Editing Orthogonal group (section)

=== Reflections ===
[[Reflection (mathematics)|Reflection]]s are the elements of {{math|O(''n'')}} whose canonical form is 
: <math>\begin{bmatrix}-1&0\\0&I\end{bmatrix},</math>
where {{mvar|I}} is the {{math|(''n'' − 1) × (''n'' − 1)}} identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its [[mirror image]] with respect to a [[hyperplane]].

In dimension two, [[Rotations and reflections in two dimensions|every rotation can be decomposed into a product of two reflections]]. More precisely, a rotation of angle {{math|''θ''}} is the product of two reflections whose axes form an angle of {{math|''θ'' / 2}}.

A product of up to {{math|''n''}} elementary reflections always suffices to generate any element of {{math|O(''n'')}}. This results immediately from the above canonical form and the case of dimension two.

The [[Cartan–Dieudonné theorem]] is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.

The [[reflection through the origin]] (the map {{math|''v'' ↦ −''v''}}) is an example of an element of {{math|O(''n'')}} that is not a product of fewer than {{math|''n''}} reflections.