{0,1} Completely positive matrices

Let S be a subset of the set of real numbers R. A is called S-factorizable if it can be factorized as A=BBT with bij∈S. The smallest possible number of columns of B in such factorization is called the S-rank of A and is denoted by rankSA. If S is a set of nonnegative numbers, then A is called S-cp. The aim of this work is to study {0,1}-cp matrices. We characterize {0,1}-cp matrices of order less than 4, and give a necessary and sufficient condition for a matrix of order 4 with some zero entries, to be {0,1}-cp. We 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. We 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.