A generalized eigenvalue problem in the max algebra
type: Item
instance of: scholarly article
title: A generalized eigenvalue problem in the max algebra
zbMATH Open document ID: 1121.15011
published in: Linear Algebra and its Applications
publication date: 2007-04-10T00:00:00Z
review text: Let \(A\) and \(B\) be two (entrywise) nonnegative \({n}\times{n}\) matrices; and define the ``max'' product \(\otimes\) by \((A\otimes{x})_{i}=Max^{n}_{1}a_{im}{x_{m}}\). The generalized eigenvalue problem consists of the problem of the existence and uniqueness of an eigenvalue \(\lambda\) such that \(A\otimes{x}=\lambda{B}\otimes{x}\), \(x\geq{0}\), \({x}\neq{0}\). In this paper, the authors use the results for the case \(A\otimes{x}=\lambda{x}\), \(x\geq{0}\), \({x}\neq{0}\), (when \(B=I\)), to consider the generalized eigenvalue problem: \(A\otimes{x}=\lambda{B}\otimes{x}\), \(x\geq{0}\), \({x}\neq{0}\), where \(A\) and \(B\) are positive \({n}\times{n}\) matrices. The authors explore the possible number of eigenvalues and corresponding eigenvectors when \(A\) and \(B\) are \({2}\times{2}\). They use degree theory and certain geometric properties of the graphs of the some specific functions corresponding to algebraic conditions on certain \({2}\times{2}\) determinants.
reviewed by: Manouchehr Misaghian
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5141619
zbMATH Keywords: Max algebra, generalized eigenvalue problem, Brouwer degree
zbMATH Keywords: maximum cycle geometric mean
zbMATH Keywords: nonnegative matrices
author: P. A. Binding
author: Hans W. Volkmer
MaRDI profile type: Publication
full work available at URL: j.laa.2006.09.023
OpenAlex ID: W2127017400
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DOI: 10.1016/J.LAA.2006.09.023
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description: scientific article; zbMATH DE number 5141619