Let $A\in \mathbb{C}^{m \times n}$ be a wide unknown complex matrix ($m<n$), and $\Sigma\in \mathbb{R}^{p \times n}$ a known rectangular diagonal matrix. $A$ has full row-rank. Is it possible to obtain a closed form solution for $A$ in terms of $\Sigma$ from the following equation:
$\frac{\left\Vert \Sigma A^H (AA^H)^{-1}\right\Vert_F^2}{\left\Vert A \right\Vert_F^2} AA^H = (AA^H)^{-1} A \Sigma^H \Sigma A^H (AA^H)^{-1}$,
where $\Vert\cdot\Vert_F$ is the Frobenius norm, $(\cdot)^H$ denotes conjugate traspose, and $(\cdot)^{-1}$ denotes matrix inverse?