# Structured Eigenvalue Condition Numbers

Karow, Michael and Kressner, Daniel and Tisseur, Françoise (2006) Structured Eigenvalue Condition Numbers. SIAM J. Matrix Anal. Appl., 28 (4). pp. 1052-1068. ISSN 0895-4798

Official URL: http://siamdl.aip.org/simax

## Abstract

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.

Item Type: Article Structured eigenvalue problem, condition number, Jordan algebra, Lie algebra, automorphism group, symplectic, perplectic, pseudo-orthogonal, pseudo-unitary, complex symmetric, persymmetric, perskew-symmetric, Hamiltonian, skew-Hamiltonian, structure preservation. MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Dr Françoise Tisseur 03 Mar 2009 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/672