Let $n,m\in\mathbb N$. Let $G \le GL(n,\mathbb C)$ and $H \le GL(m,\mathbb C)$ be matrix Lie groups with their smooth embedded-submanifold structures, identity elements $e_G,e_H$, and real Lie algebras $\mathfrak g=T_{e_G}G\subset M(n,\mathbb C)$ and $\mathfrak h=T_{e_H}H\subset M(m,\mathbb C)$ under the standard tangent-space identifications for matrix Lie groups. Let $\exp_G:\mathfrak g\to G$ and $\exp_H:\mathfrak h\to H$ denote the restrictions of the matrix exponential to the corresponding Lie algebras. If $\varphi:G\to H$ is a continuous [group homomorphism](/page/Group%20Homomorphism), then $\varphi$ is smooth as a map of smooth manifolds.