  We study the maximally supersymmetric plane wave matrix model (the BMN model)
at finite temperature, $T$, and locate the high temperature phase boundary in
the $(\mu,T)$ plane, where $\mu$ is the mass parameter. We find the first
transition, as the system is cooled from high temperatures, is from an
approximately $SO(9)$ symmetric phase to one where three matrices expand to
form fuzzy spheres. For $\mu > 3.0$ there is a second distinct transition at a
lower temperature. The two transitions approach one another at smaller $\mu$
and merge in the vicinity of $\mu=3.0$. The resulting single transition curve
then approaches the gauge/gravity prediction as $\mu$ is further decreased. We
find a rough estimate of the transition, for all $\mu$, is given by a Pad\'e
resummation of the large-$\mu$, 3-loop perturbative, predictions. We find
evidence that the transition at small $\mu$ is to an M5-brane phase of the
theory.
