kroneck
=======

Kronecker form of matrix pencil



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [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
~~~~~~~~


+ `gschur`_ generalized Schur form (obsolete).
+ `gspec`_ eigenvalues of matrix pencil (obsolete)
+ `systmat`_ system matrix
+ `pencan`_ canonical form of matrix pencil
+ `randpencil`_ random pencil
+ `trzeros`_ transmission zeros and normal rank


.. _gspec: gspec.html
.. _gschur: gschur.html
.. _trzeros: trzeros.html
.. _pencan: pencan.html
.. _systmat: systmat.html
.. _randpencil: randpencil.html