I am not familiar with the website, so I write my question in latex. I want to calculate the diagonal matrix $\boldsymbol\Theta_n$ through the following nonlinear equation: \begin{align} \mathbf{I}\odot\left[ \boldsymbol\Sigma_0 - \left( \mathbf{D} + {\boldsymbol\gamma_n\boldsymbol\gamma_n^H} - \boldsymbol\Theta_n\right)^{-1} \right] = \mathbf{O} \end{align} where $\mathbf{I}$ is identity matrix, $\boldsymbol\Sigma_0$ and $\mathbf{D}$ are all real positive definite diagonal matrices, $\boldsymbol\gamma_n$ is a complex vector with non-zero elements. $\boldsymbol\Theta_n$ is a real diagonal matrix. And $\odot$ is hadamard product. That is the diagonal elements of $\left( \mathbf{D} + {\boldsymbol\gamma_n\boldsymbol\gamma_n^H} - \boldsymbol\Theta_n\right)^{-1}$ are equal to those of $\boldsymbol\Sigma_0$.
I have tried the Sherman-Morrison lemma, but it seems useless. However, since the number of variates is equal to the equation, I think this equation may have solutions or even a unique solution. How can I calculate the $\boldsymbol\Theta_n$?