On the number of CP factorizations of a completely positive matrix

2020/09/25 by Shaked-Monderer, Naomi
#15A23 #15B48 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2009.12290

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 CP_n of n× n completely positive matrices. We also describe the minimal face of CP_n containing a completely positive A. If A has a unique CP factorization, this face is polyhedral.

Related

• Factorization of CP-rank-3 completely positive matrices
• The cone of 5× 5 completely positive matrices
• Integer completely positive matrices of order two
• Interiors of completely positive cones
• The CP-matrix completion problem