  We introduce the Fisher information in the basis of decay modes of Markovian
dynamics, arguing that it encodes important information about the behavior of
nonequilibrium systems. In particular we generalize an orthonormality relation
between decay eigenmodes of detailed balanced systems to normal generators that
commute with their time-reversal. Viewing such modes as tangent vectors to the
manifold of statistical distributions, we relate the result to the choice of a
coordinate patch that makes the Fisher-Rao metric Euclidean at the steady
distribution, realizing a sort of statistical equivalence principle. We then
classify nonequilibrium systems according to their spectrum, showing that a
degenerate Fisher matrix is the signature of the insurgence of a class of
dynamical phase transitions between nonequilibrium regimes, characterized by
level crossing and power-law decay in time of suitable order parameters. An
important consequence is that normal systems cannot manifest critical behavior.
Finally, we study the Fisher matrix of systems with time-scale separation.
