Hessenberg matrix
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In linear algebra, a Hessenberg matrix is one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal.
For example:
is upper Hessenberg and
is lower Hessenberg.
Many linear algebra algorithms require significantly less computational effort when applied to triangular matrices, and this improvement often carries over to Hessenberg matrices as well. When the constraints of a linear algebra problem do not allow a general matrix to be reduced to a triangular one, reduction to Hessenberg form is often the next best thing. In particular, many eigenvalue algorithms involve Hessenberg reduction as a first step.
[edit] External links
- Hessenberg matrix at MathWorld.
- Hessenberg matrix at PlanetMath.
