An orthogonal transformation is a linear transformation
which preserves a symmetric inner product.
In particular, an orthogonal transformation (technically, an orthonormal transformation)
preserves lengths of vectors and angles between vectors,

(1)

In addition, an orthogonal transformation is either a rigid rotation or an improper rotation (a rotation followed
by a flip). (Flipping and then rotating can be realized by first rotating in the
reverse direction and then flipping.) Orthogonal transformations correspond to and
may be represented using orthogonal matrices.

The set of orthonormal transformations forms the orthogonal group, and an orthonormal transformation can be realized by an orthogonal
matrix.