Basic matrix perturbation theory

Abstract

In this expository note, we give proofs of several results in finite-dimensional matrix perturbation theory: continuity of the spectrum, regularity of the total eigenprojectors, existence and computation of one-sided directional derivatives of semi-simple eigenvalues, and Puiseux expansions of coalescing eigenvalues. These results are all classical, at least in the case of one-dimensional, analytical perturbations; a standard reference is the treatise of T. Kato, Perturbation theory for linear operators (Springer, 1980). In contrast with Kato, we consider perturbations which are not necessarily smooth, in arbitrary finite dimension, and for coalescing eigenvalues we do not use the notion of multi-valued function. The proofs use Rouché’s theorem, representations of projectors as contour integrals, and the description of conjugacy classes of connected covering maps of the punctured disk.