Pseudospectra of matrices. (English) Zbl 0798.15005

Trefethen, L. N.

Griffiths, D. F. (ed.) et al., Numerical analysis 1991. Proceedings of the 14th Dundee conference, June 25-28, 1991, Dundee, UK. Dundee: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 260, 234-266 (1992).

The pseudospectrum of a square matrix \(A\) is the family of sets \(\Delta_ \varepsilon (A) = \{z \in \mathbb{C}:z\) is eigenvalue of \(A+E\) for some \(E\) with \(\| E \| \leq \varepsilon\}\). In this overview this concept is introduced and equivalent definitions are given. It is shown that for nonnormal matrices \(A\) this concept describes the behaviour of the mapping \(A\) more precisely than the eigenvalues alone. The remainder of the paper presents graphical examples of pseudospectra. Besides Jordan and more complicated Toeplitz matrices, other examples introduced by Wilkinson, Frank, Kahan, Demmel, and some random matrices are considered.
For the entire collection see [Zbl 0787.00029].

Reviewer: L.Elsner (Bielefeld)

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
65F15	Numerical computation of eigenvalues and eigenvectors of matrices

Keywords:

pseudospectrum; nonnormal matrices; Toeplitz matrices; random matrices