\(\{0,1\}\) completely positive matrices
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.
\(\{1,0\}\) completely positive matrices
scientific article; zbMATH DE number 2160179
scientific article