# hess

## Purpose

Hessenberg form of a matrix.

## Synopsis

    [P,H] = hess(A)
    H = hess(A)

## Description

`hess` finds the Hessenberg form of a matrix. A Hessenberg matrix is zero below the first subdiagonal. If the matrix is symmetric or Hermitian, the form is tridiagonal.

`[P,H]` =`hess(A)` produces a Hessenberg matrix `H` and a unitary matrix `P` so that`A = P*H*P'` and `P'*P` = `eye(size(A))`.

`hess(A)` returns `H`.

## Examples

The matrix `gallery(3)` is a 3-by-3 eigenvalue test matrix:

       -149    -50   -154
        537    180    546
        -27     -9    -25

Its Hessenberg form introduces a single zero in the (3,1) position:

    hess(gallery(3)) =
       -149.0000    42.2037   -156.3165
       -537.6783   152.5511   -554.9272
               0     0.0728      2.4489

This matrix has the same eigenvalues as the original, but more computation is needed to reveal them.

## Algorithm

For real matrices, `hess` uses the EISPACK routines `ORTRAN` and `ORTHES`. `ORTHES` converts a real general matrix to Hessenberg form using orthogonal similarity transformations. `ORTRAN` accumulates the transformations used by `ORTHES`.

When `hess` is used with a complex argument, the solution is computed using the QZ algorithm by the EISPACK routines `QZHES`. It has been modified for complex problems and to handle the special case B = I.

For detailed write-ups on these algorithms, see the EISPACK Guide.

## See Also

    eig, qz, schur

## References

[1] B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide, Lecture Notes in Computer Science, volume 6, second edition, Springer-Verlag, 1976.

[2] B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide Extension, Lecture Notes in Computer Science, volume 51, Springer-Verlag, 1977.

[3] C.B. Moler and G. W. Stewart, "An Algorithm for Generalized Matrix Eigenvalue Problems," SIAM J. Numer. Anal., Vol. 10, No. 2, April 1973.

(c) Copyright 1994 by The MathWorks, Inc.