Regional Stability and New Eigenvalue Perturbation Bounds

Let $A$ be an $n\times n$ symmetric matrix with eigenvalues $λ_1\geq\cdots\geqλ_n$. Let $\tilde A:=A+E$, where $E$ is a symmetric noise matrix, and denote the eigenvalues of $\tilde A$ by $\tildeλ_1\geq\cdots\geq\tildeλ_n$. Bounding the perturbation $|\tilde λ_i -λ_i| $ is a central problem in linea