  Periodic-boundary spectrum, open-boundary spectrum, as well as the
generalized Brillouin zone (GBZ) are three essential properties of a
one-dimensional non-Hermitian system. In this paper we illustrate that the deep
connections between them can be revealed by a series of special similar
transformations. This viewpoint closely connects the topological geometry of
the open-boundary spectrum with the GBZ and provides a new efficient numerical
method of calculating them accurately. We further extend these connections to
non-Hermitian systems in the symplectic symmetry class. We show that if just
the open-boundary features of a non-Hermitian system such as the spectrum and
the GBZ, are concerned, the relevant symmetry we should consider is not that of
the original system itself, but that of one which has higher symmetry and is
related to the original system by a similarity transformation.

Other Condensed Matter;Strongly Correlated Electrons