# HessenbergDecomposition

gives the Hessenberg decomposition of a numerical matrix m.

# Details and Options • The result is given in the form {p,h} where p is a unitary matrix such that p.h.ConjugateTranspose[p]==m.
• The matrix m must be square.

# Examples

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## Basic Examples(1)

Find the Hessenberg decomposition of a 4×4 matrix:

The matrix h is an upper Hessenberg matrix:

## Scope(2)

Hessenberg decomposition works for complex matrices:

A matrix with entries having 24digit precision:

The Hessenberg decomposition is computed using 24-digit precision:

## Applications(1)

A 4×4 random symmetric matrix:

Compute its Hessenberg decomposition:

Do 100 iterations of the unshifted QR algorithm:

The eigenvalues lie along the diagonal:

## Properties & Relations(1)

A random 4×4 matrix:

Compute its Hessenberg decomposition:

The matrix is unitary:

The matrix is upper Hessenberg:

The original matrix is given by p.h.ConjugateTranspose[p]:

## Possible Issues(1)

HessenbergDecomposition works only with matrices of approximate numerical values: Use JordanDecomposition for exact matrices: