= Solution
For the proposed <vector>, the first horizontal component of $Kq$ is $-k(\alpha+\beta)/2$. Since the corresponding component of $q$ is zero, a <normal mode> requires $\beta=-\alpha$. The vertical components then give $Kq=2kq$. With an arbitrary nonzero overall amplitude, choose $\alpha=1$ to obtain
$$
\boxed{q_y=(0,1,0,-1,0,-1,0,1)^T,\qquad\omega=\sqrt{\frac{2k}m}.}
$$
The analogous pattern invariant under <reflection> across the vertical axis is
$$
\boxed{q_x=(1,0,-1,0,-1,0,1,0)^T,\qquad\omega=\sqrt{\frac{2k}m}.}
$$
Direct multiplication confirms $Kq_x=2kq_x$. The two patterns give independent <eigenvectors> with the same <eigenvalue>, as expected from the square's <rotation> symmetry.
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