Editing Rotation matrix (section)

=== Lie group ===
{{main|Special orthogonal group|Rotation group SO(3)}}
The {{math|''n'' × ''n''}} rotation matrices for each {{mvar|n}} form a [[group (mathematics)|group]], the [[special orthogonal group]], {{math|SO(''n'')}}. This [[algebraic structure]] is coupled with a [[topological structure]] inherited from <math>\operatorname{GL}_n(\R)</math> in such a way that the operations of multiplication and taking the inverse are [[analytic function]]s of the matrix entries. Thus {{math|SO(''n'')}} is for each {{mvar|n}} a [[Lie group]]. It is [[compact space|compact]] and [[connected space|connected]], but not [[simply connected]]. It is also a [[semi-simple group]], in fact a [[simple group]] with the exception SO(4).<ref>{{Harvtxt|Baker|2003}}; {{Harvtxt|Fulton|Harris|1991}}</ref> The relevance of this is that all theorems and all machinery from the theory of [[analytic manifold]]s (analytic manifolds are in particular [[smooth manifold]]s) apply and the well-developed representation theory of compact semi-simple groups is ready for use.