Editing Lie group (section)

=== Matrix Lie groups ===

Let <math>\operatorname{GL}(n, \mathbb{C})</math> denote the group of <math>n\times n</math> invertible matrices with entries in {{tmath|1= \mathbb{C} }}. Any [[Closed subgroup theorem|closed subgroup]] of <math>\operatorname{GL}(n, \mathbb{C})</math> is a Lie group;<ref>{{harvnb|Hall|2015}} Corollary 3.45</ref> Lie groups of this sort are called '''matrix Lie groups.''' Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall,{{sfn|ps=|Hall|2015}} Rossmann,<ref>{{harvnb|Rossmann|2001}}</ref> and Stillwell.<ref>{{harvnb|Stillwell|2008}}</ref> 
Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.
* The [[special linear group]]s over <math>\mathbb{R}</math> and {{tmath|1= \mathbb{C} }}, <math>\operatorname{SL}(n, \mathbb{R})</math> and {{tmath|1= \operatorname{SL}(n, \mathbb{C}) }}, consisting of <math>n\times n</math> matrices with determinant one and entries in <math>\mathbb{R}</math> or <math>\mathbb{C}</math>
* The [[unitary group]]s and special unitary groups, <math>\operatorname{U}(n,\mathbb{C})</math> and {{tmath|1= \operatorname{SU}(n,\mathbb{C}) }}, consisting of <math>n\times n</math> complex matrices satisfying <math>U^*=U^{-1}</math> (and also <math>\det(U)=1</math> in the case of <math>\operatorname{SU}(n)</math>)
* The [[orthogonal group]]s and special orthogonal groups, <math>\operatorname{O}(n,\mathbb{R})</math> and {{tmath|1= \operatorname{SO}(n,\mathbb{R}) }}, consisting of <math>n\times n</math> real matrices satisfying <math>R^\mathrm{T}=R^{-1}</math> (and also <math>\det(R)=1</math> in the case of <math>\operatorname{SO}(n,\mathbb{R})</math>)
All of the preceding examples fall under the heading of the [[classical group]]s.