Question

What are proper and improper orthogonal matrices?​

Answer 1

Answer:

Three-Dimensional Proper and Improper Rotation Matrices

1. Proper and improper rotation matrices

A real orthogonal matrix R is a matrix whose elements are real numbers and satisfies

R−1 = RT (or equivalently, RRT = I, where I is the 3 × 3 identity matrix). Taking

the determinant of the equation RRT = I and using the fact that det(RT) = det R, it

follows that (det R)

2 = 1, which implies that either det R = 1 or det R = −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 nˆ. 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 det R = −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 nˆ and a mirror reflection through a plane that passes through the

origin and is perpendicular to nˆ.

In these notes, we shall explore the matrix representations of three-dimensional

proper and improper rotations. By determining the most general form for a three-

dimensional 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 op-

erator 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.

2. Properties of the 3 × 3 rotation matrix

A rotation in the x–y plane by an angle θ measured counterclockwise from the

positive x-axis is represented by the 2 × 2 real orthogonal matrix with determinant

equal to 1,

cos θ − sin θ

sin θ cos θ