Editing Orthogonal group (section)

== Of indefinite quadratic form over the reals==
{{Main|Indefinite orthogonal group}}

Over the real numbers, [[nondegenerate quadratic form]]s are classified by [[Sylvester's law of inertia]], which asserts that, on a vector space of dimension {{mvar|n}}, such a form can be written as the difference of a sum of {{mvar|p}} squares and a sum of {{mvar|q}} squares, with {{math|1=''p'' + ''q'' = ''n''}}. In other words, there is a basis on which the matrix of the quadratic form is a [[diagonal matrix]], with {{mvar|p}} entries equal to {{math|1}}, and {{mvar|q}} entries equal to {{math|−1}}. The pair {{math|(''p'', ''q'')}} called the ''inertia'', is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.

The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted {{math|O(''p'', ''q'')}}. Moreover, as a quadratic form and its opposite have the same orthogonal group, one has {{math|1=O(''p'', ''q'') = O(''q'', ''p'')}}.

The standard orthogonal group is {{math|1=O(''n'') = O(''n'', 0) = O(0, ''n'')}}. So, in the remainder of this section, it is supposed that neither {{mvar|p}} nor {{mvar|q}} is zero.

The subgroup of the matrices of determinant 1 in {{math|O(''p'', ''q'')}} is denoted {{math|SO(''p'', ''q'')}}. The group {{math|O(''p'', ''q'')}} has four connected components, depending on whether an element preserves  orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted {{math|SO{{sup|+}}(''p'', ''q'')}}.

The group {{math|O(3, 1)}} is the [[Lorentz group]] that is fundamental in [[relativity theory]]. Here the {{math|3}} corresponds to space coordinates, and {{math|1}} corresponds to the time coordinate.