Let $n\in\mathbb N$, let $G \le GL(n,\mathbb C)$ be a closed matrix Lie subgroup regarded as a real embedded Lie subgroup of $GL(n,\mathbb C)$, and let $\mathfrak g\subset M(n,\mathbb C)$ be its real [Lie algebra](/page/Lie%20Algebra), equivalently $\mathfrak g=\{X\in M(n,\mathbb C):\exp(tX)\in G\text{ for all }t\in\mathbb R\}$. Then there exist an open neighbourhood $U \subset \mathfrak g$ of $0$ and a neighbourhood $V \subset G$ of $I$ which is open in the [subspace topology](/page/Subspace%20Topology) on $G$ such that the restricted exponential map $\exp|_U:U\to V$ is bijective. Moreover, the inverse map $\log_G:V\to U$ is smooth as a map from the smooth submanifold $V\subset G$ to the finite-dimensional real [vector space](/page/Vector%20Space) $\mathfrak g$. Finally, if $E:\mathfrak g\to M(n,\mathbb C)$ denotes the ambient matrix exponential map $E(X)=\exp X$, then $dE_0:\mathfrak g\to M(n,\mathbb C)$ is the inclusion $A\mapsto A$; after identifying $\mathfrak g$ with its image in $M(n,\mathbb C)$, this differential is the identity map on $\mathfrak g$.