On Hermitian nonnegative-definite solutions to matrix equations

In this note, for a system of q matrix equations of the form $$ A_i XA_i^* = B_i B_i^* ,i = 1,2,...,q, $$ we consider the problem of the existence of Hermitian nonnegative-definite solutions. 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 proposed a revised version of Young et al. 2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement.