The necessary and sufficient conditions for dependent quadratic forms to be distributed as multivariate gamma

Let S be distributed as noncentral Wishart given by Wp(m, [Sigma], [Omega]) and let x be an n - 1 random vector distributed as N([mu], V). If qi = x'Aix + 2l'ix + ci, i = 1, 2,..., p, are p dependent second degree polynomials in the elements of x where Aj's are symmetric matrices, then the necessary and sufficient conditions for q1 , q2 ,..., qp to be distributed as the diagonal elements of S are established and this generalizes the result for [Sigma] = I. Some special cases are considered.