Perturbation theory for the spectral decomposition of Hermitian matrices

Marcus Carlsson

Published 2018-09-24Version 1

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. We also study the analogous line perturbation A + tF where t varies, which leads to explicit Gateaux type differentiability results.