Distance matrices, non-normality and pseudospectra

G. Tsiroyiannis, G. Tsachtsiris, C. Bekas, E. Kokiopoulou and E. Gallopoulos

University of Patras, Greece


Abstract

The e- pseudospectrum of a matrix, defined for example as

Le (A) = {z: z Î L(A+E) for some   E Î Cn x n with || E || £ e}
where L(A) denotes the spectrum of A, is acknowledged to be a powerful mechanism for studying the sensitivity of eigenvalues as well as for investigating the behavior of non-normal matrices and operators. See, for example, the Oxford Pseudospectra Gateway for a wealth of information and references to the topic. In this contribution we consider the case of distance matrices that originate in the course of methods used in computational electromagnetics and study their properties. We are particularly interested in ways to quantify the sensitivity based on information obtained from the pseudospectra. We consider novel algorithms that use such measures to construct non-normal distance matrices with varying levels of sensitivity.