On commuting matrices and exponentials

TL;DR

This paper investigates conditions under which matrix exponentials imply commutativity and certain algebraic properties, revealing new links between exponential identities and matrix commutation.

Contribution

It establishes that specific exponential identities imply matrix commutativity or property L, extending understanding of exponential maps in matrix algebra.

Findings

If exp(A)^k exp(B)^l=exp(kA+lB) for all integers k,l, then AB=BA.

If exp(A)^k exp(B)=exp(B)exp(A)^k=exp(kA+B) for all positive integers k, then (A,B) has property L.

Subgroups and subsemigroups with exponential homomorphisms have commuting matrices or property L.

Abstract

Let A and B be matrices of M_n(C). We show that if exp(A)^k exp(B)^l=exp(kA+lB) for all integers k and l, then AB=BA. We also show that if exp(A)^k exp(B)=exp(B)exp(A)^k=exp(kA+B)$ for every positive integer k, then the pair (A,B) has property L of Motzkin and Taussky. As a consequence, if G is a subgroup of (M_n(C),+) and M -> exp(M) is a homomorphism from G to (GL_n(C),x), then G consists of commuting matrices. If S is a subsemigroup of (M_n(C),+) and M -> exp(M) is a homomorphism from S to (GL_n(C),x), then the linear subspace Span(S) of M_n(C) has property L of Motzkin and Taussky.