Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 4 / b / v / Solution

At the formal value $Re=-10$, $q=-1$ and $A=-i{\omega }_{1}I+\left(\begin{array}{cc}0& 1\\ -1& -i\end{array}\right)$ is a skew-Hermitian matrix. Put $r_{\pm }=(-1\pm \sqrt{5})/2$. Its two eigenvalues and normalized eigenvectors areSubstitution uses $r_{\pm }^2+r_{\pm }-1=0$ and verifies $A{v}_{\pm }={\lambda }_{\pm }{v}_{\pm }$. Their Hermitian inner product issince $r_{+}{r}_{-}=-1$. This provides the requested orthogonality in the formal continuation. There is no corresponding physical $R{e}_N>0$ for the printed system.