  We study a quartic matrix model with partition function $Z=\int d\ M\exp{\rm
Tr}\ (-\Delta M^2-\frac{\lambda}{4}M^4)$. The integral is over the space of
Hermitian $(\Lambda+1)\times(\Lambda+1)$ matrices, the matrix $\Delta$, which
is not a multiple of the identity matrix, encodes the dynamics and $\lambda>0$
is a scalar coupling constant. We proved that the logarithm of the partition
function is the Borel sum of the perturbation series, hence is a well defined
analytic function of the coupling constant in certain analytic domain of
$\lambda$, by using the multi-scale loop vertex expansions. All the non-planar
graphs generated in the perturbation expansions have been taken care of on the
same footing as the planar ones.
  This model is derived from the self-dual $\phi^4$ theory on the 2 dimensional
Moyal space, also called the 2 dimensional Grosse-Wulkenhaar model. This would
also be the first fully constructed matrix model which is non-trivial and not
solvable.
