= Solution
At the formal value $Re=-10$, $q=-1$ and $A=-i\omega_1I+\begin{pmatrix}0&1\\-1&-i\end{pmatrix}$ is a <skew-Hermitian matrix>. Put $r_\pm=(-1\pm\sqrt5)/2$. Its two eigenvalues and normalized <eigenvectors> are
$$
\lambda_\pm=-i\omega_1+ir_\pm,\qquad
\boxed{v_\pm=\frac{(1,ir_\pm)^T}{\sqrt{1+r_\pm^2}}.}
$$
Substitution uses $r_\pm^2+r_\pm-1=0$ and verifies $Av_\pm=\lambda_\pm v_\pm$. Their Hermitian inner product is
$$
v_+^\dagger v_-=
\frac{1+r_+r_-}{\sqrt{(1+r_+^2)(1+r_-^2)}}=0,
$$
since $r_+r_-=-1$. This provides the requested orthogonality in the formal continuation. There is no corresponding physical $Re_N>0$ for the printed system.
Back to article page
