  Loss-induced transmission in waveguides, and reversed pump dependence in
lasers, are two prominent examples of counter-intuitive effects in
non-Hermitian systems with patterned gain and loss. By analyzing the eigenvalue
dynamics of complex symmetric matrices when a system parameter is varied, we
introduce a general set of theoretical conditions for these two effects. We
show that these effects arise in any irreducible system where the gain or loss
is added to a subset of the elements of the system, without the need for
parity-time symmetry or for the system to be near an exceptional point. These
results are confirmed using full-wave numerical simulations. The conditions
presented here vastly expand the design space for observing these effects. We
also show that a similarly broad class of systems exhibit a loss-induced
narrowing of the density of states.
