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"?

Solution 1:

In a sense, those complex eigenvalues are the rotation. One way to think of a real eigenvalue is the amount by which a matrix stretches or shrinks things along a certain axis—the associated eigenvector. With a pair of complex eigenvalues (they always come in conjugate pairs for a real matrix), there’s no axis along which things are stretched, i.e., no real eigenvector. Instead, there’s a plane in which vectors get rotated along with the stretching/shrinking that might be going on. If the vector being transformed doesn’t lie in that plane, then its component in the plane undergoes the rotation+scaling.

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.

Addendum: The complex eigenvectors associated with the complex eigenvalue pair give you the plane in which the rotation occurs. If you take the real and imaginary parts of any of these eigenvectors, you get a pair of real vectors that span this plane. The action of the matrix in this plane is encoded in the eigenvalues: the argument of the complex number gives the rotation and its norm gives the dilation. So, just as with real eigenvalues and eigenvectors, they describe a subspace of the domain and the action that the matrix has on that subspace.

Once you go beyond three dimensions, complex eigenvalues can appear with multiplicities greater than one, so the associated generalized eigenspaces can have more than two dimensions. It's hard to say much about what goes on in them beyond the fact that they're invariant with respect to the transformation, i.e., vectors in that subspace get mapped to vectors in the same subspace.