  Imaginary gauge transformation (IGT) provides a clear understanding of the
non-Hermitian skin effect by transforming the non-Hermitian Hamiltonians with
real spectra into Hermitian ones. In this work, we extend this approach to the
complex spectrum regime in a general nonreciprocal lattice model. We unveil the
validity of IGT hinges on a class of pseudo-Hermitian symmetry. The generalized
Brillouin zone of Hamiltonian respect such pseudo-Hermiticity is demonstrated
to be a circle, which enables easy access to the continuum bands, localization
length of skin modes, and relevant topological numbers. Furthermore, we
investigate the applicability of IGT and the underlying pseudo-Hermiticity
beyond nearest-neighbour hopping, offering a graphical interpretation. Our
theoretical framework is applied to establish bulk-boundary correspondence in
the nonreciprocal trimer Su-Schrieffer-Heeger model and analyze the
localization behaviors of skin modes in the two-dimensional Hatano-Nelson
model.
