Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

TL;DR

This paper investigates factorizations of invertible matrices over rings into elementary matrices and exponentials, using algebraic and topological methods, with special focus on rings of holomorphic functions.

Contribution

It introduces new approaches based on Bass stable rank and null-homotopy to factor matrices over rings, including rings of holomorphic functions.

Findings

Characterization of matrix factorizations over rings of holomorphic functions

Conditions under which matrices are products of elementary matrices

Representation of matrices as products of exponentials

Abstract

Let $R$ be a commutative unital ring. A well-known factorization problem is whether any matrix in $\mathrm{SL}_n(R)$ is a product of elementary matrices with entries in $R$. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where $R$ is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in $\mathrm{GL}_n(R)$ as a product of exponentials.