Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

• Published in: Linear algebra and its applications 2009-02, Vol.430 (4), p.1171-1195
• Main Authors: Ahmad, Sk. Safique, Alam, Rafikul
• Format: Article
• Language: English
• Subjects: Algebra; Backward error; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; Matrix polynomial; Multiple eigenvalue; Pseudospectrum; Sciences and techniques of general use
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2008.10.019

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space Lm(Cn×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in Lm(Cn×n). We present a family of natural norms on the space Lm(Cn×n) and show that the norms on the spaces Cm+1 and Cn×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space Lm(Cn×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space Lm(Cn×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.