  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.
