On commuting matrices and exponentials

Abstract:

Let $A$ and $B$ be matrices of $\operatorname {M}_n(\mathbb C)$. 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 $(\operatorname {M}_n(\mathbb C),+)$ and $M\mapsto \exp (M)$ is a homomorphism from $G$ to $(\operatorname {GL}_n(\mathbb C),\times )$, then $G$ consists of commuting matrices. If $S$ is a subsemigroup of $(\operatorname {M}_n(\mathbb {C}),+)$ and $M \mapsto \exp (M)$ is a homomorphism from $S$ to $(\operatorname {GL}_n(\mathbb {C}),\times )$, then the linear subspace $\operatorname {Span}(S)$ of $\operatorname {M}_n(\mathbb {C})$ has property L of Motzkin and Taussky.