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.
Upper Hessenberg matrix
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 .
Lower Hessenberg matrix
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. 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
Any 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 & Roger.
Let be any real or complex matrix, then let be the submatrix of constructed by removing the first row in and let be the first column of . Construct the householder matrix where
This householder matrix will map to and as such, the block matrix will map the matrix to the matrix which has only zeros below the second entry of the first column. Now construct householder matrix in a similar manner as such that maps the first column of to , where is the submatrix of constructed by removing the first row and the first column of , then let which maps to the matrix which has only zeros below the first and second entry of the subdiagonal. Now construct and then in a similar manner, but for the matrix constructed by removing the first row and first column of and proceed as in the previous steps. Continue like this for a total of steps.
Realising that the first rows of any matrix is invariant under multiplication by from the right, by construction of , and so, any matrix can be transformed to an upper Hessenberg matrix by a similarity transformation of the form .
The product of a Hessenberg matrix with a triangular matrix is again Hessenberg. More precisely, if is upper Hessenberg and is upper triangular, then and are upper Hessenberg.
A matrix that is both upper Hessenberg and lower Hessenberg is a tridiagonal matrix, of which symmetric or Hermitian Hessenberg matrices are important examples. A Hermitian matrix can be reduced to tri-diagonal real symmetric matrices.
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 , given by
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.
- Horn & Johnson (1985), page 28; Stoer & Bulirsch (2002), page 251
- Biswa Nath Datta (2010) Numerical Linear Algebra and Applications, 2nd Ed., Society for Industrial and Applied Mathematics (SIAM) ISBN 978-0-89871-685-6, p. 307
- Horn & Johnson (1985), page 35
- Ramon Garcia, Stephan; Horn, Roger (2017). A Second Course In Linear Algebra. ISBN 9781107103818.
- https://www.cs.cornell.edu/~bindel/class/cs6210-f16/lec/2016-10-21.pdf[bare URL PDF]
- "Computational Routines (eigenvalues) in LAPACK". sites.science.oregonstate.edu. Retrieved 2020-05-24.
- Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6.
- Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95452-3.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 11.6.2. Reduction to Hessenberg Form", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8