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

## See also

- schur
- spec
- systmat
- pencan
- randpencil
- trzeros