On the number of CP factorizations of a completely positive matrix

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 ${\mathcal{C}\mathcal{P}}_n$ of n × n completely positive matrices. We also describe the minimal face of ${\mathcal{C}\mathcal{P}}_n$ containing a completely positive A. If A has a unique CP factorization, this face is polyhedral.

Keywords:

• Completely positive matrix
• CP-rank
• CP factorization
• minimal CP factorization
• M-matrix
• copositive matrix

2010 Mathematics Subject Classifications:

• 15A23
• 15B48

Disclosure statement

No potential conflict of interest was reported by the author.