Semi-scalar equivalence of polynomial matrices

Abstract

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 $A(\lambda )=PB(\lambda )Q(\lambda ).$ 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.

References in corpus (1)

• Canonical matrices for linear matrix problems