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Aide de Scilab >> Algèbre Lineaire > Matrice compagnon > randpencil

# randpencil

random pencil

### Calling Sequence

`F=randpencil(eps,infi,fin,eta)`

### Arguments

eps

vector of integers

infi

vector of integers

fin

real vector, or monic polynomial, or vector of monic polynomial

eta

vector of integers

F

real matrix pencil `F=s*E-A` (`s=poly(0,'s')`)

### Description

Utility function. `F=randpencil(eps,infi,fin,eta)` returns a random pencil `F` with given Kronecker structure. The structure is given by: `eps=[eps1,...,epsk]`: structure of epsilon blocks (size eps1x(eps1+1),....) `fin=[l1,...,ln]` set of finite eigenvalues (assumed real) (possibly []) `infi=[k1,...,kp]` size of J-blocks at infinity `ki>=1` (infi=[] if no J blocks). `eta=[eta1,...,etap]`: structure ofeta blocks (size eta1+1)xeta1,...)

`epsi`'s should be >=0, `etai`'s should be >=0, `infi`'s should be >=1.

If `fin` is a (monic) polynomial, the finite block admits the roots of `fin` as eigenvalues.

If `fin` is a vector of polynomial, they are the finite elementary divisors of `F` i.e. the roots of `p(i)` are finite eigenvalues of `F`.

### Examples

```F=randpencil([0,1],[2],[-1,0,1],[3]);
[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F);
Qd, Zd
s=poly(0,'s');
F=randpencil([],[1,2],s^3-2,[]); //regular pencil
det(F)```

• kroneck — Kronecker form of matrix pencil
• pencan — canonical form of matrix pencil
• penlaur — Laurent coefficients of matrix pencil

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