Properties of perturbative solutions of unilateral matrix equations

TL;DR

This paper introduces a unified approach to solving unilateral matrix equations using the generalized Bezout theorem, analyzing perturbative solutions and their properties in a noncommutative algebra context.

Contribution

It provides a novel, unified method for analyzing perturbative solutions of unilateral matrix equations via the generalized Bezout theorem.

Findings

Describes the relation between two known unilateral matrix equations.

Extends the analysis to coefficients and unknowns in noncommutative algebra.

Offers insights into the properties of perturbative solutions.

Abstract

A left-unilateral matrix equation is an algebraic equation of the form $$ a_0+a_1 x+a_2 x^2+... +a_n x^n=0 $$ where the coefficients $a_r$ and the unknown $x$ are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra.