Consider $M=A-yB$ where $A$ is real, symmetric, positive definite, $B$ is real, symmetric, positive semi-definite, and $y$ is a positive scalar variable. I am interested in maximum value of $y$ for which $M$ is positive semi-definite. I've seen in some papers that they split $M$ into smaller matrices as \begin{equation} M=\sum_{i} (A_i-y_iB_i) \end{equation} then they find $y_i$'s for small matrices and choose $max(y_i)$ as the answer. I'm just wondering under what condition can we do this? Small matrices $M_i$ can overlap with each other.
Thanks,
Assume $B\ne0$. Take $x\ne0$. You have $$ x^T(A-yB)x \ge x^TAx - y x^TBx \ge (\lambda_{min}(A) - y \lambda_{max}(B)) \|x\|^2, $$ this gives immediately that for $$ y\le \frac{\lambda_{min}(A)}{\lambda_{max}(B)} $$ the matrix $A-yB$ is positive semi-definite. The bound is sharp, if there is a vector that is both eigenvector of $A$ to $\lambda_{min}(A)$ and eigenvector of $B$ to $\lambda_{max}(B)$.
This discussion can be repeated for those small matrices, of course. Hope this helps.