Semi-scalar equivalence of polynomial matrices

Polynomial $n\times n$ matrices $A(\lambda)$ and $B(\lambda)$ over a field $\mathbb F$ are called semi-scalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over the field $\mathbb F$ and an invertible $n\times n$ matrix $Q(\lambda)$ over the ring ${\mathbb F}[\lambda]$ such that $A(\lambda)=P B(\lambda)Q(\lambda).$ The semi-scalar equivalence of matrices over a field ${\mathbb F}$ contain the problem of similarity between two families of matrices. Therefore, these equivalences of matrices can be considered a difficult problem in linear algebra. The aim of the present paper is to present the necessary and sufficient conditions of semi-scalar 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. We also establish similarity of monic polynomial matrices $A(\lambda)$ and $B(\lambda)$ over a field.