Rotation Matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix

rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system.

Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In two-dimensional space, a rotation can be simply described by an angle θ of rotation, but it can also be represented by the four entries of a rotation matrix with two rows and two columns. In three-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem), and hence it can be simply described by an angle and a vector with three entries. However, it can also be represented by the nine entries of a rotation matrix with three rows and three columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement, which can be represented by a matrix, but not associated single axis or angle.

Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1:

.

The set of all such matrices of size n forms a (generally not commutative) group, known as the special orthogonal group SO(n).

In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant -1 (instead of +1). These combine proper rotations with reflections (which invert orientation). In this sense the members of the (general) orthogonal group O(n) may also be called rotation matrices. In other cases, where reflections are not being considered, the label proper may be dropped unless demands of clarity make it wise to specify it. This is the reason it is more frequent to associate the term rotation matrix with members of SO(n) (as in the rest of this article) while it may also be found referring to members of O(n).