Generalizations of the field of values useful in the study of polynomial functions of a matrix
type: Item
instance of: scholarly article
title: Generalizations of the field of values useful in the study of polynomial functions of a matrix
zbMATH Open document ID: 1004.15027
DOI: 10.1016/S0024-3795(01)00555-9
author: Anne Greenbaum
published in: Linear Algebra and its Applications
publication date: 2002-08-28T00:00:00Z
review text: When estimating \(\|f(A) \|\) for various functions \(f\), where \(A\) is an \(n \times n\) matrix, it is useful to associate \(A\) with some set in the complex plane and relate \(\|f(A) \|\) to the size of \(f\) on this set. For normal matrices an appropriate set is the spectrum of \(A\). For nonnormal matrices the situation is less clear. In this paper polynomials of a fixed degree \(k\) or less satisfying \[ \|p(A) \|\geq \max_{z \in \Omega} |p(z) | \] are considered and sets \(\Omega\) are looked for. The largest such set \(\Omega\) was discussed by \textit{O. Nevanlinna} [Convergence of iterations for linear equations (Birkhäuser, Basel) (1993; Zbl 0846.47008)] and referred as the polynomial numerical hull of degree \(k\). These sets are for nonnormal matrices, in some way, the analog of eigenvalues for a normal matrix. For \(k=1\), each of these sets is the field of values \(A\) and for \(k \geq m\), where \(m\) is the degree of the minimal polynomial of \(A\), they are shown to be the spectrum of \(A\). For \(1 < k < m\) the sets are intermediate between the field of values and the spectrum. Several equivalent characterizations of these sets are presented and for some examples (Jordan block, Gauss-Seidel matrix, the Ehrenfest's urn and Grcar matrix) the calculation of the polynomial numerical hull is carried out. One possible application of the analysis of polynomial functions of a matrix is in describing the convergence rate of the GMRES algorithm for solving linear systems \(Ax=b\).
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 1790235
zbMATH Keywords: polynomial numerical hull
zbMATH Keywords: normal matrix
zbMATH Keywords: nonnormal matrix
zbMATH Keywords: spectral norm
zbMATH Keywords: pseudospectrum
zbMATH Keywords: convergence
zbMATH Keywords: GMRES algorithm
describes a project that uses: SDPT3
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1790235