  This paper provides an extension of the constructive loop vertex expansion to
stable matrix models with interactions of arbitrarily high order. We introduce
a new representation for such models, then perform a forest expansion on this
representation. It allows to prove that the perturbation series of the free
energy for such models is analytic in a domain uniform in the size N of the
matrix. Our method applies to complex (rectangular) matrices. The extension to
Hermitian square matrices, which was claimed wrongly in the first arXiv version
of this paper, is postponed to a future study.
