\(\{0,1\}\) completely positive matrices
type: Item
instance of: scholarly article
title: \(\{0,1\}\) completely positive matrices
zbMATH Open document ID: 1072.15027
author: A. Berman
author: Changqing Xu
published in: Linear Algebra and its Applications
publication date: 2005-04-21T00:00:00Z
review text: This paper deals with \(\{0,1\}\) completely positive matrices. Let \(S\) be a subset of the set of real numbers \(\mathbb R\). \(A\) is called \(S\)-factorizable if it can be factorized as \(A=BB^T\), where \(b_{ij} \in S\). The smallest possible number of columns of \(B\) in this factorization is called the \(S\)-rank (rank\(_{S}A\)). For \(S= \mathbb R_{+}\), the set of \(\mathbb R_{+}\)-factorizable matrices is the set of completely positive (cp) matrices. In particular, if \(S\) is a subset of \(\mathbb R_{+}\), a \(S\)-factorizable matrix is also called \(S\)-completely positive (\(S\)-cp) matrix. The aim of this work is to study \(\{0,1\}\)-cp matrices. The authors characterize \(\{0,1\}\)-cp matrices of order less than 4, and give necessary and sufficient conditions for a matrix of order 4 with some zero entries, to be \(\{0,1\}\)-cp. They show that a nonnegative integral Jacobi matrix is \(\{0,1\}\)-cp if and only if it is diagonally dominant, and obtain a necessary condition for a 2-banded symmetric nonnegative integral matrix to be \(\{0,1\}\)-cp. The authors also give formulae for the exact value of the \(\{0,1\}\)-rank of integral symmetric nonnegative diagonally dominant matrices and some other \(\{0,1\}\)-cp matrices.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B36
zbMATH DE Number: 2160179
zbMATH Keywords: factorization
zbMATH Keywords: rank
zbMATH Keywords: diagonally dominant matrix
zbMATH Keywords: \(\{1,0\}\) completely positive matrices
zbMATH Keywords: integral Jacobi matrix
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2004.04.026
OpenAlex ID: W2069709268
cites work: Q4405832
cites work: Completely positive matrices associated withM-matrices
cites work: Q3715120
cites work: On nonnegative factorization of matrices
cites work: On the Matrix Equation X′X = A
cites work: The representation of a definite quadratic form as a sum of two others
cites work: Solving \(XX^{tr}=A\) over the integers
DOI: 10.1016/J.LAA.2004.04.026
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description: scientific article; zbMATH DE number 2160179