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| hess | Examples See Also |
Syntax
[P,H] = hess(A) H = hess(A)
Description
H = hess(A)
finds H, the Hessenberg form of matrix A.
[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)).
Definition
A Hessenberg matrix is zero below the first subdiagonal. If the matrix is symmetric or Hermitian, the form is tridiagonal. This matrix has the same eigenvalues as the original, but less computation is needed to reveal them.Examples
H is a 3-by-3 eigenvalue test matrix:
H =
-149 -50 -154
537 180 546
-27 -9 -25
Its Hessenberg form introduces a single zero in the (3,1) position:
hess(H) =
-149.0000 42.2037 -156.3165
-537.6783 152.5511 -554.9272
0 0.0728 2.4489
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] Smith, B. T., 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, Vol. 6, second edition, Springer-Verlag, 1976. [2] Garbow, B. S., J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide Extension, Lecture Notes in Computer Science, Vol. 51, Springer-Verlag, 1977. [3] Moler, C.B. and G. W. Stewart, "An Algorithm for Generalized Matrix Eigenvalue Problems," SIAM J. Numer. Anal., Vol. 10, No. 2, April 1973.