On extremal matrices of second largest exponent by Boolean rank
type: Item
instance of: scholarly article
title: On extremal matrices of second largest exponent by Boolean rank
zbMATH Open document ID: 1122.15002
author: Lihua You
author: Gexin Yu
author: Bolian Liu
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
review text: A Boolean matrix is a matrix over the binary Boolean algebra \((0,1)\). Given an \(m \times n\) Boolean matrix \(A\), its Boolean rank \(b(A)\) is defined to be the smallest integer such that for some \(m \times k\) Boolean matrix \(F\) and \(k \times n\) Boolean matrix \(G\), the equality \(A = FG\) holds. If \(A = 0\), then \(b(A)\) is defined to be \(0\). The product \(A=FG\) is called a Boolean rank factorization of \(A\). A Boolean \(n \times n\) matrix \(A\), considered as a real matrix, is primitive if there exists an integer \(k \geq 1\) such that \(A^k\) has only positive entries. The smallest such \(k\) is called the primitive exponent of \(A\), in symbols \(exp(A)\). The authors first review results by \textit{H. Wielandt} [Math. Z. 52, 642--648 (1950; Zbl 0035.29101)] and by \textit{D. A. Gregory, S. J. Kirkland}, and \textit{N. J. Pullman} [Linear Algebra Appl. 217, 101--116 (1995; Zbl 0822.15005)], and then show that for each \(3 \leq b \leq n-1\) there exist \(n \times n\) primitive Boolean matrices \(A\) with \(b(A) = b\) such that \(exp(A) = (b-1)^2+1\). Moreover, the authors explicitly describe all such matrices.
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 5132623
zbMATH Keywords: exponent
zbMATH Keywords: Boolean matrix
zbMATH Keywords: rank factorization
MaRDI profile type: Publication
full work available at URL: j.laa.2006.09.016
OpenAlex ID: W1965786185
cites work: A bound on the exponent of a primitive matrix using Boolean rank
cites work: An extension of the Dulmage-Mendelsohn theorem
cites work: Q4517538
cites work: On the exponent of a primitive digraph
cites work: Unzerlegbare, nicht negative Matrizen
DOI: 10.1016/J.LAA.2006.09.016
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description: scientific article; zbMATH DE number 5132623