On The Perturbation Theory For Unitary Eigenvalue Problems

Abstract

. 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..