` of eta blocks of size 2 x 1

`numbeta(3)` = `#` of eta blocks of size 3 x 2 etc...

The code is taken from T. Beelen (Slicot-WGS group).

Examples

```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```