Notes on completely positive matrices
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 \(G(A)\).
scientific article; zbMATH DE number 1116055