# Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations

• N. S. Dzhaliuk Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine Lviv, Ukraine
• V. M. Petrychkovych Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine Lviv, Ukraine
Keywords: polynomial matrix, matrix polynomial, matrix equation, ylvester-type matrix polynomial equation, solution, Kronecker product of matrices

### 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.

### Author Biographies

N. S. Dzhaliuk, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine Lviv, Ukraine

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine

Lviv, Ukraine

V. M. Petrychkovych, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine Lviv, Ukraine

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine
Lviv, Ukraine

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Published
2024-06-19
How to Cite
Dzhaliuk, N. S., & Petrychkovych, V. M. (2024). Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations. Matematychni Studii, 61(2), 115-122. https://doi.org/10.30970/ms.61.2.115-122
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