Structured eigenvalue condition numbers

This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo-symmetric, persymmetric, skew-symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number

Structured Eigenvalue Problems

Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices

On The Perturbation Theory For Unitary Eigenvalue Problems

. Some aspects of the perturbation theory for eigenvalues of unitary matrices are considered. Making use of the close relation between unitary and Hermitian eigenvalue problems a Courant-Fischer-type theorem for unitary matrices is derived and an inclusion theorem analogue to the Kahan theorem for Hermitian matrices is presented. Implications for the special case of unitary Hessenberg matrices are discussed. Key words. unitary eigenvalue problem, perturbation theory AMS(MOS) subject classifications. 15A18, 65F99 1. Introduction. New numerical methods to compute eigenvalues of unitary matrices have been developed during the last ten years. Unitary QR-type methods [19, 9], a divide-and-conquer method [20, 21], a bisection method [10], and some special methods for the real orthogonal eigenvalue problem [1, 2] have been presented. Interest in this task arose from problems in signal processing [11, 29, 33], in Gaussian quadrature on the unit circle [18], and in trigonometric approximation..