What do eigenvectors and eigenvalues do in rotation operations?

Question

There's some material briefly mentioned the eigenvectors and eigenvalues when it comes to rotation matrices.

Can someone give me a neat explanation for what eigenvectors and eigenvalues do in rotation operations?

Answer 1

In $\Bbb R^2$ a rotation matrix ($\neq$ identity and reflection) does not have any eigenvectors/eigenvalues in $\Bbb R$, because there is no direction, which is fixed by rotation.

In $\Bbb R^3$ a rotation matrix has the eigenvalue 1 and the corresponding eigenvector spans the axis of rotation. Again there are (up to the exceptions above) no more eigenvalues/eigenvectors since the linear transformation restricts to a rotation on the remaining two dimensional subspace.

In $\Bbb R^n$ one (afaik) usually does not speak of rotations, because there are not really axes of rotation any longer.