On Hermitian Nonnegative-Definite Solutions to Matrix Equations

For a system of $q$ matrix equations denoted by
$$ \mathbf A_i\mathbf X\mathbf A_i^*=\mathbf B_i\mathbf B_i^*,\qquad i=1,2,\dots,q, $$
the problem of the existence of Hermitian nonnegative-definite solutions is considered in this note. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with $q=2$ to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who provided a revision of Young et al. [2]. Moreover, we give a necessary condition for the general case and then pose a conjecture, for which at least some special situations are argued.