New perturbation bounds for the spectrum of a normal matrix

Abstract: 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.