Let $M \in \mathbb{S}^d_+$ be a $d$-dimensional symmetric positive definite matrix. Any element of $M$ is bounded, i.e., $|M_{ij}|\le \kappa$. I was wondering how the eigenvalues of $M$ change if I symmetrically change one row-column pair of $M$ (i.e., change the $i$-th row and $i$-th column of $M$, and the new matrix $M'$ is still symmetric positive definite).
The Cauchy's interlacing theorem seems related to my question, but it's about eigenvalue perturbation if a row-column pair is removed.