2020/03/11 by В. М. Прокіп, V. M. Prokip, Prokip, V. M.
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #math.AC
paper · pdf · doi:10.48550/arxiv.2003.05041
arxiv created 2020/03/11 · openalex publication_date 2020/03/11 · arxiv updated 2020/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Polynomial n× n matrices A(λ) and B(λ) over a field \mathbb F are called semi-scalar equivalent if there exist a nonsingular n× n matrix P over the field \mathbb F and an invertible n× n matrix Q(λ) over the ring \mathbb F[λ] such that A(λ)=P B(λ)Q(λ). 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(λ) and B(λ) 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(λ) and B(λ) over a field.