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. 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 approximations 31, 16] which can be stated as eigenvalue problems for unitary matrices, often in Hessenberg form. As those numerical methods exploit the rich mathematical structure of unitary matrices , which is closely analogous to the structure of Hermitian matrices, the methods are eecient and deliver very good approximations to the desired eigenvalues. There exist, however, only a few perturbation results for the unitary eigenvalue problem, which can be used to derive error bounds for the computed eigenvalue approximations. A thorough and complete treatment of the perturbation aspects associated with the numerical methods for unitary eigenvalue problems is still missing. The following perturbation results have been obtained so far. If U and e U are unitary matrices with spectra (U) = f j g, and (e U) = f e j g, respectively, we can arrange the eigenvalues in diagonal matrices and e , respectively, and consider as a measure for the distance of the spectra