  Celebrated fluctuation-dissipation theorem (FDT) linking the response
function to time dependent correlations of observables measured in the
reference unperturbed state is one of the central results in equilibrium
statistical mechanics. In this letter we discuss an extension of the standard
FDT to the case when multidimensional matrix representing transition
probabilities is strictly non-normal. This feature dramatically modifies the
dynamics, by incorporating the effect of eigenvector non-orthogonality via the
associated overlap matrix of Chalker-Mehlig type. In particular, the rate of
entropy production per unit time is strongly enhanced by that matrix. We
suggest, that this mechanism has an impact on the studies of collective
phenomena in neural matrix models, leading, via transient behavior, to such
phenomena as synchronisation and emergence of the memory. We also expect, that
the described mechanism generating the entropy production is generic for wide
class of phenomena, where dynamics is driven by non-normal operators. For the
case of driving by a large Ginibre matrix the entropy production rate is
evaluated analytically, as well as for the Rajan-Abbott model for neural
networks.
