  Long term optimal investment problems are studied in a factor model with
matrix valued state variables. Explicit parameter restrictions are obtained
under which, for an isoelastic investor, the finite horizon value function and
optimal strategy converge to their long-run counterparts as the investment
horizon approaches infinity. This convergence also yields portfolio turnpikes
for general utilities. By using results on large time behaviour of semi-linear
partial differential equations, our analysis extends affine models, where the
Wishart process drives investment opportunities, to a non-affine setting.
Furthermore, in the affine setting, an example is constructed where the value
function is not exponentially affine, in contrast to models with vector-valued
state variables.
