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Scilab Help >> Linear Algebra > Matrix Pencil > fstair

# fstair

computes pencil column echelon form by qz transformations

### Syntax

`[AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol)`

### Arguments

A

m x n matrix with real entries.

tol

real positive scalar.

E

column echelon form matrix

Q

m x m unitary matrix

Z

n x n unitary matrix

stair

vector of indexes (see ereduc)

rk

integer, estimated rank of the matrix

AE

m x n matrix with real entries.

EE

column echelon form matrix

QE

m x m unitary matrix

ZE

n x n unitary matrix

nblcks

is the number of submatrices having full row rank >= 0 detected in matrix `A`.

muk:

integer array of dimension (n). Contains the column dimensions mu(k) (k=1,...,nblcks) of the submatrices having full column rank in the pencil sE(eps)-A(eps)

nuk:

integer array of dimension (m+1). Contains the row dimensions nu(k) (k=1,...,nblcks) of the submatrices having full row rank in the pencil sE(eps)-A(eps)

muk0:

integer array of dimension (n). Contains the column dimensions mu(k) (k=1,...,nblcks) of the submatrices having full column rank in the pencil sE(eps,inf)-A(eps,inf)

nuk:

integer array of dimension (m+1). Contains the row dimensions nu(k) (k=1,...,nblcks) of the submatrices having full row rank in the pencil sE(eps,inf)-A(eps,inf)

mnei:

integer array of dimension (4). mnei(1) = row dimension of sE(eps)-A(eps)

### Description

Given a pencil `sE-A` where matrix `E` is in column echelon form the function `fstair` computes according to the wishes of the user a unitary transformed pencil `QE(sEE-AE)ZE` which is more or less similar to the generalized Schur form of the pencil `sE-A`. The function yields also part of the Kronecker structure of the given pencil.

`Q,Z` are the unitary matrices used to compute the pencil where E is in column echelon form (see ereduc)