New perturbation bounds for the spectrum of a normal matrix

Let $A\in\mathbb{C}^{n\times n}$ and $\widetilde{A}\in\mathbb{C}^{n\times n}$ be two normal matrices with spectra $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$, respectively. The celebrated Hoffman--Wielandt theorem states that there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that $\left(\sum_{i=1}^{n}\big|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big|^{2}\right)^{1\over 2}$ is no larger than the Frobenius norm of $\widetilde{A}-A$. However, if either $A$ or $\widetilde{A}$ is non-normal, this result does not hold in general. In this paper, we present several novel upper bounds for $\left(\sum_{i=1}^{n}\big|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big|^{2}\right)^{1\over 2}$, provided that $A$ is normal and $\widetilde{A}$ is arbitrary. Some of these estimates involving the "departure from normality" of $\widetilde{A}$ have generalized the Hoffman--Wielandt theorem. Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix.