I want to know whether there is a general solution to this problem. I want to solve the following matrix equation with respect to $X$.
$$ \mathbf{X}^{-1}-\mathbf{A}- \mathbf{X}\odot \mathbf{D} = \mathbf{0}$$
where $\odot$ is the Hadamard product, $\mathbf{D}$ is lower-triangular matrix and every other matrix is symmetric positive definite. Would there be a solution for this?