On a decomposition of conditionally positive-semidefinite matrices

A symmetric matrix C is said to be copositive if its associated quadratic form is nonnegative on the positive orthant. Recently it has been shown that a quadratic form x'Qx is positive for all x that satisfy more general linear constraints of the form Ax ⩾0, x ≠0 iff Q can be decomposed as a sum Q = A'CA +S, with C strictly copositive and S positive definite. However, if x'Qx is merely nonnegative subject to the constraints Ax ⩾0, it does not follow that Q admits such a decomposition with C copositive and S positive semidefinite. In this paper we give a characterization of those matrices A for which such a decomposition is always possible.