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Concept# Hessenberg matrix

Summary

In linear algebra, a Hessenberg matrix is a special kind of square matrix, 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 after Karl Hessenberg.
A Hessenberg decomposition is a matrix decomposition of a matrix into a unitary matrix and a Hessenberg matrix such that where denotes the conjugate transpose.
A square matrix is said to be in upper Hessenberg form or to be an upper Hessenberg matrix if for all with .
An upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero, i.e. if for all .
A square matrix is said to be in lower Hessenberg form or to be a lower Hessenberg matrix if its transpose is an upper Hessenberg matrix or equivalently if for all with .
A lower Hessenberg matrix is called unreduced if all superdiagonal entries are nonzero, i.e. if for all .
Consider the following matrices.
The matrix is an upper unreduced Hessenberg matrix, is a lower unreduced Hessenberg matrix and is a lower Hessenberg matrix but is not unreduced.
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 fact, reduction of any matrix to a Hessenberg form can be achieved in a finite number of steps (for example, through Householder's transformation of unitary similarity transforms). Subsequent reduction of Hessenberg matrix to a triangular matrix can be achieved through iterative procedures, such as shifted QR-factorization. In eigenvalue algorithms, the Hessenberg matrix can be further reduced to a triangular matrix through Shifted QR-factorization combined with deflation steps.

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