More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$.
What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ in terms of the eigenvectors of $X^T X$?
(The notation $\odot$ means $[A\odot B]_{ij}=A_{ij}B_{ij}$).