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 $A$?

Given an arbitrary symmetric N-by-N matrix A, how can its original values be calculated from $P$?

$$ P = A'A$$

Both $A$ and $P$ have \( \frac{N^2-N}{2}+N \) degrees of freedom.

Edit: added the constraint that A is symmetric

Given an arbitrary N-by-N matrix A, to what extent can it's 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?

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 $A$?