Compactification of a set of matrices with convergent infinite products
type: Item
instance of: scholarly article
title: Compactification of a set of matrices with convergent infinite products
zbMATH Open document ID: 0958.15021
DOI: 10.1016/S0024-3795(00)00080-X
author: Jianhong Shen
published in: Linear Algebra and its Applications
publication date: 2001-04-09T00:00:00Z
review text: The author generalizes and unifies some known results on a set \(\Sigma\) of square matrices (real or complex) with right-convergent-products (RCPs). Here \(\Sigma\) is said to be an RCP set if for any sequence \((A_1, A_2,\dots)\in \Sigma^\infty\), \(\lim_{n\to\infty} A_1A_2\cdots A_n\) exists. A König chain of \(\Sigma\) with respect to a given norm \(\|\cdot\|\) is a sequence \((A_1,A_2,\dots)\in \Sigma^\infty\) such that \(\|A_1A_2\cdots A_m\|^{{1\over m}}\geq \widehat\rho\), \(m= 1,2,\dots\)\ . In Section 2 the author studies some important properties that are shared by both a bounded matrix set and its closure. In Section 3 he proves the existence of a König chain in a compact set of matrices, and then shows applications by reproducing some major known results.
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1462183
zbMATH Keywords: joint spectral radius
zbMATH Keywords: compactness
zbMATH Keywords: right-convergent-products
zbMATH Keywords: König chain
reviewed by: Takayuki Nôno
MaRDI profile type: Publication
cites work: Simultaneous Contractibility
cites work: Bounded semigroups of matrices
cites work: Sets of matrices all infinite products of which converge
cites work: Q3272731
cites work: Q4513996
cites work: Simultaneous Schur stability
description: scientific article; zbMATH DE number 1462183