On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation

Abstract

We study the low-degree solution of the Sylvester matrix equation $\left(A_1\lambda +A_0\right)X\left(\lambda \right)+Y\left(\lambda \right)\left(B_1\lambda +B_0\right)=C_0$ , where $A_1\lambda +A_0$ and $B_1\lambda +B_0$ are regular. Using the substitution of parameter variables $\lambda$ , we assume that the matrices $A_0$ and $B_0$ are invertible. Thus, we prove that if the equation is solvable, then it has a low-degree solution $\left(L\left(\lambda \right),M\left(\lambda \right)\right)$ , satisfying the degree conditions $\delta L\left(\lambda \right)<\text{Ind}\left(A_0^{-1}{A}_1\right)\text{\hspace{0.17em}and\hspace{0.17em}}\delta M\left(\lambda \right)<\text{Ind}\left(B_1{B}_0^{-1}\right)$ .