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| At least some of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] of a rotation matrix must be complex valued. Recall that eigenvectors maintain their direction through a transformation. Rotation necessarily makes this impossible, unless they are allowed to take on complex values. Furthermore, note that all eigenvectors of the rotation matrix in ''R^2^'' are complex; 2 of the 3 eigenvectors of the rotation matrix in ''R^3^'' are complex. The reason for this is that the axis of rotation is the only vector which maintains its direction, and while this axis can be expressed in ''R^3^'' space, it exists outside the plane in ''R^2^'' space. |
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 an orthogonal matrix. It follows that RT = R-1.
The determinant of a rotation matrix is either 1 or -1.
At least some of the eigenvectors of a rotation matrix must be complex valued. Recall that eigenvectors maintain their direction through a transformation. Rotation necessarily makes this impossible, unless they are allowed to take on complex values.
Furthermore, note that all eigenvectors of the rotation matrix in R2 are complex; 2 of the 3 eigenvectors of the rotation matrix in R3 are complex. The reason for this is that the axis of rotation is the only vector which maintains its direction, and while this axis can be expressed in R3 space, it exists outside the plane in R2 space.