Three-Dimensional Proper and Improper Rotation Matrices

A real orthogonal matrix R is a matrix whose elements are real numbers and satisfies R−1 = R (or equivalently, RR = I, where I is the 3 × 3 identity matrix). Taking the determinant of the equation RR = I and using the fact that det(R) = det R, it follows that (det R) = 1, which implies that either detR = 1 or detR = −1. A real orthogonal matrix with detR = 1 provides a matrix representation of a proper rotation. The most general rotation matrix represents a counterclockwise rotation by an angle θ about a fixed axis that lies along the unit vector n̂. The rotation matrix operates on vectors to produce rotated vectors, while the coordinate axes are held fixed. In typical parlance, a rotation refers to a proper rotation. Thus, in the following sections of these notes we will often omit the adjective proper when referring to a proper rotation. A real orthogonal matrix with detR = −1 provides a matrix representation of an improper rotation. To perform an improper rotation requires mirrors. That is, the most general improper rotation matrix is a product of a proper rotation by an angle θ about some axis n̂ and a mirror reflection through a plane that passes through the origin and is perpendicular to n̂. In these notes, we shall explore the matrix representations of three-dimensional proper and improper rotations. By determining the most general form for a threedimensional proper and improper rotation matrix, we can then examine any 3 × 3 orthogonal matrix and determine the rotation and/or reflection it produces as an operator acting on vectors. If the matrix is a proper rotation, then the axis of rotation and angle of rotation can be determined. If the matrix is an improper rotation, then the reflection plane and the rotation, if any, about the normal to that plane can be determined.