  Several important properties of positive semidefinite processes of
Ornstein--Uhlenbeck type are analysed. It is shown that linear operators of the
form $X\mapsto AX+XA^{\mathrm{T}}$ with $A\in M_d(\mathbb{R})$ are the only
ones that can be used in the definition provided one demands a natural
non-degeneracy condition. Furthermore, we analyse the absolute continuity
properties of the stationary distribution (especially when the driving matrix
subordinator is the quadratic variation of a $d$-dimensional L\'{e}vy process)
and study the question of how to choose the driving matrix subordinator in
order to obtain a given stationary distribution. Finally, we present results on
the first and second order moment structure of matrix subordinators, which is
closely related to the moment structure of positive semidefinite
Ornstein--Uhlenbeck type processes. The latter results are important for method
of moments based estimation.
