I recently stumbled across the following eigenvalue problem, where $\omega$ is the desired eigenvalue, $\Omega$ is a parameter:
$$ \begin{pmatrix} A & B \\ B^* & C \end{pmatrix} \begin{pmatrix} \vec{u} \\ \vec{v} \end{pmatrix}= \begin{pmatrix} (\omega - \Omega)^2 & 0 \\ 0 & (\omega + \Omega)^2 \end{pmatrix} \begin{pmatrix} \vec{u} \\ \vec{v} \end{pmatrix}. $$
$A, B, C$ are square, non-commuting matrices. Of course this is a quadratic eigenvalue problem and can be solved by standard methods. However, I was wondering if the special structure maybe allowed further simplifications.