New upper bounds for the spectral variation of a general matrix

Let $A\in\mathbb{C}^{n\times n}$ be a normal matrix with spectrum $\{\lambda_{i}\}_{i=1}^{n}$, and let $\widetilde{A}=A+E\in\mathbb{C}^{n\times n}$ be a perturbed matrix with spectrum $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$. If $\widetilde{A}$ is still normal, the celebrated Hoffman--Wielandt theorem states that there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that $\big(\sum_{i=1}^{n}|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}|^{2}\big)^{1/2}\leq\|E\|_{F}$, where $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix. This theorem reveals the strong stability of the spectrum of a normal matrix. However, if $A$ or $\widetilde{A}$ is non-normal, the Hoffman--Wielandt theorem does not hold in general. In this paper, we present new upper bounds for $\big(\sum_{i=1}^{n}|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}|^{2}\big)^{1/2}$, provided that both $A$ and $\widetilde{A}$ are general matrices. Some of our estimates improve or generalize the existing ones.