What are the complex eigenvalues of the matrix $A$ that represents a rotation of $\mathbb{R}^3$ through the angle $\theta$ about a vector $u$?
I know that an eigenvalue should be $1$, and the other $2$ should be complex, where one is the conjugate of the other. I can't seem to get it however.
The eigenvalues of a plane rotation of angle $\theta$ are the eigenvalues of $\left(\begin{smallmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{smallmatrix}\right)$: $e^{\pm i\theta}$. So, since a rotation around $u$ with angle $\theta$ maps $u$ into itself (and therefore $u$ is an engenvector with eigenvalue $1$), its eigenvalues are $1$ and $e^{\pm i\theta}$.