Let $P,Q$ be $n$ by $n$ invertible matrices. Suppose further that $P$ and $Q$ satisfies the following equation :

$$(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$$

where $\circ$ denotes the Hadamard matrix product, which is simply the entrywise product.

Then what can be said about $P$ and $Q$? More precisely, I want to know if there are additional relations between $P$ and $Q$. For example, one can show that the condition $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ implies

$$tr(P^{-1}DPE) = tr(Q^{-1}DQE)$$ for all diagonal matrices $D$ and $E$.

References in the litterature about matrices of the form $(P^{-1})^T \circ P$ would help too. Thank you, Malik

Topics in Matrix Analysis(not to be confused withMatrix Analysisby the same authors). – Federico Poloni May 22 '13 at 8:44gain arraymatrix associated with $P$. It was studied by C. R. Johnson & H. Shapiro. – Denis Serre May 22 '13 at 11:44