Inverse of a Hadamard product of a matrix and its traspose

Question

I have to obtain the inverse of a Hadamard product of a matrix and its transpose, i.e. $$B=(A∘A^T)^{-1}$$ I would like to know if a simple expression of this inverse is known, in terms or the $A$ matrix, which is not Hermitian in general. I am interested in the case where $A$ and $B$ are invertible. Any help will be welcome.

Answer 1

There cannot be an expression in terms of $A$, at least not one simpler than explicitly writing the adjugate transpose of $A\circ A^T$. Consider $$A=\begin{bmatrix}1&t\\0&1\end{bmatrix}.$$ Then $$ A\circ A^T=I, $$ so $B=I$, this, for any $t$. The key point is that $A\circ A^T$ can kill essential information about $A$.