Generalized eigenvalue equations describe the band structures of various metamaterials, where complex bands can emerge even if the involved matrices are Hermitian. In this paper, we provide a geometrical understanding of the real-complex transition of the band structures. Specifically, our analysis, based on auxiliary eigenvalues, clarifies the correspondence between the real-complex transition of the generalized eigenvalue equations and the Lifshitz transition in electron systems. Furthermore, we demonstrate that real (complex) bands of a photonic system correspond to the Fermi surfaces of type-II (type-I) Dirac cones in electron systems when the permittivity $\varepsilon$ and the permeability $\mu$ are independent of frequency. In addition, our analysis reveals that exceptional points are induced by the frequency dependence of permittivity $\varepsilon$ and permeability $\mu$ in our photonic system.

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• Classical optics
• Non-Hermitian systems
• Band structure methods
• Maxwell's equations