Abstract
A square matrix A is completely positive if A = BB T , where B is a (not necessarily square) nonnegative matrix. In general, a completely positive matrix may have many, even infinitely many, such CP factorizations. But in some cases a unique CP factorization exists. We prove a simple necessary and sufficient condition for a completely positive matrix whose graph is triangle free to have a unique CP factorization. This implies uniqueness of the CP factorization for some other matrices on the boundary of the cone of n × n completely positive matrices. We also describe the minimal face of containing a completely positive A. If A has a unique CP factorization, this face is polyhedral.
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