= Solution
The upper $2\times2$ block has trace zero and determinant $-\cos^22\theta-\sin^22\theta=-1$, so its characteristic polynomial is $\lambda^2-1$. The third coordinate has eigenvalue two. Angle-addition identities give the normalized <eigenvectors>
$$
\boxed{\lambda_+=1:\ \mathbf u_+=(\cos\theta,\sin\theta,0),\qquad\lambda_-=-1:\ \mathbf u_-=(-\sin\theta,\cos\theta,0),\qquad\lambda_3=2:\ \mathbf u_3=(0,0,1).}
$$
These formulas apply to every real $\theta$, including values where an expression obtained by dividing by $\sin2\theta$ would fail. Geometrically the planar block is a reflection: it fixes the axis at angle $\theta$ and reverses its perpendicular direction.
Back to article page
