  Motivated by [9] we study the existence of the inverse of infinite Hermitian
moment matrices associated with measures with support on the complex plane. We
relate this problem to the asymptotic behaviour of the smallest eigenvalues of
finite sections and we study it from the point of view of infinite transition
matrices associated to the orthogonal polynomials. For Toeplitz matrices we
introduce the notion of weakly asymptotic Toeplitz matrix and we show that,
under certain assumptions, the inverse of a Toeplitz moment matrix is weakly
asymptotic Toeplitz. Such inverses are computed in terms of some limits of the
coefficients of the associated orthogonal polynomials. We finally show that the
asymptotic behaviour of the smallest eigenvalue of a moment Toeplitz matrix
only depends on the absolutely part of the associated measure.
