  We investigate the spontaneous breaking of the SO(D) symmetry in matrix
models, which can be obtained by the zero-volume limit of pure SU(N) super
Yang-Mills theory in D = 6, 10 dimensions. The D = 10 case corresponds to the
IIB matrix model, which was proposed as a non-perturbative formulation of type
IIB superstring theory, and the spontaneous breaking corresponds to the
dynamical compactification of space-time suggested in that model. First we
study the D = 6 case by the Gaussian expansion method, which turns out to yield
clearer results than the previous results for the D = 10 case for certain
technical reasons. By comparing the free energy of the SO(d) symmetric vacua
for d = 2, 3, 4, 5, we conclude that the breaking SO(6) \to SO(3) actually
occurs. We find that the extent of space-time in the shrunken directions is
almost independent of d. In units of this universal scale, the extended
directions seem to have large but still finite extents depending on d. We show
that these results for the extent of space-time can be explained quantitatively
by an argument based on the low-energy effective theory. With these new
insights, we reconsider the previous results for the IIB matrix model, and find
that they are also consistent with our argument based on the low-energy
effective theory. Thus we arrive at comprehensive understanding and some
quantitative predictions concerning the nature of the spontaneous symmetry
breaking taking place in these models. The space-time picture that emerges from
the IIB matrix model and its implication on possible interpretations of the
model are also discussed.
