Characterization of joint spectral radius via trace
type: Item
instance of: scholarly article
title: Characterization of joint spectral radius via trace
zbMATH Open document ID: 0970.15020
DOI: 10.1016/S0024-3795(00)00149-X
author: Quande Chen
author: Xinlong Zhou
published in: Linear Algebra and its Applications
publication date: 2001-10-15T00:00:00Z
review text: For a bounded set \(\Sigma\) of square matrices the joint spectral radius \(\rho(\Sigma)\) is defined as the limit superior of the supremum over the set \({\mathcal {L}}_n\) of all matrix products of length \(n\) from the collection \(\Sigma\) of the \(n^{th}\) root of the product matrix norm, while the joint generalized spectral radius \(\widetilde \rho(\Sigma)\) is the same expression involving the spectral radii of the matrix products instead. Inequalities such as \(\sup \rho (A)^{1/n} \leq \widetilde \rho(\Sigma) \leq \rho(\Sigma) \leq \sup \|A\|^{1/n}\) are known. For a finite set \(\Sigma\) of matrices this paper establishes that \(\rho(\Sigma) = \limsup\limits _{n \to \infty} \sup \limits_{A \in {\mathcal {L}}_n} |{trace}(A)|^{1/n}\) and gives estimates for the joint spectral radius for sets generated by two matrices in terms of the trace. These concepts apply to cascade algorithms for wavelets with compact support.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 1523269
zbMATH Keywords: joint spectral radius
zbMATH Keywords: trace
zbMATH Keywords: wavelet
zbMATH Keywords: matrix products
zbMATH Keywords: inequalities
zbMATH Keywords: cascade algorithms
reviewed by: Frank Uhlig
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1523269