Given an arbitrary $N$-by-$N$ matrix $A$, to what extent can its original values be estimated from $P$?

$$ P = A'A$$

$A$ has \\(N^2\\) degrees of freedom, whereas $P$ has \\( \frac{N^2-N}{2}+N \\) degrees of freedom.  What is the best way to estimate the original values in $A$?