Orthogonal transformation

A linear transformation $A$ of a Euclidean space preserving the lengths (or, equivalently, the scalar product) of vectors. Orthogonal transformations and only they can transfer an orthonormal basis to an orthonormal one. The equality $A^*=A^{-1}$ is also a necessary and sufficient condition of orthogonality, where $A^*$ is the conjugate and $A^{-1}$ the inverse linear transformation.

With respect to an orthonormal basis, orthogonal matrices correspond to orthogonal transformations and only to them. The eigen values of an orthogonal transformation are equal to $\pm1$, while the eigen vectors which correspond to different eigen values are orthogonal. The determinant of an orthogonal transformation is equal to $+1$ (special orthogonal transformation) or $-1$ (non-special orthogonal transformation). In the Euclidean plane, every special orthogonal transformation is a rotation, and its matrix in an appropriate orthonormal basis has the form

$$\begin{Vmatrix}\cos\phi&-\sin\phi\\\sin\phi&\hphantom{-}\cos\phi\end{Vmatrix},$$

where $\phi$ is the angle of the rotation; and every non-special orthogonal transformation is a reflection with respect to a straight line through the origin, and its matrix in an appropriate orthonormal basis has the form

$$\begin{Vmatrix}1&\hphantom{-}0\\0&-1\end{Vmatrix}.$$

In three-dimensional space, every special orthogonal transformation is a rotation around an axis, while every non-special orthogonal transformation is the product of such a rotation and a reflection in a perpendicular plane. In an arbitrary $n$-dimensional Euclidean space, orthogonal transformations also reduce to rotations and reflections (see Rotation).

The set of all orthogonal transformations in a Euclidean space is a group with respect to multiplication of transformations — the orthogonal group of the given Euclidean space. The special orthogonal transformations form a normal subgroup in this group (the special orthogonal group).

Comments

See also Orthogonal matrix and Orthogonal group, and the references therein.