Remarks on Non-Linear Spectral Perturbation

We are interested in some aspects of the perturbation effects in the spectrum of a real nonnormal matrix A under linear perturbations. We discuss some known results and we use them to justify some recent experimental observations. Moreover, we demonstrate that the qualitative behavior of the eigenvalues of A under linear perturbations may be predicted by inspecting the spectral radius of a related matrix. Then, we show how this information can be used to analyze the quality of the approximation of a projection method and to justify the presence of unexpected approximate eigenvalues.