# 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)

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