[proofplan] The essential point is that the words “matrix Lie group” include closedness in the ambient general linear group. We therefore apply [Cartan's closed subgroup theorem](/theorems/8813) to the closed subgroup $G\le GL(n,\mathbb C)$, obtaining an embedded Lie subgroup structure on $G$. Since $GL(n,\mathbb C)$ is an open submanifold of the real [vector space](/page/Vector%20Space) $M(n,\mathbb C)$, embeddedness in $GL(n,\mathbb C)$ immediately implies embeddedness in $M(n,\mathbb C)$. Finally, the Lie subgroup structure gives smooth multiplication and inversion on $G$. [/proofplan] [step:Use the definition of a matrix Lie group to obtain closedness in $GL(n,\mathbb C)$] By hypothesis, $G\le GL(n,\mathbb C)$ is a matrix Lie group. In the convention used here, this means that $G$ is a subgroup of $GL(n,\mathbb C)$ and is closed in the topology inherited from the ambient [matrix space](/page/Matrix%20Space) $M(n,\mathbb C)$. Since $GL(n,\mathbb C)$ carries the [subspace topology](/page/Subspace%20Topology) from $M(n,\mathbb C)$, the same condition says exactly that $G$ is a closed subgroup of the Lie group $GL(n,\mathbb C)$. [guided] The statement is not true for an arbitrary subgroup of $GL(n,\mathbb C)$: dense proper subgroups can fail to be embedded submanifolds. Thus the first mathematical input is the definition of “matrix Lie group.” In this theorem, a matrix Lie group $G\le GL(n,\mathbb C)$ means a subgroup $G$ that is closed in the topology inherited from $M(n,\mathbb C)$. Because $GL(n,\mathbb C)$ itself is given the subspace topology from $M(n,\mathbb C)$, closedness of $G$ as a subset of $GL(n,\mathbb C)$ is exactly the closedness needed to apply the closed subgroup theorem inside the Lie group $GL(n,\mathbb C)$. The subgroup condition supplies closure under products and inverses, while the topological closedness supplies the analytic hypothesis of Cartan's theorem. [/guided] [/step] [step:Apply Cartan's closed subgroup theorem inside $GL(n,\mathbb C)$] By [citetheorem:8772], $GL(n,\mathbb C)$ is an open subset of $M(n,\mathbb C)$ and hence is a smooth manifold of real dimension $2n^2$. By the previous step, $G$ is a closed subgroup of this Lie group. Applying [citetheorem:8813] to the Lie group $GL(n,\mathbb C)$ and its closed subgroup $G$, we obtain a smooth structure on $G$ for which $G$ is an embedded Lie subgroup of $GL(n,\mathbb C)$. In particular, $G$ is an embedded submanifold of $GL(n,\mathbb C)$. [/step] [step:Pass embeddedness from $GL(n,\mathbb C)$ to $M(n,\mathbb C)$] Let \begin{align*} j:GL(n,\mathbb C)\to M(n,\mathbb C),\quad A\mapsto A \end{align*} denote the open inclusion. By [citetheorem:8772], $GL(n,\mathbb C)$ is open in $M(n,\mathbb C)$, so $j$ is an open embedding of smooth manifolds. Since the inclusion \begin{align*} \iota_G:G\to GL(n,\mathbb C),\quad A\mapsto A \end{align*} is an embedded-submanifold inclusion by the previous step, the composite \begin{align*} j\circ \iota_G:G\to M(n,\mathbb C),\quad A\mapsto A \end{align*} is also an embedded-submanifold inclusion. Therefore $G$ is an embedded submanifold of $M(n,\mathbb C)$. [/step] [step:Restrict the ambient group operations to $G$] By [citetheorem:8773], the multiplication map \begin{align*} m_{GL}:GL(n,\mathbb C)\times GL(n,\mathbb C)\to GL(n,\mathbb C),\quad (A,B)\mapsto AB \end{align*} and the inversion map \begin{align*} i_{GL}:GL(n,\mathbb C)\to GL(n,\mathbb C),\quad A\mapsto A^{-1} \end{align*} are smooth. Since $G$ is a subgroup, $AB\in G$ for all $A,B\in G$, and $A^{-1}\in G$ for all $A\in G$. Thus the restrictions \begin{align*} m_G:G\times G\to G,\quad (A,B)\mapsto AB \end{align*} and \begin{align*} i_G:G\to G,\quad A\mapsto A^{-1} \end{align*} are precisely the group operations on $G$. Moreover, the smooth structure on $G$ obtained from [citetheorem:8813] makes $G$ an embedded Lie subgroup of $GL(n,\mathbb C)$, so these restricted operations are smooth as maps into $G$. Hence $G$ is a smooth embedded submanifold of both $GL(n,\mathbb C)$ and $M(n,\mathbb C)$, and its multiplication and inversion maps are smooth. [/step]