  So far, topological band theory is discussed mainly for systems described by
eigenvalue problems. Here, we develop a topological band theory described by a
generalized eigenvalue problem (GEVP). Our analysis elucidates that
non-Hermitian topological band structures may emerge for systems described by a
GEVP with Hermitian matrices. The above result is verified by analyzing a
two-dimensional toy model where symmetry-protected exceptional rings (SPERs)
emerge although the matrices involved are Hermitian. Remarkably, these SPERs
are protected by emergent symmetry, which is unique to the systems described by
the GEVP. Furthermore, these SPERs elucidate the origin of the characteristic
dispersion of hyperbolic metamaterials which is observed in experiments.
