I am looking for a closed-form solution for the inverse $J_p^{-1}(v)$ of the following matrix:
$J_p(v) = \text{diag}(v)Y+\text{diag}(Yv)$,
where $v\in\mathbb{R}^{n}$ is arbitrary and $Y=Y^T\in\mathbb{R}^{n\times n}$ is known.
Background: The power flow equations for a DC-Microgrid are given by $p=v\circ Yv$, where $Y$ is the nodal admittance matrix and $\circ$ the Hadamard product.
Does somebody have some hints or ideas? Thanks.