 Solution

The upper $2\times 2$ block has trace zero and determinant $-{\cos }^{2}2\theta -{\sin }^{2}2\theta =-1$, so its characteristic polynomial is ${\lambda }^2-1$. The third coordinate has eigenvalue two. Angle-addition identities give the normalized eigenvectorsThese formulas apply to every real $\theta$, including values where an expression obtained by dividing by $\sin 2\theta$ would fail. Geometrically the planar block is a reflection: it fixes the axis at angle $\theta$ and reverses its perpendicular direction.