A lower bound for the product of eigenvalues of solutions to matrix equations

The following matrix equations: A T X B + B T X T A = C and A T X B + B T X A = C , are encountered in many systems and control applications, and these matrix equations contain several linear matrix equations as special cases. In the present work, we introduce the inequalities for the determinant of the solutions of these matrix equations, separately. Then using these inequalities, we introduce a lower bound for the product of eigenvalues of the solutions to the matrix equations.