On the period and base of a sign pattern matrix
type: Item
instance of: scholarly article
title: On the period and base of a sign pattern matrix
zbMATH Open document ID: 0821.15017
DOI: 10.1016/0024-3795(94)90398-0
author: Zhongshan Li
author: Frank J. Hall
author: Carolyn A. Eschenbach
published in: Linear Algebra and its Applications
publication date: 1995-02-09T00:00:00Z
review text: Let \(Q_ n\) denote the set of all \(n \times n\) sign pattern matrices (each entry is \(+\), \(-\), or \(0)\). A pattern \(A \in Q_ n\) is powerful provided all powers \(A^ 1\), \(A^ 2\), \(A^ 3, \dots\) are unambiguously defined (no two walks \(a_{i_ 0 i_ 1} a_{i_ 1 i_ 2} \dots a_{i_{k - 1} i_ k}\) of length \(k\) from \(i_ 0\) to \(i_ k\) have opposite sign). For a powerful pattern \(A\), if \(A^ l = A^{l + p}\) with \(l\) and \(p\) minimal, then \(l\) is called the base of \(A\) and \(p\) the period of \(A\). Irreducible powerful sign pattern matrices are characterized, and the period and base of powerful patterns are investigated. The paper contains a good review of the basic concepts and several previously known results. It concludes by producing families of examples of powerful patterns by using directed graphs known as negative \(C_ r\)- cockades studied in \textit{J. R. Lundgren} and \textit{J. S. Maybee} [J. Comb. Inf. Syst. Sci., 16, No. 1, 29-36 (1991; Zbl 0764.05032)].
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 720471
zbMATH Keywords: sign pattern matrix
zbMATH Keywords: base
zbMATH Keywords: period
zbMATH Keywords: powerful patterns
zbMATH Keywords: directed graphs
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90398-0
OpenAlex ID: W2027541388
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description: scientific article; zbMATH DE number 720471