Editing Orthogonal matrix (section)

==Spin and pin==
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not [[Connected space|connected]] to each other, even the +1 component, {{math|SO(''n'')}}, is not [[simply connected space|simply connected]] (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a [[covering map|covering group]] of SO(''n''), the [[spinor group|spin group]], {{math|Spin(''n'')}}. Likewise, {{math|O(''n'')}} has covering groups, the [[pin group]]s, Pin(''n''). For {{math|''n'' > 2}}, {{math|Spin(''n'')}} is simply connected and thus the universal covering group for {{math|SO(''n'')}}. By far the most famous example of a spin group is {{math|Spin(3)}}, which is nothing but {{math|SU(2)}}, or the group of unit [[quaternion]]s.

The Pin and Spin groups are found within [[Clifford algebra]]s, which themselves can be built from orthogonal matrices.