The minimal measurement number for low-rank matrices recovery

The paper presents several results that address a fundamental question in low-rank matrices recovery: how many measurements are needed to recover low rank matrices? We begin by investigating the complex matrices case and show that $4nr-4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-$r$ matrices in $\C^{n\times n}$ by algebraic tools. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound $4nr-4r^2$ is tight provided $n=2^k+r, k\in \Z_+$. Motivated by Vinzant's work, we construct $11$ matrices in $\R^{4\times 4}$ by computer random search and prove they define injective measurements on rank-$1$ matrices in $\R^{4\times 4}$. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than $2n-1$ orthogonal projections are possible for the recovery of $x\in \R^n$ from the norm of them.