New upper bounds for the spectral variation of a general matrix

TL;DR

This paper derives new upper bounds for the spectral variation of general matrices under perturbations, extending classical results like Hoffman–Wielandt to non-normal matrices and improving existing bounds.

Contribution

It introduces novel upper bounds for spectral variation applicable to non-normal matrices, generalizing and improving upon classical spectral stability results.

Findings

New bounds improve existing estimates for spectral variation.

Results extend spectral stability analysis to non-normal matrices.

Some bounds are tighter or more general than previous ones.

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.

Tables

Table 1. Table 1. The upper bounds in ( 1.6 ) and ( 3.1 )–( 3.3 ).

Estimate	Upper bound for ${𝔻}_2$
(1.6)	0.455931801780
(3.1)	0.228717520806
(3.2)	0.044693805777
(3.3)	0.044693805018

Table 2. Table 2. The upper bounds in ( 1.6 ) and ( 3.6 )–( 3.8 ).

Estimate	Upper bound for ${𝔻}_2$
(1.6)	2.330923594272
(3.6)	1.129360191939
(3.7)	0.325303334100
(3.8)	0.325303282160

Equations

$$\delta(Y):=\bigg{(}\|Y\|_{F}^{2}-\frac{1}{n}|\operatorname*{tr}(Y)|^{2}\bigg{)}^{\frac{1}{2}}.$$

$$\mathbb{D}_{2}:=\Bigg{(}\sum_{i=1}^{n}\big{|}\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big{|}^{2}\Bigg{)}^{\frac{1}{2}}.$$

$${\mathbb{D}}_2\le \Vert E{\Vert }_F.$$

$${\mathbb{D}}_2\le \sqrt{n}\Vert E{\Vert }_F.$$

$${\mathbb{D}}_2\le \sqrt{\Vert E{\Vert }_F^2+(n-1)\delta (E{)}^2},$$

$$Q^{-1}AQ=\operatorname*{diag}\big{(}J_{1},\ldots,J_{p}\big{)},$$

$$m=\underset{1\le i\le p}{\max }{m}_i\text{and}{E}_Q=Q^{-1}EQ.$$

$$\mathbb{D}_{2}\leq\begin{cases}\sqrt{n}\big{(}\sqrt{n-p}+1\big{)}\|E_{Q}\|_{F}^{\frac{1}{m}},&\text{if $\|E_{Q}\|_{F}<1$},\\ \sqrt{n}\big{(}\sqrt{n-p}+1\big{)}\|E_{Q}\|_{F},&\text{if $\|E_{Q}\|_{F}\geq 1$}.\end{cases}$$

$$\mathbb{D}_{2}\leq\begin{cases}\sqrt{n\Big{(}n-p+2\sqrt{n-p}\,\delta(E_{Q})+\frac{\delta(E_{Q})^{2}}{\|E_{Q}\|_{F}^{2}}\Big{)}\|E_{Q}\|_{F}^{\frac{2}{m}}+\frac{1}{n}|\operatorname*{tr}(E)|^{2}},&\text{if $\|E_{Q}\|_{F}<1$},\\ \sqrt{n\big{(}\sqrt{n-p}+\delta(E_{Q})\big{)}^{2}+\frac{1}{n}|\operatorname*{tr}(E)|^{2}},&\text{if $\|E_{Q}\|_{F}\geq 1$}.\end{cases}$$

$$\Vert \mathcal{L}(M){\Vert }_F^2+\Vert \mathcal{U}(M){\Vert }_F^2\le \delta (M{)}^2,$$

$${\mathbb{D}}_2\le \sqrt{\Vert E{\Vert }_F^2+(n-1)\delta (E{)}^2},$$

$$A=Q\operatorname*{diag}\big{(}J_{1},\ldots,J_{p}\big{)}Q^{-1},$$

$$J_i=\left(\begin{array}{ccccc}{\lambda }_i& 1& 0& \cdots & 0\\ 0& {\lambda }_i& 1& \cdots & 0\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& \cdots & {\lambda }_i& 1\\ 0& 0& \cdots & 0& {\lambda }_i\end{array}\right).$$

$$T=\operatorname*{diag}\big{(}T_{1},\ldots,T_{p}\big{)},$$

$$T^{-1}Q^{-1}AQT=\operatorname*{diag}\big{(}T_{1}^{-1}J_{1}T_{1},\ldots,T_{p}^{-1}J_{p}T_{p}\big{)}=\Lambda+\Omega,$$

