On solutions of generalized Sylvester equation in polynomial matrices

This paper deals with the generalized Sylvester equation in polynomial matrices A(λ)X(λ)+Y(λ)B(λ)=C(λ), where A(λ) and B(λ) are monic. If the equation has solutions, then it has a solution satisfying a natural degree constraint condition. It is shown that the generalized Sylvester equation in polynomial matrices can be reduced to the linear matrix equation ARnY+∑i=0n−1ARiYBi=C, where AR is the second block-companion matrix of A(λ).