More on pseudospectra for polynomial eigenvalue problems and applications in control theory

Definitions and characterizations of pseudospectra are given for rectangular matrix polynomials expressed in homogeneous form: P(α, β) = αAd + αd−1βAd−1 + · · · + βA0. It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospectra. Lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system (with equality for standard state-space systems) are obtained in terms of pseudospectra, showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory. How and why to incorporate linear structure into pseudospectra is also discussed by example. © 2002 Elsevier Science Inc. All rights reserved. AMS classification: 65F15; 15A22