Editing Orthogonal group (section)

=== As algebraic groups ===
The orthogonal group {{math|O(''n'')}} can be identified with the group of the matrices {{mvar|A}} such that {{math|1=''A''{{sup|T}}''A'' = ''I''}}. 
Since both members of this equation are [[symmetric matrices]], this provides {{math|''n''(''n'' + 1) / 2}} equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.

This proves that {{math|O(''n'')}} is an [[algebraic set]]. Moreover, it can be proved{{cn|date=July 2022}} that its dimension is
: <math>\frac{n(n - 1)}{2} = n^2 - \frac{n(n + 1)}{2},</math> 
which implies that {{math|O(''n'')}} is a [[complete intersection]]. This implies that all its [[irreducible component]]s have the same dimension, and that it has no [[embedded prime|embedded component]].
In fact, {{math|O(''n'')}} has two irreducible components, that are distinguished by the sign of the determinant (that is {{math|1=det(''A'') = 1}} or {{math|1=det(''A'') = −1}}). Both are [[singular point of an algebraic variety|nonsingular algebraic varieties]] of the same dimension {{math|''n''(''n'' − 1) / 2}}. The component with {{math|1=det(''A'') = 1}} is {{math|SO(''n'')}}.