Matrix Lie group

Remarks

Example: $SL_n(\R) = \set{A \in GL_n(\C) : A \text{ real}, \det A = 1}$ is a subgroup, and it is closed because the entries being real and $\det$ being $1$ are conditions preserved under limits, as $\det$ is continuous.

Non-example: let $\alpha$ be irrational and $G = \set{\operatorname{diag}(e^{it}, e^{i\alpha t}) : t \in \R}$. It is a subgroup of $GL_2(\C)$, but it is not closed: its closure is the set of all $\operatorname{diag}(e^{i\theta}, e^{i\varphi})$, which contains $\operatorname{diag}(1, -1) \notin G$.

The condition is closedness in $GL_n(\C)$, not in the space of all matrices: $GL_n(\C)$ itself is a matrix Lie group. Closed subgroups of $GL_n(\R)$ are matrix Lie groups too. By the closed subgroup theorem, a matrix Lie group is an embedded Lie subgroup of $GL_n(\C)$. Its Lie algebra is $\mathfrak{g} = \set{X \in M_n(\C) : e^{tX} \in G \text{ for all } t \in \R}$, with the matrix exponential and the bracket $[X, Y] = XY - YX$.