Nonnegative definite solutions to matrix equations with applications to multivariate test statistics

LetX ∼Np, n (M, Σ), where the meanM is a matrix of orderp × n and the covariance matrixΣ is a nnd of orderp n. In this paper we first obtain a version of Cochran’s theorem. Basically, the theorem reduces the problem of verifying Wishartness and independence of matrix quadratic forms inX to solving matrix equations inΣ. Next, we obtain characterizations of the class of nnd solutionsΣ to those matrix equations. As an application we give a simple description of the class of nnd matrices such that the distributions of common multivariate test statistics are invariant except for a scale factor.