On the joint spectral radius of matrices of order 2 with equal spectral radius
type: Item
instance of: scholarly article
title: On the joint spectral radius of matrices of order 2 with equal spectral radius
zbMATH Open document ID: 1197.15007
author: Bernhard Mößner
published in: Advances in Computational Mathematics
publication date: 2010-07-24T00:00:00Z
review text: Many applications like stability analysis of stochastic dynamical systems, or in approximation theory use the joint spectral radius to verify the convergence or smoothness of subdivision algorithms. The joint spectral radius of two matrices \(A_1,A_2\in \mathbb{R}^{d\times d}\) is \(\mathrm{jsr}(A_1,A_2):=\lim_{n \to \infty}\max \{\| T_1T_2\cdots T_n\|^{1/n} \; | \; T_i=A_1 \text{ or } A_2 \}\) and it measures the growth (or decrease) rate of products of matrices taken from a set. In the Introduction of this paper, an accurate presentation of the problem of computing spectral radius and of the literature around this subject is given. The author shows the difficulty of determining the joint spectral radius and the few results that were produced in this direction starting from 1990 by Kozyakin that proved that \(\{(A,B)\in (\mathbb{R}^{2\times 2})^2 \; | \; \mathrm{jsr}(A,B)<1\}\) is not a semi-algebraic set. Some of the known results are about the polynomial-time approximation algorithms for the joint spectral radius (\textit{V. D. Blondel} and \textit{J. N. Tsitsiklis} showed in 1997 that until \(P=NP\) there is no such an approximation algorithm [Math. Control Signals Syst. 10, No.~1, 31--40 (1997; Zbl 0888.65044)]). The joint spectral radius seems to be known in the case of pairs of matrices of order 2 both when one has negative determinant [the author and \textit{X. Zhou}, SIAM J. Matrix Anal. Appl. 22, No.~1, 242--257 (2000; Zbl 0964.42017)] and when of the matrix is singular [\textit{R. Lima} and \textit{M. Rahibe}, J. Phys. A, Math. Gen. 27, No.~10, 3427--3437 (1994; Zbl 0830.15018)]. Other examples were given by \textit{T. N. T. Goodman, C. A. Micchelli} and \textit{J. D. Ward} [in: Recent advances in wavelet analysis. Boston, MA: Academic Press, Inc. Wavelet Anal. Appl. 3, 335--360 (1994; Zbl 0840.42020)] and by \textit{L. Villemoes} [SIAM J. Math. Anal. 25, No.~5, 1433--1460 (1994; Zbl 0809.42016)]. The main result of the paper under review is the computati…
reviewed by: Alessandra Bernardi
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 5760442
zbMATH Keywords: joint spectral radius
MaRDI profile type: Publication
full work available at URL: s10444-009-9130-y
OpenAlex ID: W2144019097
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DOI: 10.1007/S10444-009-9130-Y
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description: scientific article; zbMATH DE number 5760442