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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $GL_n(\\C)$ be the [[general linear group|0FCHE902D7]] of invertible complex $n \\times n$ matrices, with the topology of $\\C^{n^2}$.\nA *matrix Lie group* is a [[subgroup|0X1JXT47GA]] $G$ of $GL_n(\\C)$ that is a [[closed subset|0ZCK8DQYN1]] of $GL_n(\\C)$.\nThat is, if $A_k \\in G$ and $A_k \\to A$ entrywise with $A \\in GL_n(\\C)$, then $A \\in G$.\n\n## Remarks\n\nExample: $SL_n(\\R) = \\set{A \\in GL_n(\\C) : A \\text{ real}, \\det A = 1}$ 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.\n\nNon-example: let $\\alpha$ be irrational and $G = \\set{\\operatorname{diag}(e^{it}, e^{i\\alpha t}) : t \\in \\R}$.\nIt 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 $\\operatorname{diag}(1, -1) \\notin G$.\n\nThe condition is closedness in $GL_n(\\C)$, not in the space of all matrices: $GL_n(\\C)$ itself is a matrix Lie group.\nClosed subgroups of $GL_n(\\R)$ are matrix Lie groups too.\nBy the [[closed subgroup theorem|3NJGY8QSTE]], a matrix Lie group is an embedded Lie subgroup of $GL_n(\\C)$.\nIts 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|020M04FN1H]] and the bracket $[X, Y] = XY - YX$.\n",
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