Rotation Matrix
A rotation matrix represents the linear transformation of rotation.
Contents
Description
A rotation matrix is defined for a specific number of dimensions. In R2, a rotation matrix is:
where θ is the angle of counter-clockwise rotation. Note that a negative θ represents clockwise rotation.
As an example, a counter-clockwise rotation of 90 degrees (or π/2 radians) is represented by:
In R3, there are separate rotation matrices for each dimension.
where α, β, and γ represent yaw, pitch, and roll in the Z, Y, and X dimensions respectively. A complete rotation matrix in three dimensions can then be calculated as R = Rz(α)Ry(β)Rx(γ).
Properties
A rotation matrix is always orthonormal. It follows that RT = R-1.
The determinant of a pure rotation matrix, i.e., one that rotates and does not stretch, is either 1 or -1.
Decomposition of Rotation and Stretching
A 2-dimensional matrix that rotates and stretches can be decomposed between the two components.
Consider the example matrix 
:
The eigenvalues (λ) are a + bi and a - bi
Following from the formula for distance of complex vectors, the stretching factor is ||λ|| = √(a2 + b2).
Therefore the decomposition is:
where:
a = ||λ|| cos(θ) and b = ||λ|| sin(θ), or
θ = tan-1(b/a)
Eigenvectors
A matrix that rotates will always have complex eigenvectors. In fact, a complex eigenpair means that the transformation involves rotation.
Consider the 2-dimensional case again. There clearly cannot be any vectors which do not change direction through rotation.
Consider now the 3-dimensional rotation matrix for rotation around the Z axis. This is fundamentally the same rotation. But now all vectors along the axis of rotation are unchanged by rotation.