  Topological phases of Hermitian systems are known to exhibit intriguing
properties such as the presence of robust boundary states and the famed
bulk-boundary correspondence. These features can change drastically for their
non-Hermitian generalizations, as exemplified by a general breakdown of
bulk-boundary correspondence and a localization of all states at the boundary,
termed the non-Hermitian skin effect. In this article, we present a completely
analytical unifying framework for studying these systems using generalized
transfer matrices -- a real-space approach suitable for systems with periodic
as well as open boundary conditions. We show that various qualitative
properties of these systems can be easily deduced from the transfer matrix. For
instance, the connection between the breakdown of the conventional
bulk-boundary correspondence and the existence of a non-Hermitian skin effect,
previously observed numerically, is traced back to the transfer matrix having a
determinant not equal to unity. The vanishing of this determinant signals
real-space exceptional points, whose order scales with the system size. We also
derive previously proposed topological invariants such as the biorthogonal
polarization and the Chern number computed on a complexified Brillouin zone.
Finally, we define an invariant for and thereby clarify the meaning of
topologically protected boundary modes for non-Hermitian systems.
