# Definition A proper definition requires the machinery of [[Differentiable Manifold|differentiable manifolds]]. But we can formulate a simpler definition. A *matrix Lie group* is a [[subgroup]] $G$ of the [[general linear group]] over complex numbers $GL(n,\mathbb{C})$ that is closed in the following sense: for any sequence of matrices $A_n \in G$ that converges to a limit matrix $A$, either $A \in G$ or $A \notin GL(n, \mathbb{C})$. That is, the limit of matrices must either be in $G$ or [[Non-singular Matrix|non-invertible]]. # Example: $O(n)$ Consider the [[Orthogonal Group]] $O(n)$. Consider an [[continuous map]] from $GL(n, \mathbb{R})$ to itself, defined by $f: A \mapsto A^T A$. Now consider a sequence $R_i$ in $O(n)$ that converges to some limit matrix $R$. Then: $ \begin{align} f(R) &= f\left(\lim_{i \rightarrow \infty} R_i \right)\\ &= \lim_{i\rightarrow \infty} f(R_i)\\ &= \lim_{i\rightarrow \infty} R_i^T R_i\\ &= \lim_{i\rightarrow \infty} I\\ &= I, \end{align} $ where the second line follows from continuity. Thus, $f(R) = R^T R = I$ and $R \in O(n)$