Orthogonal transformation

Inlinear algebra,anorthogonal transformationis alinear transformationT:V→Von arealinner product spaceV,that preserves theinner product.That is, for each pairu,vof elements ofV,we have

$\langle u,v\rangle =\langle Tu,Tv\rangle \,.$

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 maporthonormal basesto orthonormal bases.

Orthogonal transformations areinjective:if$Tv=0$then$0=\langle Tv,Tv\rangle =\langle v,v\rangle$,hence$v=0$,so thekernelof$T$is trivial.

Orthogonal transformations in two- or three-dimensionalEuclidean spaceare stiffrotations,reflections,or combinations of a rotation and a reflection (also known asimproper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. Thematricescorresponding to proper rotations (without reflection) have adeterminantof +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 anorthonormal basis) of an orthogonal transformation is anorthogonal matrix.Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis ofV.The columns of the matrix form another orthonormal basis ofV.

If an orthogonal transformation isinvertible(which is always the case whenVis finite-dimensional) then its inverse$T^{-1}$is another orthogonal transformation identical to the transpose of$T$:$T^{-1}=T^{\mathtt {T}}$.

Examples

Consider the inner-product space$(\mathbb {R} ^{2},\langle \cdot,\cdot \rangle )$with the standard Euclidean inner product and standard basis. Then, the matrix transformation

$T={\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$

is orthogonal. To see this, consider

${\begin{aligned}Te_{1}={\begin{bmatrix}\cos(\theta )\\\sin(\theta )\end{bmatrix}}&&Te_{2}={\begin{bmatrix}-\sin(\theta )\\\cos(\theta )\end{bmatrix}}\end{aligned}}$

Then,

${\begin{aligned}&\langle Te_{1},Te_{1}\rangle ={\begin{bmatrix}\cos(\theta )&\sin(\theta )\end{bmatrix}}\cdot {\begin{bmatrix}\cos(\theta )\\\sin(\theta )\end{bmatrix}}=\cos ^{2}(\theta )+\sin ^{2}(\theta )=1\\&\langle Te_{1},Te_{2}\rangle ={\begin{bmatrix}\cos(\theta )&\sin(\theta )\end{bmatrix}}\cdot {\begin{bmatrix}-\sin(\theta )\\\cos(\theta )\end{bmatrix}}=\sin(\theta )\cos(\theta )-\sin(\theta )\cos(\theta )=0\\&\langle Te_{2},Te_{2}\rangle ={\begin{bmatrix}-\sin(\theta )&\cos(\theta )\end{bmatrix}}\cdot {\begin{bmatrix}-\sin(\theta )\\\cos(\theta )\end{bmatrix}}=\sin ^{2}(\theta )+\cos ^{2}(\theta )=1\\\end{aligned}}$

The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on$(\mathbb {R} ^{3},\langle \cdot,\cdot \rangle )$:

${\begin{bmatrix}\cos(\theta )&-\sin(\theta )&0\\\sin(\theta )&\cos(\theta )&0\\0&0&1\end{bmatrix}},{\begin{bmatrix}\cos(\theta )&0&-\sin(\theta )\\0&1&0\\\sin(\theta )&0&\cos(\theta )\end{bmatrix}},{\begin{bmatrix}1&0&0\\0&\cos(\theta )&-\sin(\theta )\\0&\sin(\theta )&\cos(\theta )\end{bmatrix}}$

See also

• Improper rotation
• Linear transformation
• Orthogonal matrix
• Unitary transformation