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 π of $\{1,\dots ,n\}$ such that $(\sum _{i=1}^n|{\tilde{\lambda }}_{\pi \left(i\right)}-{\lambda }_i{|}^2{)}^{1/2}\le \parallel E{\parallel }_F$, where $\parallel \cdot {\parallel }_F$ 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 $(\sum _{i=1}^n|{\tilde{\lambda }}_{\pi \left(i\right)}-{\lambda }_i{|}^2{)}^{1/2}$, provided that both A and $\tilde{A}$ are general matrices. Some of our estimates improve or generalize the existing ones.

让$\mathrm{一个}\in {\mathbb{C}}^{n\times n}$是具有谱的正规矩阵$\{{\lambda }_{\mathrm{一世}}{\}}_{\mathrm{一世}=1}^n$， 然后让$\widetilde{\mathrm{一个}}=\mathrm{一个}+乙\in {\mathbb{C}}^{n\times n}$是一个有谱的扰动矩阵$\{{\tilde{\lambda }}_{\mathrm{一世}}{\}}_{\mathrm{一世}=1}^n$. 如果$\widetilde{\mathrm{一个}}$仍然是正常的，著名的Hoffman-Wielandt 定理指出存在一个置换π$\{1,\dots ,n\}$这样$(\sum _{\mathrm{一世}=1}^n|{\tilde{\lambda }}_{\pi \left(\mathrm{一世}\right)}-{\lambda }_{\mathrm{一世}}{|}^2{)}^{1/2}\le \parallel 乙{\parallel }_F$， 在哪里$\parallel \cdot {\parallel }_F$表示矩阵的 Frobenius 范数。该定理揭示了正规矩阵谱的强稳定性。但是，如果A或$\widetilde{\mathrm{一个}}$是非正态的，Hoffman-Wielandt 定理一般不成立。在本文中，我们提出了新的上限$(\sum _{\mathrm{一世}=1}^n|{\tilde{\lambda }}_{\pi \left(\mathrm{一世}\right)}-{\lambda }_{\mathrm{一世}}{|}^2{)}^{1/2}$，前提是A和$\widetilde{\mathrm{一个}}$是一般矩阵。我们的一些估计改进或概括了现有的估计。