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

The paper deals with the two-parameter matrix eigenvalue problems T_jv_j- λ₁A_{j 1}v_j- λ₂A_{j 2}v_j= 0 and T~ _jv~ _j- λ₁A~ _{j 1}v~ _j- λ₂A~ _{j 2}v~ _j= 0 , where λ_j, λ~ _j∈ C; Tj,Ajk,T~j,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.