A generalized eigen value problem, that is described by two hermitian Hamiltonian/overlap matrices,
intrisically posesses a conic structure that may induce complex eigen values. The story is geometically described in our recent paper, “Band structures of the generalized eigenvalue equation and conic section” by Takuma Isobe, Tsuneya Yoshida, and Yasuhiro Hatsugai, which has been accepted for publication in Phys. Rev. A – Accepted 16 June, 2025. See also arXiv:2409.01191