Eigenvalues may not be always real for generalized eigenvalue problems even if the Hamiltonian and overlap matrices are both Hermitian. When the overlap is associated with a norm of physical state, the overlap matrix is always positive definite. Then, as is well known, we can reduce the problem to that of a simple eigenvalue problem of the Hermitian Hamiltonian. Generically speaking, the generalized eigenvalue problem is reduced to that of the non-Hermitian Hamiltonian with some (emergent) symmetry. A typical example is a photonic crystal where the dielectric constant is not uniform. Have a look at our recent paper, “Topological band theory of a generalized eigenvalue problem with Hermitian matrices: Symmetry-protected exceptional rings with emergent symmetry” accepted for publication in Phys. Rev. B (Lett.) (Aug.18, 2021) and published in Phys. Rev. B 104, L121105 – Published 7 September 2021 by Takuma Isobe, Tsuneya Yoshida, and Yasuhiro Hatsugai. See also arXiv:2105.01283.