  We address some theoretical guarantees for Schatten-$p$ quasi-norm
minimization ($p \in (0,1]$) in recovering low-rank matrices from compressed
linear measurements. Firstly, using null space properties of the measurement
operator, we provide a sufficient condition for exact recovery of low-rank
matrices. This condition guarantees unique recovery of matrices of ranks equal
or larger than what is guaranteed by nuclear norm minimization. Secondly, this
sufficient condition leads to a theorem proving that all restricted isometry
property (RIP) based sufficient conditions for $\ell_p$ quasi-norm minimization
generalize to Schatten-$p$ quasi-norm minimization. Based on this theorem, we
provide a few RIP-based recovery conditions.
