  Non-Hermitian dynamics in quantum systems preserves the rank of the state
density operator. We use this insight to develop its geometrical description.
In particular, we identify mutually orthogonal coherent and incoherent
directions and give their physical interpretation. This understanding allows us
to optimize the success rate for implementing non-Hermitian driving along
prescribed trajectories. We show its significance for shortcuts to
adiabaticity. We introduce the geometrical interpretation of a speed limit for
non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit
expression that saturates such a speed limit and illustrate our results on a
minimal example of a dissipative qubit.
