Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations

Abstract

We investigate the solutions of the Sylvester-type matrix polynomial equation $$A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda),$$ where\ $A(\lambda),$ \ $ B(\lambda),$\ and \ $C(\lambda)$ are the polynomial matrices with elements in a ring of polynomials \ $\mathcal{F}[\lambda],$ \ $\mathcal{F}$ is a field,\ $X(\lambda)$\ and \ $Y(\lambda)$ \ are unknown polynomial matrices. Solving such a matrix equation is reduced to the solving a system of linear equations $$G \left\|\begin{array}{c}\mathbf{x} \\ \mathbf{y} \end{array} \right\|=\mathbf{c}$$ over a field $\mathcal{F}.$ In this case, the Kronecker product of matrices is applied. In terms of the ranks of matrices over a field $\mathcal{F},$ which are constructed by the coefficients of the Sylvester-type matrix polynomial equation,the necessary and sufficient conditions for the existence of solutions \ $X_0(\lambda)$\ and \ $Y_0(\lambda)$ \ 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 \ $X_0$\ and \ $Y_0,$ \ whose entries are elements in a field $\mathcal{F},$ to the Sylvester-type matrix polynomial equation.