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--Johnson--Loewy conjecture [Linear Multilinear Algebra, 37 (1994), pp. 303--310] 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.