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Wikipedia / Hessenberg matrix

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 into a unitary matrix denotes the conjugate transpose.

Definitions

Upper Hessenberg matrix

is said to be in upper Hessenberg form or to be an upper Hessenberg matrix it has the following structure:

An upper Hessenberg matrix is called unreduced.

Lower Hessenberg matrix

is said to be in lower Hessenberg form or to be a lower Hessenberg matrix it has the following structure:

A lower Hessenberg matrix is called unreduced.

Examples

is a lower Hessenberg matrix but is not unreduced.

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 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. Reducing a general matrix to a Hessenberg matrix and then reducing further to a triangular matrix, instead of directly reducing a general matrix to a triangular matrix, often economizes the arithmetic involved in the QR algorithm for eigenvalue problems.

Reduction to Hessenberg matrix

Householder transformations

matrix can be transformed into a Hessenberg matrix by a similarity transformation using Householder transformations. The following procedure for such a transformation is adapted from A Second Course In Linear Algebra by Garcia & Horn.

steps.

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Jacobi (Givens) rotations

A Jacobi rotation (also called Givens rotation) is an orthogonal matrix transformation in the form

, is the Jacobi rotation matrix with all matrix elements equal zero except for

to satisfy the equation

to the lower Hessenberg form.

Properties

matrix is both upper Hessenberg, and lower Hessenberg.

are upper Hessenberg. The Hessenberg matrix can be presented in a Jordan canonical form, with confluent Vandermonde matrix as the similarity matrix (chapter 1.4.2 of ).

A matrix that is both upper Hessenberg and lower Hessenberg is a tridiagonal matrix, of which the Jacobi matrix is an important example. This includes the symmetric or Hermitian Hessenberg matrices. A Hermitian matrix can be reduced to tri-diagonal real symmetric matrices.

Hessenberg operator

The Hessenberg operator is an infinite dimensional Hessenberg matrix. It commonly occurs as the generalization of the Jacobi operator to a system of orthogonal polynomials for the space of square-integrable holomorphic functions over some domain—that is, a Bergman space. In this case, the Hessenberg operator is the right-shift operator

The eigenvalues of each principal submatrix of the Hessenberg operator are given by the characteristic polynomial for that submatrix. These polynomials are called the Bergman polynomials, and provide an orthogonal polynomial basis for Bergman space.

See also

Notes

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References

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