On the Collatz-Wielandt sets associated with a cone-preserving map
type: Item
instance of: scholarly article
title: On the Collatz-Wielandt sets associated with a cone-preserving map
zbMATH Open document ID: 0684.15008
DOI: 10.1016/0024-3795(89)90033-5
author: Shiow-Fang Wu
author: Bit-Shun Tam
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let K be a proper cone in \(R^ n\) and \(\pi\) (K) be the set of all \(n\times n\) real matrices A such that AK\(\subseteq K\). \textit{G. P. Barker} and \textit{H. Schneider} [ibid. 11, 219-233 (1975; Zbl 0311.15011)] have introduced four sets \(\Omega,\Omega_ 1,\Sigma\) and \(\Sigma_ 1\) associated with A in \(\pi\) (K). The authors refer to these sets as Collatz-Wielandt sets. They determine the values sup \(\Omega\), sup \(\Omega_ 1\), Inf \(\Sigma\), and Inf \(\Sigma_ 1\). The given proofs are elementary and geometric and make use of the Perron-Frobenius theorem. Some known results for a nonnegative matrix relating to spectral radius to its upper and lower Collatz-Wielandt numbers are extended to the setting of a cone-preserving map. Applications of the results obtained to the nonnegativity of solutions of linear inequalities with a nonnegative matrix are also considered.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 4120325
zbMATH Keywords: cone
zbMATH Keywords: Collatz-Wielandt sets
zbMATH Keywords: Perron-Frobenius theorem
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: spectral radius
zbMATH Keywords: Collatz-Wielandt numbers
zbMATH Keywords: cone-preserving map
zbMATH Keywords: nonnegativity of solutions
zbMATH Keywords: linear inequalities
reviewed by: Prem Narain
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90033-5
OpenAlex ID: W2061409307
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description: scientific article; zbMATH DE number 4120325