How to solve an equation with Hadamard product

Question

For positive definite matrices $A$ and $B$, I have the following identity:

$$ C \circ [A (C \circ X)B] = C \circ Z $$

How can I solve this equation for $X$?

Note: $C$ is a matrix with 1s and zeros.

Edit:

Is it correct to argue that

$A(C \circ X) B = Z$ so $C \circ X = A^{-1}ZB^{-1}$. However, $A^{-1}ZB^{-1}$ corresponds to a dense matrix which contradicts with $C \circ X$ term

Answer 1

You may rewrite the equation $C\circ[A(C\circ X)B]=C\circ Z$ as $$ \operatorname{diag}(\operatorname{vec}(C))(B^T\otimes A)\operatorname{diag}(\operatorname{vec}(C))\operatorname{vec}(X)=\underbrace{\operatorname{vec}(C\circ Z)}_{=\,\operatorname{diag}(\operatorname{vec}(C))\operatorname{vec}(Z)} $$ and solve it, where $\operatorname{diag}(v)$ means a diagonal matrix using the vector $v$ as its main diagonal.