  In this paper, we provide a dissipative Hamiltonian (DH) characterization for
the set of matrices whose eigenvalues belong to a given LMI region. This
characterization is a generalization of that of Choudhary et al. (Numer. Linear
Algebra Appl., 2020) to any LMI region. It can be used in various contexts,
which we illustrate on the nearest $\Omega$-stable matrix problem: given an LMI
region $\Omega \subseteq \mathbb{C}$ and a matrix $A \in \mathbb{C}^{n,n}$,
find the nearest matrix to $A$ whose eigenvalues belong to $\Omega$. Finally,
we generalize our characterization to more general regions that can be
expressed using LMIs involving complex matrices.
