On solutions of matrix equations V−AVF=BWV−AVF=BW and V−AV̄F=BW

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BWV−AVF=BW, with FF in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B)(A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.