A matrix is nonnormal if it lacks an orthogonal basis of eigenvectors; equivalently, a nonnormal matrix does not commute with its adjoint. This nonnormality adds complexity to eigenvalue perturbation theory, as small changes to matrix coefficients can change the spectrum dramatically. This eigenvalue sensitivity has important practical implications, for example, to stability analysis of dynamical systems and numerical methods for linear systems and eigenvalue problems.
Appreciation of the fact that "eigenvalues are not enough" for analysis of nonnormal matrices has grown in the applied mathematics community over the past dozen years. As an alternative to eigenvalues, we study "epsilon-pseudospectra", sets in the complex plane where the resolvent norm is bounded below by 1/epsilon. These sets, popularized by Nick Trefethen, have proved interesting in both theory and applications. In this talk, we will survey the field, including applications involving matrix iterations and non-Hermitian quantum mechanics, based on collaborations with Nick Trefethen and Marco Contedini.