New perturbation bounds for the spectrum of a normal matrix

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