  We present and analyze an efficient implementation of an iteratively
reweighted least squares algorithm for recovering a matrix from a small number
of linear measurements. The algorithm is designed for the simultaneous
promotion of both a minimal nuclear norm and an approximatively low-rank
solution. Under the assumption that the linear measurements fulfill a suitable
generalization of the Null Space Property known in the context of compressed
sensing, the algorithm is guaranteed to recover iteratively any matrix with an
error of the order of the best k-rank approximation. In certain relevant cases,
for instance for the matrix completion problem, our version of this algorithm
can take advantage of the Woodbury matrix identity, which allows to expedite
the solution of the least squares problems required at each iteration. We
present numerical experiments that confirm the robustness of the algorithm for
the solution of matrix completion problems, and demonstrate its competitiveness
with respect to other techniques proposed recently in the literature.
