New upper bounds for the spectral variation of a general matrix

Abstract: 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.