The characterization of symmetric primitive matrices with exponent \(n-1\).
type: Item
instance of: scholarly article
title: The characterization of symmetric primitive matrices with exponent \(n-1\).
zbMATH Open document ID: 1033.15018
DOI: 10.1016/S0024-3795(02)00543-8
published in: Linear Algebra and its Applications
publication date: 2003-05-04T00:00:00Z
review text: The authors characterize completely the symmetric primitive matrices with exponent \(n-1\). An \(n\times n\) nonnegative matrix \( A=(a_{ij}) \) is said to be primitive if \(A^{k}>0\) for some positive integer \(k\), and the least such \(k\) is called the exponent of the matrix \(A\). The associated graph to an \(n\times n\)\ symmetric matrix \(A\), \(G(A)\), has \(n\) vertices and there exists an edge from \(i\) to \(j\) in \(G(A)\) if and only if \( a_{ij}>0\). A graph \(G\) is called primitive if there exists an integer \(k>0\) such that for all ordered pairs of vertices \(i,j\) of \(G\) there is a walk from \(i\) to \( j \) of length \(k\), and the least such \(k\) is called the exponent of the graph \(G.\) To prove the characterization the authors use graphs instead of matrices since a symmetric matrix \(A\) is primitive if and only if its associated graph \(G(A)\) is primitive, and moreover their exponents coincide. \textit{J. Shao} [Sci. Sin., Ser. A 30, 348--358 (1987; Zbl 0628.15017)] proved that the possible exponents for a symmetric primitive matrix are \(\{1,2,\dots ,2n-2\}-S\), where \(S\) are all odd numbers among \(\{n,n+1,\dots,2n-2\}\). In an earlier paper \textit{J. L. Cai} and \textit{K. M. Zhang} [Linear Multilinear Algebra 39, 391--396 (1995; Zbl 0843.15008)] characterize the symmetric primitive matrices with exponent \(2n-2r\) \((\geq n)\) and in this paper the authors give the first characterization with exponent less than \(n\).
reviewed by: M. Ladra
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 1903582
zbMATH Keywords: primitive graph
zbMATH Keywords: primitive exponent
zbMATH Keywords: symmetric primitive matrices
author: Junliang Cai
author: Bo Ying Wang
MaRDI profile type: Publication
cites work: Q5422499
cites work: The characterization of symmetric primitive matrices with exponent 2n−2r(n)∗
cites work: Q4029099
cites work: The exponent set of symmetric primitive (0,1) matrices with zero trace
full work available at URL: s0024-3795(02)00543-8
OpenAlex ID: W2001182398
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description: scientific article; zbMATH DE number 1903582