All matrices I consider are complex hermitian $N\times N$ matrices. None of them are Diagonal matrices. Consider a positive semi definite matrix $A$ and a matrix $B$. I am interested in the sum $C=A-\gamma B$ which $\gamma$ is a non-negative scalar.
Question
What is the smallest gamma such that $C$ is negative semi-definite when $B$ is 1) positive definite 2) positive semi-definite?
If $\gamma=0$, $C$ is positive semidefinite, as $\gamma$ increases, the "positive semi-definiteness" of $C$ decreases. At some point, it becomes, indefinite, and after a while, it will become a negative semidefinite matrix. I want to know for what gamma this happens.
Solution of Case 1
It is quite straightforward to prove that the solution of case 1 is $\gamma=\lambda_{max}(B^{-1}A)$ which follows from expressing $\gamma$ as the solution to a generalized eigenvalue problem of $A$ and $B$ and using the fact $B$ is invertible.
(please see this MSE question for a distantly related problem, but from which solution can't be deduced for this one).