  In this paper, we establish the following perturbation result concerning the
singular values of a matrix: Let $A,B \in \mathbb{R}^{m\times n}$ be given
matrices, and let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a concave function
satisfying $f(0)=0$. Then, we have $$ \sum_{i=1}^{\min\{m,n\}} \big|
f(\sigma_i(A)) - f(\sigma_i(B)) \big| \le \sum_{i=1}^{\min\{m,n\}}
f(\sigma_i(A-B)), $$ where $\sigma_i(\cdot)$ denotes the $i$--th largest
singular value of a matrix. This answers an open question that is of interest
to both the compressive sensing and linear algebra communities. In particular,
by taking $f(\cdot)=(\cdot)^p$ for any $p \in (0,1]$, we obtain a perturbation
inequality for the so--called Schatten $p$--quasi--norm, which allows us to
confirm the validity of a number of previously conjectured conditions for the
recovery of low--rank matrices via the popular Schatten $p$--quasi--norm
heuristic. We believe that our result will find further applications,
especially in the study of low--rank matrix recovery.
