TITLE: Is there a formula for the inverse of Hadamard product?
QUESTION [17 upvotes]: Say $A$ and $B$ are two square, positive-semidefinite matrices. Is there an expression in terms of matrix product, transpose, and inverse for the Hadamard product $A∘B$? 
For example, "$(A∘B)^{-1} = A^{-1} ∘ B^{-1}$" (which is not true).
Edit: I understand that $A∘B$ may not be invertible, but is there any expression if invertibility is given?

REPLY [2 votes]: Yes, I think there is a formula for that, in terms of the Hadamard inverse of the first matrix and the inverse of the second one. For example, if:
$\Sigma = A \circ B$
then the inverse of $\Sigma$ can be written as:
$\Sigma^{-1} = A^{\circ (-1)} \circ B^{-1}$
where $A^{\circ (-1)}$ is the Hadamard inverse of $A$, which is defined as
$[A^{\circ (-1)}]_{i,j} = 1/[A]_{i,j}$
Thus B must be invertible, and if $\Sigma$ is invertible as well (as in your question), it follows that $A^{\circ (-1)}$ is well defined. 
Also, a sufficient though not necessary condition is then that $B$ and $A$ be invertible, as Reams proofs that if $A$ is invertible then $A^{\circ (-1)}$ is positive definite.