  In this paper we investigate the reconstruction conditions of nuclear norm
minimization for low-rank matrix recovery. We obtain sufficient conditions
$\delta_{tr}<t/(4-t)$ with $0<t<4/3$ to guarantee the robust reconstruction
$(z\neq0)$ or exact reconstruction $(z=0)$ of all rank $r$ matrices
$X\in\mathbb{R}^{m\times n}$ from $b=\mathcal{A}(X)+z$ via nuclear norm
minimization. Furthermore, we not only show that when $t=1$, the upper bound of
$\delta_r<1/3$ is the same as the result of Cai and Zhang \cite{Cai and Zhang},
but also demonstrate that the gained upper bounds concerning the recovery error
are better. Moreover, we prove that the restricted isometry property condition
is sharp. Besides, the numerical experiments are conducted to reveal the
nuclear norm minimization method is stable and robust for the recovery of
low-rank matrix.
