Let $A_1,\ldots,A_m \in R^{n\times n}$ be symmetric and positive semidefinite, and suppose that their sum $A$ is positive definite. For some nonzero vector $u\in R^n$ with $u^TA_ju>0$ for all $j$, put
$$v_j:= \frac{1}{\sqrt{u^TA_ju}} A_ju$$
Can one bound the eigenvalues of the symmetric, positive semidefinite matrix $B$ with components $B_{jk}:=v_j^TA^{-1}v_k$ from above, uniformly in $m$ and $n$? Perhaps even by $1$ (which is a trivial lower bound for the requested upper bound attained by taking all $A_j$ as identity)?
