REFINED PERTURBATION BOUNDS FOR EIGENVALUES OF HERMITIAN AND NON-HERMITIAN MATRICES

摘要

We present eigenvalue bounds for perturbations of Hermitian matrices and expressthe change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace,rather than in terms of the full perturbation. The perturbations we consider are Hermitian ofrank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specificeigenvalue. Applications include principal component analysis under a spiked covariance model, andpseudo-arclength continuation methods for the solution of nonlinear systems.