The Difference Between 5× 5 Doubly Nonnegative and Completely Positive Matrices

The convex cone of n × n completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for n ≤ 4 only, every DNN matrix is CPP. In this paper, we investigate the difference between 5×5 DNN and CPP matrices. Defining a bad matrix to be one which is DNN but not CPP, we: (i) characterize all 5× 5 extreme DNN matrices, in particular bad ones; (ii) design a finite procedure to decompose any n×n DNN matrix into the sum of a CPP matrix and a bad matrix, which itself cannot be further decomposed; (iii) show that every bad 5×5 DNN matrix is the sum of a CPP matrix and a single bad extreme matrix; and (iv) demonstrate how to separate bad extreme matrices from the cone of 5× 5 CPP matrices.