Mathematics-Online lexicon: Hessenberg Form

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	Mathematics-Online lexicon:
	Hessenberg Form

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overview

For an $n \times n$ matrix

, the entries $a_{j,k}$ with

can be annihilated with

Householder similarity transformations:

$\displaystyle A \mapsto B= Q_{n-2} \cdots Q_1 A Q_1 \cdots Q_{n-2}.$

The transformation $Q_\ell = Q_\ell^{\operatorname t}$ produces zeros below position $(\ell +1, \ell)$ .

For symmetric , is also symmetric, hence tridiagonal. Since eigenvalues are preserved, transformation to the so-called Hessenberg form is a useful preprocessing step for any eigenvalue routine.

(Authors: Höllig/Pfeil/Walter)

Annotation:

Proof: Hessenberg Form

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automatically generated 4/24/2007