Max-plus definite matrix closures and their eigenspaces
type: Item
instance of: scholarly article
title: Max-plus definite matrix closures and their eigenspaces
zbMATH Open document ID: 1131.15009
author: N. E. Zubov
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0506177
review text: This paper provides a contribution to the geometrical understanding of earlier algebraic results on max-plus eigenspaces [cf. \textit{P. Butkovič}, Discrete Appl. Math. 105, 73--86 (2000; Zbl 0976.15013); \textit{P. Butkovič} and \textit{R. A. Cuninghame-Green}, Linear Algebra Appl. 421, No. 2--3, 370--381 (2007; Zbl 1131.15008)] by regarding the max-plus semiring \(\mathbb R _{\max}:= \mathbb R \cup \{-\infty\}\) with its simplifying total order and using basic tools of max-algebra. Definite forms for a given matrix \(A= (a_{ij}) \in \mathbb R^{n\times n}_{\max}\) with nonzero (finite) permanent are considered and it is shown that the closures of all of them coincide, so resulting in the ``definite closure'' operation. The author presents a description of the eigenspace \(eig(A)\) and the faces of it for a definite matrix \(A\). Employing a representation (op. cit.), he concludes that \(eig(A)\) has a non-empty interior if and only if \(A\) has a strong permanent. Finally, this interior is fully detailed by means of Hilbert distances.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5132590
zbMATH Keywords: max-algebra
zbMATH Keywords: max-plus geometry
zbMATH Keywords: definite matrix
zbMATH Keywords: max-plus eigenspace
zbMATH Keywords: Hilbert distance
zbMATH Keywords: max-plus semiring
zbMATH Keywords: permanent
reviewed by: George F. Nassopoulos
MaRDI profile type: Publication
OpenAlex ID: W2127947324
arXiv ID: math/0506177
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DOI: 10.1016/J.LAA.2006.02.038
description: scientific article