Orthogonal Transformation

a linear transformation of a Euclidean vector space that preserves the lengths or (equivalently) the scalar products of vectors. In an orthonormal basis an orthogonal transformation corresponds to an orthogonal matrix. Orthogonal transformations form a group, the group of rotations of the given Euclidean space about the origin. In three-dimensional space an orthogonal transformation reduces to a rotation through a certain angle about some axis passing through the origin O, if the determinant of the corresponding orthogonal matrix is +1. If the determinant is —1, then the rotation must be supplemented by a reflection in the plane passing through O perpendicular to the axis of rotation. In two-dimensional space, that is, in a plane, an orthogonal transformation defines a rotation through a certain angle about O or a reflection relative to some line passing through O. Orthogonal transformations are used to reduce a quadratic form to the principal axes.