  In this paper, we theoretically investigate the low-rank matrix recovery
problem in the context of the unconstrained regularized nuclear norm
minimization (RNNM) framework. Our theoretical findings show that, the RNNM
method is able to provide a robust recovery of any matrix $X$ (not necessary to
be exactly low-rank) from its few noisy measurements
$\textbf{b}=\mathcal{A}(X)+\textbf{n}$ with a bounded constraint
$\|\textbf{n}\|_{2}\leq\epsilon$, provided that the $tk$-order restricted
isometry constant (RIC) of $\mathcal{A}$ satisfies a certain constraint related
to $t>0$. Specifically, the obtained recovery condition in the case of $t>4/3$
is found to be same with the sharp condition established previously by Cai and
Zhang (2014) to guarantee the exact recovery of any rank-$k$ matrix via the
constrained nuclear norm minimization method. More importantly, to the best of
our knowledge, we are the first to establish the $tk$-order RIC based
coefficient estimate of the robust null space property in the case of
$0<t\leq1$.
