HessenbergDecomposition

HessenbergDecomposition[m]

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 24‐digit 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:

Wolfram Research (2004), HessenbergDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/HessenbergDecomposition.html.

Text

Wolfram Research (2004), HessenbergDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/HessenbergDecomposition.html.

CMS

Wolfram Language. 2004. "HessenbergDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HessenbergDecomposition.html.

APA

Wolfram Language. (2004). HessenbergDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HessenbergDecomposition.html

BibTeX

@misc{reference.wolfram_2021_hessenbergdecomposition, author="Wolfram Research", title="{HessenbergDecomposition}", year="2004", howpublished="\url{https://reference.wolfram.com/language/ref/HessenbergDecomposition.html}", note=[Accessed: 28-May-2022 ]}

BibLaTeX

@online{reference.wolfram_2021_hessenbergdecomposition, organization={Wolfram Research}, title={HessenbergDecomposition}, year={2004}, url={https://reference.wolfram.com/language/ref/HessenbergDecomposition.html}, note=[Accessed: 28-May-2022 ]}