$$ \text{Let}\\ A, B \in \mathbb{C}^{n \times n} \text{ and } \hat{\alpha}, \hat{\beta}, \hat{f} \in \mathbb{C}^n \\ \text{Find }\\ \underline{\mathbf{x}} \in \mathbb{C}^{2n} \text{ such that}\\ \begin{bmatrix} A & -B\\ \hat{A} & \hat{B} \end{bmatrix} \cdot \underline{\mathbf{x}} = \hat{f}\\ \hat A = \text{diag}(\hat\alpha) ,\quad \hat B = \text{diag}(\hat\beta) $$$$ \text{Let}\\ A, B \in \mathbb{C}^{n \times n} \text{ and } \hat{\alpha}, \hat{\beta} \in \mathbb{C}^{n}, \hat{f} \in \mathbb{C}^{2n} \\ \text{Find }\\ \underline{\mathbf{x}} \in \mathbb{C}^{2n} \text{ such that}\\ \begin{bmatrix} A & -B\\ \hat{A} & \hat{B} \end{bmatrix} \cdot \underline{\mathbf{x}} = \hat{f}\\ \hat A = \text{diag}(\hat\alpha) ,\quad \hat B = \text{diag}(\hat\beta) $$ SoFor all rows, only one of $\hat{\alpha}$ and $\hat{\beta}$ in this row can be zero.
So basically we are dealing with a block matrix, where the upper row consists of two fully populated, dense, complex square matrices and the lower row consists of two diagonal complex matrices. I need to solve this kind of equation system, where $n$ can be anywhere between $\approx3000$ and $\approx 20000$.
I have thought about using an augmented GMRES (adapted multiplication for the lower half of the block matrix), but I think I would need a solid preconditioner, because the system might be poorly conditioned.
Are there any direct algorithms that would profit from the fact that the lower half of the block matrix consists of two diagonal matrices?