Consider the following matrix, with some entries unknown:
$$ M = \begin{pmatrix} ? & 2 & ? \\ 2 & 4 & ? \\ 3 & ? & 3 \end{pmatrix} $$What is the full matrix ? That’s an absurd question, right ? The only way to get the full matrix is to know all of its entries, after all… Or is it ? For instance if you also know $\text{rank}M = 1$, then you know actually know that
$$ M = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 3 & 6 & 3 \end{pmatrix} $$Generalizing the generalization of the generalization of the above argument gives
[Candes]
Oh and also, all of this is effective and we can actually compute the completion of $M$ very quickly. Wtf, right ?