Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications

Necessary and sufficient conditions are given for the existence of a nonnegative definite solution, a Re-nonnegative definite solution, a positive definite solution and a Re-positive definite solution to the system of matrix equations AXA*=CandBXB*=D,respectively. The expressions for these special solutions are given when the consistent conditions are satisfied. Based on the new results, the characterization of the covariance matrix such that a pair of multivariate quadratic forms are distributed as independent noncentral Wishart random matrices is derived. Many results existing in the literature are extended.