Let $A,D\in\mathbb{C}^{n\times n}$ with eigenvalues $\alpha_i,\delta_i$ and let $b,c\in\mathbb{C}$. Without loss of generality, I've been able to prove the following statements.
- The eigenvalues of $K =\begin{bmatrix}A & bI\\ cI & A\end{bmatrix}$ are given by $\kappa_i=\alpha_i\pm\sqrt{bc}$.
- The eigenvalues of $L=\begin{bmatrix}A & bI\\ cI & 0\end{bmatrix}$ are given by $\lambda_i=\frac{\alpha_i}{2}\pm\sqrt{\left(\frac{\alpha_i}{2}\right)^2+bc}$.
Given these facts, I would like to find a bound (or closed form) for the eigenvalues $\mu_i$ of $M=\begin{bmatrix}A & bI\\ cI & D\end{bmatrix}$.
Numerically, I've observed that all $\mu_i$ lie within the convex hull of all $\alpha_i\pm\sqrt{bc}$ and all $\delta_i\pm\sqrt{bc}$. However, I'm struggling to prove this mathematically. Does anyone have a suggestion on how to proceed?