Basic matrix perturbation theory (Q2327743)

Summary: In this expository note, we give proofs of several results in fi nite-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, \textit{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.