Perturbation theory for the spectral decomposition of Hermitian matrices

Let A and E be Hermitian self-adjoint matrices, where A is fixed and E a small perturbation. We study how the eigenvalues and eigenvectors of A+E depend on E, with the aim of obtaining first order formulas (and when possible also second order) that are explicitly computable in terms of the spectral decomposition of A and the entries in E. In particular we provide explicit Frechet type differentiability results. The findings can be seen as an extension of the Rayleigh-Schr\"odinger coefficients for analytic expansions of one-dimensional perturbations.