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