  We discuss dynamics obtained by increasing powers of non-normal matrices that
are roots of the identity, and therefore have all eigenvalues on the unit
circle. Naively, one would expect that the expectation value of such powers
cannot grow as one increases the power. We demonstrate that, rather
counterintuitively, a completely opposite behavior is possible. In the limit of
infinitely large matrices one can have an exponential growth. For finite
matrices this exponential growth is a part of repeating cycles of exponential
growths followed by exponential decays. The effect can occur if the spectrum is
different than the pseudospectrum, with the exponential growth rate being given
by the pseudospectrum. We show that this effect appears in a class of transfer
matrices appearing in studies of two-dimensional non-interacting systems, for a
matrix describing the Ehrenfest urn, as well as in previously observed purity
dynamics in a staircase random circuit.
