The rank of a normally distributed matrix and positive definiteness of a noncentral Wishart distributed matrix

If X similar to N-nxk(M, I-n circle times Sigma), then S = X'X has the noncentral Wishart distribution W-k(')(n, Sigma; A), where Lambda = M'M. Here Sigma is allowed to be singular. It is well known that if Lambda = 0, then S has a (central) Wishart distribution and. S is positive definite with probability 1 if and only if n >= k and Sigma is positive definite. We show that if S has a noncentral Wishart distribution, then S is positive definite with probability 1 if and only if n,>= k and Sigma + Lambda is positive definite. This is a consequence of the main result that rank X = min(n, rank(Sigma + Lambda)) with probability 1. (c) 2007 Elsevier B.V. All rights reserved.