Let $A,B$ be two positive semi-definite matrices (we don't necessarily have $A\geq B$ or $A \leq B$), and let $\lambda_1(A)\geq \cdots \geq \lambda_n(A)$, $\lambda_1(B)\geq \cdots \geq \lambda_n(B)$, and we also know that $\lambda_i(A)\geq \lambda_i(B)$. I want to know if we want to have $\lambda_i(G^\top A G)\geq \lambda_i(G^\top BG)$, what condition do we have to impose on $G$?
Or can we have some inequality such as $\lambda_i(G^\top A G)+\epsilon\geq \lambda_i(G^\top BG)$ or $\lambda_i(G^\top A G)\geq (1-\epsilon)\lambda_i(G^\top BG)$?