Independence distribution preserving joint covariance structures for the multivariate two-group case

We characterize the general nonnegative-definite and positive-definite joint observation covariance structures for the two-group case such that the two sample mean vectors are independent of the two corresponding sample covariance matrices. Also, the sample covariance matrices are distributed as independent noncentral or central Wishart random matrices. We derive and utilize a representation of the general common non-negative-definite solution to a particular system of matrix equations with idempotent coefficient matrices.