On solutions of the matrix equations XF − AX = C and XF-AX¯=C

With the help of the concept of Kronecker map, an explicit solution for the matrix equation XF − AX = C   is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation XF-AX¯=C is also studied. An explicit solution for this equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.