A linear algebraic view of partition regular matrices
type: Item
instance of: scholarly article
title: A linear algebraic view of partition regular matrices
zbMATH Open document ID: 1218.15013
author: Leslie Hogben
author: Jillian McLeod
published in: Linear Algebra and its Applications
publication date: 2010-11-05T00:00:00Z
full work available at URL: https://lib.dr.iastate.edu/math_pubs/66
review text: Let \(A=[a_1, a_2, \dots, a_u]\in {\mathbb Q}^{v\times u}\). We say that \(A\) has the columns condition, written \(A\in CC(m)\), if there exists a partition \(\{I_1,\dots,I_m\}\) of \(\{1,\dots,u\}\) of \(A\), such that for all \(t=1,\dots,m\), \(I_t\neq 0\) and \[ \sum_{i\in I_{t}}a_{i} \in\text{Span}\{a_{j} : j\in I_{1}\cup I_2\cup\cdots\cup I_{t-1}\}. \] The authors investigate the properties of the columns condition, especially for oriented incidence matrices of directed graphs and for sign pattern matrices. They show that the oriented incidence matrix of a graph has the columns condition for some \(m\) if and only if it is strongly connected. They also show that a sign pattern matrix has the columns condition for some \(m\) if and only if it has the columns conditions for all \(m\) if and only if each row is either all zeros or contains both \(+\) and \(-\).
Mathematics Subject Classification ID: 15B35
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 5811008
zbMATH Keywords: columns condition
zbMATH Keywords: partition regular matrix
zbMATH Keywords: oriented incidence matrix
zbMATH Keywords: directed graphs
zbMATH Keywords: sign pattern matrices
MaRDI profile type: Publication
OpenAlex ID: W1969914407
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DOI: 10.1016/J.LAA.2010.06.041
description: scientific article; zbMATH DE number 5811008