This question follows this previous one: Maximize $\gamma$ such that $A+\gamma B\succeq 0$
This time, $A\succ 0$, $B$ is indefinite and $C = \begin{bmatrix} 0 & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & 1\end{bmatrix}$.
The goal is to optimize the following problem: $$ \max_{\lambda,\gamma} \gamma:A+\lambda B-\gamma C\succeq 0$$
For the previous question, an answer was provided: $$\gamma=-\frac{1}{\lambda_{\min}(A^{-1}B)}$$ where $\lambda_{\min}(A)$ refers to the minimum eigenvalue of $A$.
I don't know if it is possible to obtain such a result for the current problem. Surely, it is optimizable using a SDP solver, but I would like to avoid that.