Some remarks on unilateral matrix equations

TL;DR

This paper reviews perturbative solutions of specific unilateral matrix equations, which are algebraic equations involving matrices or algebra elements with coefficients on the left, relevant in generalized Born-Infeld theories.

Contribution

It provides a unified approach to solving certain unilateral matrix equations using the generalized Bezout theorem, extending previous results.

Findings

Analysis of two specific unilateral matrix equations

Derivation of their perturbative solutions

Establishment of relations between the equations

Abstract

We briefly review the results of our paper hep-th/0009013: we study certain perturbative solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.