New results on the cp rank and related properties of co(mpletely)positive matrices

TL;DR

This paper explores the properties of copositive and completely positive matrices, establishing new relations and improving bounds on the cp-rank, especially for matrices of order six, which advances understanding in matrix theory and optimization.

Contribution

It introduces new relations between boundary matrices of copositive and completely positive cones and improves the upper bounds on the cp-rank for general and specific matrix orders.

Findings

Improved upper bound on cp-rank for general matrices

Further improved bound for matrices of order six

New relations between boundary matrices of the cones

Abstract

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order, and a further improvement for such matrices of order six.