= Solution
Define $D=\sqrt{\kappa^2/4+q^2}$. Normalizing the first <displacement> <amplitude> to one, either row of the <eigenvector> equation gives
$$
\boxed{v_\pm=\begin{pmatrix}1\\r_\pm\end{pmatrix},\qquad r_\pm=\frac{\kappa/2\mp D}{q}.}
$$
The two <normal modes> are $x(t)=A_\pm v_\pm\cos(\omega_\pm t+\phi_\pm)$, and arbitrary small motion is their <superposition>. The lower-frequency mode has $r_->0$, so the <masses> move in phase; the upper-frequency mode has $r_+<0$, so they move in opposite phase. The identity $r_+r_-=-1$ verifies <orthogonality> of these <eigenvectors> for equal <masses>.
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