Conservation of the number of the eigenvalues of two-parameter matrix problems in bounded domains under perturbations

Abstract

The paper deals with the two-parameter matrix eigenvalue problems $$T_jv_j - \lambda _1 A_{j1}v_j - \lambda _2 A_{j2}v_j=0$$ and $$\tilde{T}_j\tilde{v}_j - \lambda _1 \tilde{A}_{j1}\tilde{v}_j - \lambda _2 \tilde{A}_{j2}\tilde{v}_j=0$$, where $$\lambda _j, \tilde{\lambda }_j\in \mathbb {C}$$; $$T_j, A_{jk},\tilde{T}_j, \tilde{A}_{jk}\;\;(j, k=1,2)$$ are matrices. We derive the conditions, under which the problems have the same number of the eigenvalues in a given bounded domain. A two-parameter matrix eigenvalue problem is called a Schur–Cohn stable if its spectrum is inside the unit circle. It is said to be asymptotically stable if its spectrum is in the open left half-plane. As a consequence of the main result of the paper we obtain the tests for the Schur–Cohn and asymptotic stabilities. In addition, we suggest new localization results for the eigenvalues of the considered problems, which are sharp if the matrix coefficients are ”close” to triangular ones. Our main tool is a new perturbation result for the determinants of linear matrix pencils.