  We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$
real or complex matrix measures defined on the interval $[0,1]$. The asymptotic
properties of the first $k$ components of a uniformly distributed vector
$(S_{1,n}, ..., S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n
\to \infty$. In particular, it is shown that an appropriately centered and
standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$ converges weakly
to a vector of $k$ independent $p \times p$ Gaussian ensembles. For the proof
of our results we use some new relations between ordinary moments and canonical
moments of matrix measures which are of their own interest. In particular, it
is shown that the first $k$ canonical moments corresponding to the uniform
distribution on the real or complex moment space $\mathcal{M}_n$ are
independent multivariate Beta distributed random variables and that each of
these random variables converge in distribution (if the parameters converge to
infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary
ensemble, respectively.
