kroneck

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