Rotation matrix

A rotation matrix is a square matrix with pure complex eigenvalues. Each pair of complex conjugate eigenvalues corresponds to a rotation about a 2D plane. The angle of rotation in this subspace is given by

The inverse of a rotation matrix rotates in the opposite direction of A. This makes intuitive sense, since for all matrices for which exists, $AA^{-1}=\mathbf{I}$$. That is, if you apply a rotation matrix, we would expect that applying its inverse would “undo” the rotation.

All rotation matrices are also orthogonal matrices.