= Solution
The upper-left block is a planar <rotation matrix>. Its real <characteristic polynomial> is $(\lambda-\cos\theta)^2+\sin^2\theta$, which is strictly positive for real $\lambda$ because the angle excludes integer multiples of $\pi$. Hence any real <eigenvector> has $x=y=0$. The remaining equation is $5z=\lambda z$, giving
$$
\boxed{\lambda=5,\qquad (x,y,z)=(0,0,z),\quad z\in\mathbb R\setminus\{0\}.}
$$
There are infinitely many nonzero solutions, spanning \b[one independent eigenvector direction].
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