General Perturbations. Stability of Diagonalizable Matrices

Abstract

If A is an H-selfadjoint matrix, a “general perturbation” of the pair (A, H) results in a pair (B, G), in which B is G-selfadjoint and is close to the unperturbed pair (A, H) in an appropriate sense. A similar convention applies to the perturbations of H-unitary matrices considered here.

Identification of a quantity which is invariant under such perturbations is one of the main results of the chapter. This general theorem will admit the characterization of all diagonalizable H-selfadjoint matrices with real spectrum which retain these properties after a general perturbation. Also a description of those cases in which analytic perturbations of H-selfadjoint matrices retain spectral properties which are familiar from the classical hermitian case is obtained. Analogous results for perturbations of H-unitary matrices are also discussed.