  We study the ergodic properties of a class of measures on
$\Sigma^{\mathbb{Z}}$ for which $\mu_{\mathcal{A},t}[x_{0}\cdots
x_{n-1}]\approx e^{-nP}\left \|A_{x_{0}}\cdots A_{x_{n-1}}\right \| ^{t}$,
where $\mathcal{A}=(A_{0}, \ldots , A_{M-1})$ is a collection of matrices. The
measure $\mu_{\mathcal{A},t}$ is called a matrix Gibbs state. In particular we
give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli
property. We employ a number of techniques to understand these measures
including a novel approach based on Perron-Frobenius theory. We find that when
$t$ is an even integer the ergodic properties of $\mu_{\mathcal{A} ,t}$ are
readily deduced from finite dimensional Perron-Frobenius theory. We then
consider an extension of this method to $t>0$ using operators on an infinite
dimensional space. Finally we use a general result of Bradley to prove the main
theorem.
