KRONECKER PRODUCT OF MATRICES AND SOLUTIONS OF SYLVESTER-TYPE 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... 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 Sylvestertype 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. [ABSTRACT FROM AUTHOR]