Polynomial Solution of Sylvester Matrix Equation

where X ∈ M(n,m)(K), play a central role in many areas of applied mathematics and in particular in systems and control theory. It is well known that if K is an algebraically closed field then the matrix equation (1) possesses a unique solution if and only if the matrices A and B have no common eigenvalues (see [[3]] and [11]). In this work we give a brief survey of methods used to solve the (SME) and we study the solvability of the Sylvester matrix equation (1) over an arbitrary field K. By using some new results on Sylvester operator (see [[6]]) we show that the Sylvester matrix equation admits a unique solution if and only if the characteristic polynomial of A and the characteristic polynomial of B have no common prime factors. In this case, a wonderful polynomial solution of the Sylvester matrix equation is described.