On matrix powers in max-algebra
type: Item
instance of: scholarly article
title: On matrix powers in max-algebra
zbMATH Open document ID: 1131.15008
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
review text: The max-algebra, an analogue of classical linear algebra, is developed over the semiring \(\bar{\mathbb R}:= \mathbb R \cup \{- \infty\}\) (alias \(\mathbb R _{\max}\)) with respect to the max-plus operators \(a \oplus b= \max(a, b)\) and \(a\otimes b= a + b\) for all \(a, b \in \bar{\mathbb R}\). A square matrix \(A= (a_{ij}) \in \bar{\mathbb R}^{n \times n}\) is called robust if for every \(x \in \bar{\mathbb R}^n\) with at least one finite component, \(A^{(k)} \otimes x\) is an eigenvector of \(A\) for some natural number \(k\), \(A^{(k)}\) being the max-algebraic power of \(A\). By studying the eigenvalue-eigenvector problem for powers of irreducible matrices, the authors characterise robust irreducible matrices. In particular, a strongly irreducible matrix \(A\) is robust if and only if the eigenspaces of all powers of \(A\) coincide.
Mathematics Subject Classification ID: 15A80
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5132604
zbMATH Keywords: max-algebra
zbMATH Keywords: matrix powers
zbMATH Keywords: irreducible matrix
zbMATH Keywords: eigenspace
zbMATH Keywords: semiring
zbMATH Keywords: max-plus operators
zbMATH Keywords: eigenvector
zbMATH Keywords: eigenvalue
author: Peter Butkovič
author: R. A. Cuninghame-Green
reviewed by: George F. Nassopoulos
MaRDI profile type: Publication
full work available at URL: j.laa.2006.09.027
OpenAlex ID: W2095988315
cites work: Q4304869
cites work: Combinatorial matrix theory
cites work: Q3964346
cites work: Minimax algebra and applications
cites work: Minimal (max,+) Realization of Convex Sequences
cites work: Linear matrix period in max-plus algebra
cites work: A characterization of the minimum cycle mean in a digraph
DOI: 10.1016/J.LAA.2006.09.027
description: scientific article; zbMATH DE number 5132604