On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

We show that the maximal cp-rank of n×n completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of n×n completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of 5×5 matrices equals six, which proves the famous Drew-JohnsonLoewy conjecture (1994) for matrices of this order. In addition we present a simple scheme for generating completely positive matrices of high cp-rank and investigate the structure of a minimal cp factorization.