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

If X∼Nn×k(M,In⊗Σ), then S=X′X has the noncentral Wishart distribution Wk′(n,Σ;Λ), where Λ=M′M. Here Σ is allowed to be singular. It is well known that if Λ=0, then S has a (central) Wishart distribution and S is positive definite with probability 1 if and only if n⩾k and Σ 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 Σ+Λ is positive definite. This is a consequence of the main result that rankX=min(n,rank(Σ+Λ)) with probability 1.