  We develop a theoretical description of non-Hermitian time evolution that
accounts for the break- down of the adiabatic theorem. We obtain closed-form
expressions for the time-dependent state amplitudes, involving the complex
eigen-energies as well as inter-band Berry connections calculated using basis
sets from appropriately-chosen Schur decompositions. Using a two-level system
as an example, we show that our theory accurately captures the phenomenon of
"sudden transitions", where the system state abruptly jumps from one eigenstate
to another.
