  The IIB matrix model proposes a mechanism for dynamically generating four
dimensional space--time in string theory by spontaneous breaking of the ten
dimensional rotational symmetry $\textrm{SO}(10)$. Calculations using the
Gaussian expansion method (GEM) lend support to this conjecture. We study a
simple $\textrm{SO}(4)$ invariant matrix model using Monte Carlo simulations
and we confirm that its rotational symmetry breaks down, showing that lower
dimensional configurations dominate the path integral. The model has a strong
complex action problem and the calculations were made possible by the use of
the factorization method on the density of states $\rho_n(x)$ of properly
normalized eigenvalues $\tilde\lambda_n$ of the space--time moment of inertia
tensor. We study scaling properties of the factorized terms of $\rho_n(x)$ and
we find them in agreement with simple scaling arguments. These can be used in
the finite size scaling extrapolation and in the study of the region of
configuration space obscured by the large fluctuations of the phase. The
computed values of $\tilde\lambda_n$ are in reasonable agreement with GEM
calculations and a numerical method for comparing the free energy of the
corresponding ansatze is proposed and tested.
