Editing Orthogonal group (section)

== In Euclidean geometry ==

The orthogonal {{math|O(''n'')}} is the subgroup of the [[general linear group]] {{math|GL(''n'', '''R''')}}, consisting of all [[endomorphisms]] that preserve the [[Euclidean norm]]; that is, endomorphisms {{math|''g''}} such that <math>\|g(x)\| = \|x\|.</math>

Let {{math|E(''n'')}} be the group of the [[Euclidean isometry|Euclidean isometries]] of a [[Euclidean space]] {{math|''S''}} of dimension {{math|''n''}}. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are [[isomorphic]]. The [[stabilizer subgroup]] of a point {{math|''x'' ∈ ''S''}} is the subgroup of the elements {{math|''g'' ∈ E(''n'')}} such that {{math|1=''g''(''x'') = ''x''}}. This stabilizer is (or, more exactly, is isomorphic to) {{math|O(''n'')}}, since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural [[group homomorphism]] {{math|''p''}} from {{math|E(''n'')}} to {{math|O(''n'')}}, which is defined by
: <math>p(g)(y-x) = g(y)-g(x),</math>
where, as usual, the subtraction of two points denotes the [[translation (geometry)|translation]] vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by {{math|''g''}} (for details, see ''{{slink|Affine space|Subtraction and Weyl's axioms}}'').

The [[kernel (algebra)|kernel]] of {{math|''p''}} is the vector space of the translations. So, the translations form a [[normal subgroup]] of {{math|E(''n'')}}, the stabilizers of two points are [[conjugate subgroup|conjugate]] under the action of the translations, and all stabilizers are isomorphic to {{math|O(''n'')}}.

Moreover, the Euclidean group is a [[semidirect product]] of {{math|O(''n'')}} and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of {{math|O(''n'')}}.

=== Special orthogonal group ===
By choosing an [[orthonormal basis]] of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of [[orthogonal matrices]], which are the matrices such that 
: <math> Q Q^\mathsf{T} = I. </math>

It follows from this equation that the square of the [[determinant]] of {{mvar|Q}} equals {{math|1}}, and thus the determinant of {{mvar|Q}} is either {{math|1}} or {{math|−1}}. The orthogonal matrices with determinant {{math|1}} form a subgroup called the ''special orthogonal group'', denoted {{math|SO(''n'')}}, consisting of all [[Euclidean group#Direct and indirect isometries|direct isometries]] of {{math|O(''n'')}}, which are those that preserve the [[orientation (vector space)|orientation]] of the space.

{{math|SO(''n'')}} is a normal subgroup of {{math|O(''n'')}}, as being the [[kernel (algebra)|kernel]] of the determinant, which is a group homomorphism whose image is the multiplicative group {{math|{{mset|−1, +1}}}}. This implies that the orthogonal group is an internal [[semidirect product]] of {{math|SO(''n'')}} and any subgroup  formed with the identity and a [[reflection (geometry)|reflection]].

The group with two elements {{math|{{mset|±''I''}}}} (where {{mvar|I}} is the identity matrix) is a [[normal subgroup]] and even a [[characteristic subgroup]] of {{math|O(''n'')}}, and, if {{math|''n''}} is even, also of {{math|SO(''n'')}}. If {{math|''n''}} is odd, {{math|O(''n'')}} is the internal [[direct product of groups|direct product]] of {{math|SO(''n'')}} and {{math|{{mset|±''I''}}}}.

The group {{math|SO(2)}} is [[abelian group|abelian]] (whereas {{math|SO(''n'')}} is not abelian when {{math|''n'' > 2}}). Its finite subgroups are the [[cyclic group]] {{math|''C''<sub>''k''</sub>}} of [[rotational symmetry|{{math|''k''}}-fold rotations]], for every positive integer {{mvar|k}}. All these groups are  normal subgroups of {{math|O(2)}} and {{math|SO(2)}}.

=== Canonical form ===
For any element of {{math|O(''n'')}} there is an orthogonal basis, where its matrix has the form
: <math>\begin{bmatrix}
  \begin{matrix}
    R_1 &        & \\
        & \ddots & \\
        &        & R_k
  \end{matrix} & 0 \\
  0 & \begin{matrix}
    \pm 1 &        & \\
          & \ddots & \\
          &        & \pm 1
  \end{matrix}\\
\end{bmatrix},</math>
where there may be any number, including zero, of ±1's; and where the matrices {{math|''R''<sub>1</sub>, ..., ''R''<sub>''k''</sub>}} are 2-by-2 rotation matrices, that is matrices of the form 
: <math>\begin{bmatrix}a&-b\\b&a\end{bmatrix},</math>
with {{math|1=''a''{{sup|2}} + ''b''{{sup|2}} = 1}}.

This results from the [[spectral theorem]] by regrouping [[eigenvalues]] that are [[complex conjugate]], and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to {{math|1}}.

The element belongs to {{math|SO(''n'')}} if and only if there are an even number of {{math|−1}} on the diagonal. A pair of eigenvalues {{math|−1}} can be identified with a rotation by {{math|π}} and a pair of eigenvalues {{math|+1}} can be identified with a rotation by {{math|0}}.

The special case of {{math|1=''n'' = 3}} is known as [[Euler's rotation theorem]], which asserts that every (non-identity) element of {{math|SO(3)}} is a [[rotation]] about a unique axis–angle pair.

=== 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.

=== Symmetry group of spheres ===
The orthogonal group {{math|O(''n'')}} is the [[symmetry group]] of the [[n-sphere|{{math|(''n'' − 1)}}-sphere]] (for {{math|1=''n'' = 3}}, this is just the [[sphere]]) and all objects with spherical symmetry, if the origin is chosen at the center.

The [[symmetry group]] of a [[circle]] is {{math|O(2)}}.<!--
[[Dihedral group|Dih]]('''S'''<sup>1</sup>), where '''S'''<sup>1</sup> denotes the multiplicative group of [[complex number]]s of [[absolute value]]&nbsp;1.
--> The orientation-preserving subgroup {{math|SO(2)}}  is isomorphic (as a ''real'' Lie group) to the [[circle group]], also known as {{math|[[unitary group|U]](1)}}, the multiplicative group of the [[complex number]]s of absolute value equal to one. This isomorphism sends the complex number {{math|1=exp(''φ'' ''i'') = cos(''φ'') + ''i'' sin(''φ'')}} of [[absolute value]]&nbsp;{{math|1}} to the special orthogonal matrix
: <math>\begin{bmatrix}
  \cos(\varphi) & -\sin(\varphi) \\
  \sin(\varphi) &  \cos(\varphi)
\end{bmatrix}.</math>

In higher dimension, {{math|O(''n'')}} has a more complicated structure (in particular, it is no longer commutative). The [[topological]] structures of the {{mvar|n}}-sphere and {{math|O(''n'')}} are strongly correlated, and this correlation is widely used for studying both [[topological space]]s.