hess

Hessenberg form of matrix

Syntax

H = hess(A)
[P,H] = hess(A)
[AA,BB,Q,Z] = hess(A,B)

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)) .

[AA,BB,Q,Z] = hess(A,B) for square matrices A and B, produces an upper Hessenberg matrix AA, an upper triangular matrix BB, and unitary matrices Q and Z such that Q*A*Z = AA and Q*B*Z = BB.

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

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Hessenberg Matrix

A Hessenberg matrix contains zeros below the first subdiagonal. If the matrix is symmetric or Hermitian, then the form is tridiagonal. This matrix has the same eigenvalues as the original, but less computation is needed to reveal them.