  Previous work regarding low-rank matrix recovery has concentrated on the
scenarios in which the matrix is noise-free and the measurements are corrupted
by noise. However, in practical application, the matrix itself is usually
perturbed by random noise preceding to measurement. This paper concisely
investigates this scenario and evidences that, for most measurement schemes
utilized in compressed sensing, the two models are equivalent with the central
distinctness that the noise associated with (\ref{eq.3}) is larger by a factor
to $mn/M$, where $m,~n$ are the dimension of the matrix and $M$ is the number
of measurements. Additionally, this paper discusses the reconstruction of
low-rank matrices in the setting, presents sufficient conditions based on the
associating null space property to guarantee the robust recovery and obtains
the number of measurements. Furthermore, for the non-Gaussian noise scenario,
we further explore it and give the corresponding result. The simulation
experiments conducted, on the one hand show effect of noise variance on
recovery performance, on the other hand demonstrate the verifiability of the
proposed model.
