Editing Orthogonal transformation

{{Short description|Linear algebra operation}}
In [[linear algebra]], an '''orthogonal transformation''' is a [[linear transformation]] ''T''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' on a [[real number|real]] [[inner product space]] ''V'', that preserves the [[Inner product space|inner product]]. That is, for each pair {{nowrap|1=''u'', ''v''}} of elements of&nbsp;''V'', we have<ref>{{cite web|last=Rowland|first=Todd|title=Orthogonal Transformation |url=http://mathworld.wolfram.com/OrthogonalTransformation.html|publisher=MathWorld|access-date=4 May 2012}}</ref>

: <math>\langle u,v \rangle = \langle Tu,Tv \rangle \, .</math>

Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map [[orthonormal basis|orthonormal bases]] to orthonormal bases.

Orthogonal transformations are [[injective]]: if <math>Tv = 0</math> then <math>0 = \langle Tv,Tv \rangle = \langle v,v \rangle</math>, hence <math>v = 0</math>, so the [[Kernel (linear algebra)|kernel]] of <math>T</math> is trivial.

Orthogonal transformations in two- or three-[[dimension (vector space)|dimensional]] [[Euclidean space]] are stiff [[Rotation (mathematics)|rotations]], [[Reflection (mathematics)|reflections]], or combinations of a rotation and a reflection (also known as [[improper rotation]]s). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The [[matrix (mathematics)|matrices]] corresponding to proper rotations (without reflection) have a [[determinant]] of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions.

In finite-dimensional spaces, the matrix representation (with respect to an [[orthonormal basis]]) of an orthogonal transformation is an [[orthogonal matrix]]. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of&nbsp;''V''. The columns of the matrix form another orthonormal basis of&nbsp;''V''.

If an orthogonal transformation is [[invertible function|invertible]] (which is always the case when ''V'' is finite-dimensional) then its inverse <math>T^{-1}</math> is another orthogonal transformation identical to the transpose of <math>T</math>: <math>T^{-1} = T^{\mathtt{T}}</math>.

==Examples==
Consider the inner-product space <math>(\mathbb{R}^2,\langle\cdot,\cdot\rangle)</math> with the standard Euclidean inner product and standard basis. Then, the matrix transformation
:<math>
T = \begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix} : \mathbb{R}^2 \to \mathbb{R}^2
</math>
is orthogonal. To see this, consider
:<math>
\begin{align}
Te_1 = \begin{bmatrix}\cos(\theta) \\ \sin(\theta)\end{bmatrix} && Te_2 = \begin{bmatrix}-\sin(\theta) \\ \cos(\theta)\end{bmatrix}
\end{align}
</math>
Then,
:<math>
\begin{align}
&\langle Te_1,Te_1\rangle = \begin{bmatrix} \cos(\theta) & \sin(\theta) \end{bmatrix} \cdot \begin{bmatrix} \cos(\theta) \\ \sin(\theta) \end{bmatrix} = \cos^2(\theta) + \sin^2(\theta) = 1\\
&\langle Te_1,Te_2\rangle = \begin{bmatrix} \cos(\theta) & \sin(\theta) \end{bmatrix} \cdot \begin{bmatrix} -\sin(\theta) \\ \cos(\theta) \end{bmatrix} = \sin(\theta)\cos(\theta) - \sin(\theta)\cos(\theta) = 0\\
&\langle Te_2,Te_2\rangle = \begin{bmatrix} -\sin(\theta) & \cos(\theta)  \end{bmatrix} \cdot \begin{bmatrix} -\sin(\theta) \\ \cos(\theta) \end{bmatrix} = \sin^2(\theta) + \cos^2(\theta) = 1\\
\end{align}
</math>

The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on <math>(\mathbb{R}^3,\langle\cdot,\cdot\rangle)</math>:
:<math>
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) & 0 \\
\sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 1
\end{bmatrix},
\begin{bmatrix}
\cos(\theta) & 0 & -\sin(\theta) \\
0 & 1 & 0 \\
\sin(\theta) & 0 & \cos(\theta)
\end{bmatrix},
\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos(\theta) & -\sin(\theta) \\
0 & \sin(\theta) & \cos(\theta) 
\end{bmatrix}
</math>

==See also==
*[[Geometric transformation]]
*[[Improper rotation]]
*[[Linear transformation]]
*[[Orthogonal matrix]]
*[[Rigid transformation]]
*[[Unitary transformation]]

== References ==
{{Reflist}}

[[Category:Linear algebra]]