Closed-form solutions to the nonhomogeneous Yakubovich-conjugate matrix equation (Q833132)

The matrix equation \(X-A\overline{X}F=BY+R\), where \(X\) and \(Y\) are matrices to be determined, is studied. By using the Smith normal form, a general parametric solution in terms of the original matrices is obtained. The solution is represented in a finite series form and the matrices \(F\) and \(B\) appear in it, fact that permits a flexibility in their choices. Also an equivalent solution in terms of controllability matrices and observability matrices is presented.