New Remarks on the Factorization and Equivalence Problems for a Class of Multivariate Polynomial Matrices

TL;DR

This paper introduces new criteria and an algorithm for factorizing and determining the equivalence of multivariate polynomial matrices, with proofs of uniqueness and practical implementation in Maple.

Contribution

It provides the first necessary and sufficient conditions for matrix equivalence and a constructive factorization algorithm for a specific class of multivariate polynomial matrices.

Findings

New criteria for matrix factorization and equivalence

A factorization algorithm with proven uniqueness

Successful implementation and illustrative examples in Maple

Abstract

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain a necessary and sufficient condition for the equivalence of a square polynomial matrix and a diagonal matrix. Based on the constructive proof of the new criteria, we give a factorization algorithm and prove the uniqueness of the factorization. We implement the algorithm on Maple, and two illustrative examples are given to show the effectiveness of the algorithm.