Matrix for arbitrary rotation in N dimensions (Rodrigues' rotation formula)

Generated by Claude AI; links and some syntactic refinement are my own

To derive a parametrized expression for a rotation matrix about an arbitrary axis in -dimensional space, we can use the concept of the exponential map from Lie algebra to Lie groups. The resulting matrix is known as the Rodrigues’ rotation formula.

Let be a unit vector representing the axis of rotation, and let be the angle of rotation. The rotation matrix is given by:

where is the skew-symmetric matrix representation of , defined as:

Using the Taylor series expansion of the matrix exponential, we can write:

where is the identity matrix.

Expanding , we get:

Substituting this back into the expression for , we obtain the final parametrized expression:

This expression allows you to compute the rotation matrix for any arbitrary axis and angle in -dimensional space.