# Complex eigenvalues of a rotation matrix

I am struggling with understanding the meaning of complex eigenvalues of a rotation matrix.

1 is always an eigenvalue - that is clear, since all the vectors on the axis of rotation are not effected by the rotation. But what about complex eigenvalues and the corresponding eigenvectors? A rotation around a "complex axis"?

• take a closer look at $\Bbb R^2$ and extrapolate to $\Bbb R^3$, maybe – janmarqz Nov 25 '15 at 16:58
• Note that $1$ is guaranteed to be an eigenvalue only if the space has odd dimension. – amd Nov 25 '15 at 18:42

In two dimensions, there’s only one plane, so a matrix with complex eigenvalues represents a rotation+scaling of the entire space. In three dimensions, there’s only one dimension left after you’ve defined the plane of rotation, so that’s going to be the axis of rotation that corresponds to the eigenvalue of $$1$$. In higher dimensions, things get wacky. In four dimensions, for instance, you can have simultaneous rotations in two different planes.