Criteria for commutative factorization of a class of algebraic matrices

TL;DR

This paper investigates conditions under which certain algebraic matrices can be factorized commutatively, introducing analytical tools and criteria to determine when such factorizations are possible.

Contribution

It provides necessary conditions and an analytical procedure for identifying commutatively factorizable matrices within a specific class.

Findings

Derived analytical continuation formulas for matrix solutions

Established necessary conditions for commutative factorization

Provided a practical method to test matrix factorizability

Abstract

The problem of matrix factorization motivated by diffraction or elasticity is studied. A powerful tool for analyzing its solutions is introduced, namely analytical continuation formulae are derived. Necessary condition for commutative factorization is found for a class of "balanced" matrices. Together with Moiseyev's method and Hurd's idea, this gives a description of the class of commutatively solvable matrices. As a result, a simple analytical procedure is described, providing an answer, whether a given matrix is commutatively factorizable or not.