Scaling relationship between the copositive cone and Parrilo's first level approximation

We investigate the relation between the cone of n x n copositive matrices and the approximating cone introduced by Parrilo. While these cones are known to be equal for n a parts per thousand currency sign 4, we show that for n a parts per thousand yen 5 they are not equal. This result is based on the fact that is not invariant under diagonal scaling. We show that for any copositive matrix which is not the sum of a nonnegative and a positive semidefinite matrix we can find a scaling which is not in . In fact, we show that if all scaled versions of a matrix are contained in for some fixed r, then the matrix must be in . For the 5 x 5 case, we show the more surprising result that we can scale any copositive matrix X into and in fact that any scaling D such that for all i yields . From this we are able to use the cone to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of in terms of . We end the paper by formulating several conjectures.