Proof (or reference) about $\lambda_i(A+\epsilon e_je_j^*) = \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2).$

Question

I'm looking for a proof (or a reference in a textbook) about the fact that $$ \lambda_i(A+\epsilon e_je_j^*) =_{\epsilon \to 0} \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2), $$ where $A$ is an hermitian matrix, $\lambda_i(A)$ is an eigenvalue of $A$, $e_j \in \bf{R}^n$ is defined by $(e_j)_i = \delta_{i,j}$, $v_{i,j}$ is the $j-$th component of a unit eigenvector of $\lambda_i(A)$.

This theorem is from perturbation theory, a field I'm not very familiar with.

This is used in : Peter B. Denton, Stephen J. Parke, Terence Tao and Xining Zhang. $\textit{Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra}, 2021;$ arXiv:1908.03795 (page $13$).