On the Boundary of the Pseudospectrum and Its Fault Points

The study of pseudospectra of linear transformations has become a significant part of numerical linear algebra and related areas. A large body of research activity has focused on how to compute these sets for a given spectral problem with, possibly, certain underlying structure. The theme of this paper was motivated by the question: How effective are path-following procedures for tracing the pseudospectral boundary? The present study of the mathematical properties of the boundary of the pseudospectrum is the result. Although this boundary is generally made up smooth curves, it is shown how the Schur triangular form of the matrix can be used to analyse the singular points of the boundary. 1. Preliminaries In this manuscript we discuss regularity properties of the boundary of the pseudospectrum of a matrix A ∈ Cn×n. This boundary turns out to be a piecewise smooth curve. Our main concern is how the structure of the Schur triangular form of A determines the singular points on this curve. Let us begin by adapting the results of [2] on general matrix polynomial to the particular case of the linear monic polynomial. Below, SpecA denotes the spectrum of A and ‖ · ‖ denotes the (maximum) norm of A as a linear operator in the Euclidean space C. For a given δ ≥ 0, the pseudospectrum of A is the set Specδ A := ⋃