A New Algorithm for the Inverse of Matrices with Noncommuting Entries

TL;DR

This paper introduces a new algorithm for inverting matrices with noncommuting entries, inspired by deformation quantization, which simplifies physical applications and incorporates noncommutative corrections.

Contribution

It presents a novel inversion algorithm that is more suitable for physical contexts and aligns with quasi-determinant methods, adding noncommutative corrections.

Findings

The new algorithm emulates the commutative case.

It provides corrections from noncommutativity.

It is equivalent to the quasi-determinant construction.

Abstract

By using the quasi-determinant the construction of Gel'fand et al. leads to the inverse of a matrix with noncommuting entries. In this work we offer a new method that is more suitable for physical purposes and motivated by deformation quantization, where our constructed algorithm emulates the commutative case and in addition gives corrections coming from the noncommutativity of the entries. Furthermore, we provide an equivalence of the introduced algorithm and the construction via quasi-determinants.