First order spectral perturbation theory of square singular matrix pencils

Abstract

Let H(λ)=A0+λA1 be a square singular matrix pencil, and let λ0∈C be an eventually multiple eigenvalue of H(λ). It is known that arbitrarily small perturbations of H(λ) can move the eigenvalues of H(λ) anywhere in the complex plane, i.e., the eigenvalues are discontinuous functions of the entries of A0 and A1. Therefore, it is not possible to develop an eigenvalue perturbation theory for arbitrary perturbations of H(λ). However, if the perturbations are restricted to lie in an appropriate set then the eigenvalues change continuously. We prove that this set of perturbations is generic, i.e., it contains almost all pencils, and present sufficient conditions for a pencil to be in this set. In addition, for perturbations in this set, explicit first order perturbation expansions of λ0 are obtained in terms of the perturbation pencil and bases of the left and right null spaces of H(λ0), both for simple and multiple eigenvalues. Infinite eigenvalues are also considered. Finally, information on the eigenvectors of the generically regular perturbed pencil is presented. We obtain, as corollaries, results for regular pencils that are also new.