Completely positive matrices

An n×n real matrix A is called completely positive (CP) if it can be factored as A=B′B (“′” stands for transpose) where B is an m×n entrywise nonnegative matrix for some integer m. The smallest such number m is called the cprank of A. In this paper we present a necessary and sufficient condition for any entrywise nonnegative and positive semidefinite matrix to be CP. We also present an expression for the cprank of any CP matrix.