Orthogonal transformation

In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, 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 map orthonormal bases to orthonormal bases.

Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that exchange left and right, similar to mirror images. The matrices corresponding to proper rotations (without reflection) have determinant +1. Transformations with reflection are represented by matrices with determinant −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 V. The columns of the matrix form another orthonormal basis of V.

The inverse of an orthogonal transformation is another orthogonal transformation. Its matrix representation is the transpose of the matrix representation of the original transformation.

See also

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