  The stability of low-rank matrix reconstruction with respect to noise is
investigated in this paper. The $\ell_*$-constrained minimal singular value
($\ell_*$-CMSV) of the measurement operator is shown to determine the recovery
performance of nuclear norm minimization based algorithms. Compared with the
stability results using the matrix restricted isometry constant, the
performance bounds established using $\ell_*$-CMSV are more concise, and their
derivations are less complex. Isotropic and subgaussian measurement operators
are shown to have $\ell_*$-CMSVs bounded away from zero with high probability,
as long as the number of measurements is relatively large. The $\ell_*$-CMSV
for correlated Gaussian operators are also analyzed and used to illustrate the
advantage of $\ell_*$-CMSV compared with the matrix restricted isometry
constant. We also provide a fixed point characterization of $\ell_*$-CMSV that
is potentially useful for its computation.
