Perturbations of Eigenvalues of Non-Normal Matrices

Abstract

The problem considered is to give bounds for finite perturbations of simple and multiple eigenvalues λi of nonnormal matrices, where these bounds are in terms of the eigenvalues {λi}, the departure from normality σ, and the Frobenius norm ‖ ΔA ‖ F of the perturbation matrix, but not in terms of the eigensystem. The bounds which are derived are shown to be almost attainable for any set of all matrices of given {λi} and σ. One conclusion is that, very roughly speaking, a simple eigenvalue λ1 is perturbed by |Δλ1| ≲ ‖ ΔA ‖F · ∏ (σ/θj) where θj is of the order of magnitude of |λ1 - λj|, the product being extended over all j where θj ≲ σ. © 1975, ACM. All rights reserved.