Commmuting exponentials in dimension at most 3

TL;DR

This paper characterizes when the exponential of a linear combination of two matrices equals the product of their exponentials, focusing on matrices of dimension at most 3 and involving conditions related to property L.

Contribution

It establishes equivalences between commutation conditions of matrix exponentials and the property L for matrices of dimension at most 3.

Findings

Conditions for exponential commutation hold outside a finite set of integers.

Equivalence between exponential commutation and property L for small matrices.

Provides a characterization of matrix exponential relations in low dimensions.

Abstract

Let A,B be two square complex matrices of dimension at most 3. We show that the following conditions are equivalent i) There exists a finite subset U included in {2,3,4,...} such that for every positive integer t that is not in U, exp(tA+B)=exp(tA)exp(B)=exp(B)exp(tA). ii) The pair (A,B) has property L of Motzkin and Taussky and exp(A+B)=exp(A)exp(B)=exp(B)exp(A).