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 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.

Keywords:

• Hoffman–Wielandt theorem
• spectral variation
• perturbation
• upper bound

2010 Mathematics Subject Classifications:

• 15A18
• 47A55
• 65F15

Disclosure statement

No potential conflict of interest was reported by the author(s).