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 of elements of V, we have
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. Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like mirrors do. The matrices corresponding to proper rotations 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 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.

Examples

Consider the inner-product space with the standard euclidean inner product and standard basis. Then, the matrix transformation
is orthogonal. To see this, consider
Then,
The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on :