Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations

We investigate the solutions of the Sylvester-type matrix polynomial equation A(λ)X(λ)+Y(λ)B(λ)=C(λ), where\ A(λ), \ B(λ),\ and \ C(λ) are the polynomial matrices with elements in a ring of polynomials \ F[λ], \ F is a field,\ X(λ)\ and \ Y(λ) \ are unknown polynomial matrices. Solving such a matrix equation is reduced to the solving a system of linear equations G \|arraycx \\ y array \|=c over a field F. In this case, the Kronecker product of matrices is applied. In terms of the ranks of matrices over a field F, which are constructed by the coefficients of the Sylvester-type matrix polynomial equation,the necessary and sufficient conditions for the existence of solutions \ X₀(λ)\ and \ Y₀(λ) \ of given degrees to the Sylvester-type matrix polynomial equation are established. The solutions of this matrix polynomial equation are constructed from the solutions of the linear equations system.As a consequence of the obtained results, we give the necessary and sufficient conditions for the existence of the scalar solutions \ X₀\ and \ Y₀, \ whose entries are elements in a field F, to the Sylvester-type matrix polynomial equation.