# Factorization of symplectic matrices into elementary factors

## TL;DR

This paper proves that symplectic matrices over certain rings can be decomposed into elementary factors, extending to null-homotopic matrices in Banach algebras and continuous functions.

## Contribution

It establishes a factorization result for symplectic matrices over rings with Bass stable rank one, including Banach algebras and continuous functions.

## Key findings

- Symplectic matrices over rings with Bass stable rank one can be factored into elementary matrices.
- The factorization extends to null-homotopic symplectic matrices in Banach algebras.
- Applicable to matrices with entries in continuous functions on finite-dimensional spaces.

## Abstract

We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of complex valued continuous functions on a finite dimensional normal topological space.