An invariance property of common statistical tests

Let A be a symmetric matrix and B be a nonnegative definite (nnd) matrix. We obtain a characterization of the class of nnd solutions Σ for the matrix equation AΣA = B. We then use the characterization to obtain all possible covariance structures under which the distributions of many common test statistics remain invariant, that is, the distributions remain the same except for a scale factor. Applications include a complete characterization of covariance structures such that the chi-squaredness and independence of quadratic forms in ANOVA problems is preserved. The basic matrix theoretic theorem itself is useful in other characterizing problems in linear algebra.