An orthogonal transformation is a linear transformation T:V->V which preserves a symmetric inner product. In particular, an orthogonal transformation (technically, an orthonormal transformation) preserves lengths of vectors and angles between vectors, 〈v, w〉 = 〈T v, T w〉. 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.
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