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}$^{n\times n}$beaperturbedmatrixwithspectrum${\widetilde{\lambda}{i}}{i=1}^{n}$\mathrm{.}If$\widetilde{A}$ is still normal, the celebrated Hoffman–Wielandt theorem states that there exists a permutation $\pi$ of $\{1,\dots ,n\}$ such that $({\sum }_{i=1}^n\mathrm{\mid }{\tilde{\lambda }}_{\pi (i)}-{\lambda }_i{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\le \mathrm{\parallel }E{\mathrm{\parallel }}_F$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_F$ 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 $({\sum }_{i=1}^n\mathrm{\mid }{\tilde{\lambda }}_{\pi (i)}-{\lambda }_i{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, provided that both $A$ and $\tilde{A}$ are general matrices. Some of our estimates improve or generalize the existing ones.