kroneck - Kronecker form of matrix pencil
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).
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