Hessenberg

Hessenberg refers to a class of square matrices in linear algebra that are almost triangular. There are two related forms: upper Hessenberg and lower Hessenberg. An upper Hessenberg matrix has nonzero entries on and above the main diagonal and on the first subdiagonal, with all entries below that subdiagonal equal to zero. Equivalently, a(i,j) = 0 for i > j+1. A lower Hessenberg matrix has nonzero entries on and below the main diagonal and on the first superdiagonal, with zeros above that superdiagonal; equivalently a(i,j) = 0 for i < j-1.

In numerical linear algebra, Hessenberg matrices are important because many algorithms exploit their sparse structure to reduce computational effort. Any square matrix can be reduced to upper Hessenberg form by a sequence of orthogonal (or unitary) similarity transformations, typically using Householder reflections. This reduction preserves eigenvalues and decreases the cost of subsequent eigenvalue computations. The QR algorithm, a standard method for finding eigenvalues, is particularly efficient when applied to a Hessenberg form, since multiplications involve far fewer nonzero entries than for a dense matrix.

There are practical distinctions between the two forms: an upper Hessenberg matrix is almost upper triangular, zero below the first subdiagonal, while a lower Hessenberg matrix is almost lower triangular, zero above the first superdiagonal. The Hessenberg concept provides a balance between structural simplicity and numerical stability, making it a common intermediate representation in algorithms for eigenvalues and in certain matrix factorizations.

The term Hessenberg is named after the mathematician associated with the concept. The form is widely used in computational linear algebra, control theory, and related fields due to its favorable balance of sparsity and versatility.