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Hessenberg Form

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For an $ n \times n$ matrix $ A$, the entries $ a_{j,k}$ with $ j>k+1$ can be annihilated with $ n-2$ 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 $ A$, $ B$ 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)

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  automatically generated 4/24/2007