Handbook of Linear Algebra, 2nd Edition

Chapter 23

Pseudospectra

Mark Embree

Rice University

Eigenvalues often provide great insight into the behavior of matrices, precisely explaining, for example, the asymptotic character of functions of matrices like Ak and etA. Yet many important applications produce matrices whose behavior cannot be explained by eigenvalues alone. In such circumstances further information can be gleaned from broader sets in the complex plane, such as the numerical range (see Chapter 25), the polynomial numerical hull [Nev93], [Gre02], and the subject of this chapter, pseudospectra.

The ε-pseudospectrum of a square matrix is a subset of the complex plane that always includes the spectrum, but can potentially contain points far from any eigenvalue. Unlike the spectrum, ...