Null spaces of correlation matrices

Let R be the real numbers and Rn the vector space of all column vectors of length n. Let be the convex set of all real correlation matrices of size n. If V is a subspace of Rn of dimension k, we consider the face FV of consisting of all such that , i.e., AV=0. If FV is nonempty, we say that V is realizable. We give complete geometric descriptions of FV in the cases k=1, n=4, and k=2, n=5. For k=2, n=5, we provide a simple algebraic method for describing FV.