  Since the 1970s there has been a rich theory of equilibrium states over shift
spaces associated to H\"older-continuous real-valued potentials. The
construction of equilibrium states associated to matrix-valued potentials is
much more recent, with a complete description of such equilibrium states being
achieved in 2011 by D.-J. Feng and A. K\"aenm\"aki. In a recent article the
author investigated the ergodic-theoretic properties of these matrix
equilibrium states, attempting in particular to give necessary and sufficient
conditions for mixing, positive entropy, and the property of being a Bernoulli
measure with respect to the natural partition, in terms of the algebraic
properties of the semigroup generated by the matrices. Necessary and sufficient
conditions were successfully established for the latter two properties but only
a sufficient condition for mixing was given. The purpose of this note is to
complete that investigation by giving a necessary and sufficient condition for
a matrix equilibrium state to be mixing.
