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. They are named for Karl Hessenberg.

For example:

$\begin{bmatrix} 1 & 4 & 2 & 3 \\ 3 & 4 & 1 & 7 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 \\ \end{bmatrix}$

is upper Hessenberg and

$\begin{bmatrix} 1 & 2 & 0 & 0 \\ 5 & 2 & 3 & 0 \\ 3 & 4 & 3 & 7 \\ 5 & 6 & 1 & 1 \\ \end{bmatrix}$

is lower Hessenberg.

[edit] Computer programming

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. If the constraints of a linear algebra problem do not allow a general matrix to be conveniently reduced to a triangular one, reduction to Hessenberg form is often the next best thing. In particular, many eigenvalue algorithms reduce their input matrix to Hessenberg form as a first step.