Notes on completely positive matrices
type: Item
instance of: scholarly article
title: Notes on completely positive matrices
zbMATH Open document ID: 0889.15018
DOI: 10.1016/S0024-3795(97)00278-4
author: Shuhuang Xiang
author: Shuwen Xiang
published in: Linear Algebra and its Applications
publication date: 1998-06-22T00:00:00Z
review text: Some notes on completely positive matrices are presented. The main conclusions are: 1. If \(A\) is a positive semidefinite \((0,1)\) matrix, then \(A\) is completely positive and \(\text{cp-rank }A=\text{rank }A\), where cp-rank \(A\) is the smallest number of columns in a nonnegative matrix \(B\) such that \(A= BB^T\). 2. If \(A\in \overline{{\mathcal M}^{-1}}\) and is symmetric, then \(A\) can be factored as \(A= L_0 L^T_0\), where \(L_0\) is lower triangular, \(L_0\in \overline{{\mathcal M}^{-1}}\), and \(\text{cp-rank }A\leq n\). Here \(\overline{{\mathcal M}^{-1}}\) is the closure of the inverse M-matrices which are nonsingular and of order \(n\). For the estimation of the cp-rank of H-matrices, the authors obtain the following results: 3. If \(A\) is a nonnegative symmetric irreducible H-matrix, then \(A\) is completely positive. In that case, if \(A\) is nonsingular, then \(\text{cp-rank }A\leq{1\over 2}n(n- 1)- N+1\); if \(A\) is singular, then \(\text{cp-rank }A\leq{1\over 2}n(n- 1)-N\), where \(2N\) is the number of zeros in off-diagonal positions of \(A\). Against that for nonnegative symmetric reducible H-matrices with form \(\text{PAP}^T= \text{diag}(A_{11}, A_{22},\dots, A_{kk})\), where \(P_i\) is a permutation matrix and \(A_{ii}\) \((i=1, 2,\dots,k)\) are nonnegative symmetric irreducible H-matrices, the results will be: \(A\) is completely positive and \(\text{cp-rank }A= {n(n- 1)\over 2}- N+k\), where \(k\) is the number of connected components of the graph \(G(A)\).
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 1116055
zbMATH Keywords: completely positive matrix
zbMATH Keywords: cp-rank
zbMATH Keywords: inverse M-matrices
zbMATH Keywords: irreducible H-matrix
zbMATH Keywords: reducible H-matrices
MaRDI profile type: Publication
cites work: Q3843714
cites work: Topological proofs for certain theorems on matrices with non-negative elements
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cites work: Q5328177
cites work: Nonnegative factorization of completely positive matrices
cites work: Inverse M-matrices
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cites work: Q5605274
description: scientific article; zbMATH DE number 1116055