The general common Hermitian nonnegative-definite solution to the matrix equations AXA∗=BB∗ and CXC∗=DD∗ with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA∗=BB∗ and CXC∗=DD∗ to have a common Hermitian nonnegative-definite solution and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).