Perturbations of eigenvalues of non-normal matrices

Abstract

The problem considered is to give bounds for finite perturbations of simple and multiple eigenvalues λ i of nonnormal matrices, where these bounds are in terms of the eigenvalues {λ i }, the departure from normality σ , and the Frobenius norm ‖ Δ A ‖ 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 {λ i } and σ . One conclusion is that, very roughly speaking, a simple eigenvalue λ 1 is perturbed by |Δλ 1 | ≲ ‖ Δ A ‖ F · ∏ ( σ / θ j ) where θ j is of the order of magnitude of |λ 1 - λ j |, the product being extended over all j where θ j ≲ σ .