# A note on trigonometric identities involving non-commuting matrices

## TL;DR

This paper introduces an algorithm for approximating trigonometric functions of sums of non-commuting matrices, useful in quantum mechanics, involving nested commutators and converging within the Zassenhaus formula domain.

## Contribution

It presents a novel iterative method to compute trigonometric functions of non-commuting matrices with convergence guarantees.

## Key findings

- Algorithm converges within the Zassenhaus domain
- Expressions involve nested commutators
- Applicable to perturbative quantum mechanics

## Abstract

An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure is shown to converge in the convergent domain of the Zassenhaus formula and can be useful in the perturbative treatment of quantum mechanical problems, where exponentials of sums of non-commuting skew-Hermitian matrices frequently appear.