On the Pseudospectra of Matrix Polynomials

Abstract

The pseudospectra of a matrix polynomial $P(\lambda)$ are sets of complex numbers that are eigenvalues of matrix polynomials which are near to $P(\lambda)$; i.e., their coefficients are within some fixed magnitude of the coefficients of $P(\lambda)$. Pseudospectra provide important insights into the sensitivity of eigenvalues under perturbations and have several applications. First, qualitative properties concerning boundedness and connected components of pseudospectra are obtained. Then an accurate continuation algorithm for the numerical determination of the boundary of pseudospectra of matrix polynomials is devised and illustrated. This algorithm is based on a prediction-correction scheme.