Primitive matrices over polynomial semirings
type: Item
instance of: scholarly article
title: Primitive matrices over polynomial semirings
zbMATH Open document ID: 1242.15030
author: Shigeki Akiyama
author: Horst Brunotte
published in: Linear Algebra and its Applications
publication date: 2012-05-14T00:00:00Z
review text: The authors investigate the behavior of powers of matrices over a commutative semiring in which a subset is considered as the `nonnegative' elements that in turn contains a subset as the `positive' elements. They extend the notion of primitivity of a real nonnegative matrix to matrices with univariate polynomial entries and give an analogue of the classical characterization of real nonnegative primitive matrices by irreducibility and aperiodicity for matrices with univariate polynomial entries. They characterize univariate polynomials with nonnegative coefficients which admit a power with strictly positive coefficients and provide a primitivity criterion based on almost linear periodic matrices over dioids. They present a formal treatment of the characterization of primitive real matrices in the vein of \textit{V. Pták} [Czech. Math. J. 8(83), 487--495 (1958; Zbl 0082.24402)], and \textit{J. C. Holladay} and \textit{R. S. Varga} [Proc. Am. Math. Soc. 9, 631--634 (1958; Zbl 0096.00805)].
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A80
Mathematics Subject Classification ID: 16Y60
zbMATH DE Number: 6034478
zbMATH Keywords: primitive matrices
zbMATH Keywords: irreducible matrices
zbMATH Keywords: real polynomials
zbMATH Keywords: max-plus algebra
zbMATH Keywords: commutative semiring
zbMATH Keywords: real nonnegative matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2011.12.029
OpenAlex ID: W1968152542
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DOI: 10.1016/J.LAA.2011.12.029
description: scientific article; zbMATH DE number 6034478