`hess`

Hessenberg form of a square matrix.

Syntax

H = hess(A)

Returns the upper Hessenberg matrix similar to A.

[P, H] = hess(A)

Also returns the unitary matrix P with A = P*H*P'.

Description

An upper Hessenberg matrix is nearly triangular: all entries below the first subdiagonal are zero. Any square matrix can be reduced to this form by a sequence of orthogonal reflections, so H has the same eigenvalues as A.

The reduction is the first step of the QR algorithm for eigenvalues: working with H makes each iteration much cheaper. For a symmetric A the result H is tridiagonal up to rounding.

Input arguments

A: Square matrix.

Output arguments

H: Matrix with zeros below the first subdiagonal.
P: Orthogonal (unitary) transformation matrix.

Example

Reducing a 3-by-3 matrix

A = [1 2 3; 4 5 6; 7 8 10];
[P, H] = hess(A);
H
disp(norm(P*H*P' - A) < 1e-10)

Output

H = 3×3

    1.0000   -3.5970   -0.2481
   -8.0623   14.8000    2.4000
         0    0.4000    0.2000
   1

hess