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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.

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  • $\begingroup$ Are you sure that you've written the matrix correctly? It looks strikingly similar to a matrix called the "Relative Gain Array", which is defined as $A\circ A^{-T}$. In fact, when $A$ is symmetric, it is the RGA. $\endgroup$
    – greg
    Commented Nov 19, 2016 at 4:39
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    $\begingroup$ Too late to edit my previous comment. Your expression isn't equal to the RGA when A is symmetric -- got excited and misspoke. $\endgroup$
    – greg
    Commented Nov 19, 2016 at 4:48

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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 $A$.

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