Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and let $a > 0$ be some constant.
I am interested in the following quantity $$ \hat x = {\rm arg} \max_{x \in R^n}\ x' \left( \frac{1}{a + \|x\|^2_2} A \right) x $$ Is it possible to relate $\|\hat x\|_2^2$ to the eigenvalue and $\hat x$ to the eigenvector of $A$.
Furthermore, suppose that $E \in \mathbb{R}^{p \times p}$ is such that $A + E$ is still positive definite and $\|E\|$ is small in some norm (say, spectral). Consider $$ \hat y = {\rm arg} \max_{y \in R^n}\ y' \left( \frac{1}{a + \|y\|^2_2} (A+E) \right) y $$ Are there known results that can be used to quantify $| \|\hat x\|_2^2 - \|\hat y\|_2^2 |$? I suppose the norms should be close if the perturbation $E$ is small, but don't know how to quantify this notion.