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.