A note on semiscalar equivalence of polynomial matrices

Polynomial matrices $A(\lambda)$ and $B(\lambda)$ of size $n\times n$ over a field $\mathbb {F}$ are semiscalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over $\mathbb F$ and an invertible $n\times n$ matrix $Q(\lambda)$ over $\mathbb F[\lambda]$ such that $A(\lambda)=PB(\lambda)Q(\lambda)$. The aim of this article is to present necessary and sufficient conditions for the semiscalar equivalence of nonsingular matrices $A(\lambda)$ and $ B(\lambda) $ over a field ${\mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations.