Solution
= Solution
The <characteristic polynomial> of the <rotation matrix> is
$$
\det(M-zI)=(\cos U-z)^2+\sin^2U
=z^2-2z\cos U+1.
$$
Its roots are $\cos U\pm i\sin U$, so
$$
\boxed{e^{i\Theta^+}=e^{iU},\qquad e^{i\Theta^-}=e^{-iU},
\qquad\Theta^\pm=\pm U\pmod{2\pi}.}
$$
For example $(1,i)^T$ is an <eigenvector> for $e^{iU}$ and $(1,-i)^T$ for $e^{-iU}$. At $U=0$ or $\pi$ the two <eigenvalues> coincide; these events have <probability> zero under the <uniform distribution>.