$${\Omega }_i=\left(\begin{array}{ccccc}0& \epsilon & 0& \cdots & 0\\ 0& 0& \epsilon & \cdots & 0\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& \cdots & 0& \epsilon \\ 0& 0& \cdots & 0& 0\end{array}\right)\in {\mathbb{C}}^{m_i\times m_i}.$$

$$\Lambda=\operatorname*{diag}\big{(}\lambda_{1}I_{m_{1}},\ldots,\lambda_{p}I_{m_{p}}\big{)}\quad\text{and}\quad T=\operatorname*{diag}\big{(}T_{1},\ldots,T_{p}\big{)},$$

$$\Vert T^{-1}{Q}^{-1}\tilde{A}QT-\Lambda {\Vert }_F^2\le \mathcal{V}(\epsilon ),$$

$$\mathcal{V}(\epsilon ):={\epsilon }^{2(1-m)}\delta (E_Q{)}^2+2{\epsilon }^2\sqrt{n-p}\delta (E_Q)+(n-p){\epsilon }^2+\frac{1}{n}|\mathrm{tr}(E){|}^2$$

$$T^{-1}{Q}^{-1}\tilde{A}QT-\Lambda =T^{-1}{E}_{Q}T+\Omega ,$$

$$\Vert T^{-1}{Q}^{-1}\tilde{A}QT-\Lambda {\Vert }_F^2=\Vert T^{-1}{E}_{Q}T{\Vert }_F^2+2\mathrm{Retr}({\Omega }^{\ast }{T}^{-1}{E}_{Q}T)+\Vert \Omega {\Vert }_F^2.$$

$$\Vert T^{-1}{E}_{Q}T{\Vert }_F^2=\sum _{i=1}^p\sum _{j=1}^p\Vert T_i^{-1}{\hat{E}}_{ij}{T}_j{\Vert }_F^2.$$

$$\Vert T^{-1}{E}_{Q}T{\Vert }_F^2$$

$$\displaystyle\leq\varepsilon^{2(1-m)}\sum_{i\neq j}\|\widehat{E}_{ij}\|_{F}^{2}+\sum_{i=1}^{p}\big{(}\|\mathcal{D}(\widehat{E}_{ii})\|_{F}^{2}+\varepsilon^{2}\|\mathcal{U}(\widehat{E}_{ii})\|_{F}^{2}+\varepsilon^{2(1-m_{i})}\|\mathcal{L}(\widehat{E}_{ii})\|_{F}^{2}\big{)}$$

$$\displaystyle\leq\varepsilon^{2(1-m)}\Bigg{(}\sum_{i\neq j}\|\widehat{E}_{ij}\|_{F}^{2}+\sum_{i=1}^{p}\|\mathcal{U}(\widehat{E}_{ii})\|_{F}^{2}+\sum_{i=1}^{p}\|\mathcal{L}(\widehat{E}_{ii})\|_{F}^{2}\Bigg{)}+\|\mathcal{D}(E_{Q})\|_{F}^{2}$$

$$\displaystyle=\varepsilon^{2(1-m)}\big{(}\|E_{Q}\|_{F}^{2}-\|\mathcal{D}(E_{Q})\|_{F}^{2}\big{)}+\|\mathcal{D}(E_{Q})\|_{F}^{2}$$

$$\displaystyle=\varepsilon^{2(1-m)}\|E_{Q}\|_{F}^{2}-\big{(}\varepsilon^{2(1-m)}-1\big{)}\|\mathcal{D}(E_{Q})\|_{F}^{2}.$$

$$\Vert \mathcal{D}(E_Q){\Vert }_F^2\ge \frac{1}{n}|\mathrm{tr}(E){|}^2.$$

$$\Vert T^{-1}{E}_{Q}T{\Vert }_F^2\le {\epsilon }^{2(1-m)}\delta (E_Q{)}^2+\frac{1}{n}|\mathrm{tr}(E){|}^2.$$

$$\mathrm{Retr}({\Omega }^{\ast }{T}^{-1}{E}_{Q}T)=\mathrm{Re}\sum _{i=1}^p\mathrm{tr}({\Omega }_i^{\ast }{T}_i^{-1}{\hat{E}}_{ii}{T}_i)=\mathrm{Re}\sum _{i=1}^p\sum _{j=2}^{m_i}\epsilon (T_i^{-1}{\hat{E}}_{ii}{T}_i{)}_{j-1,j}.$$

$$\mathrm{Retr}({\Omega }^{\ast }{T}^{-1}{E}_{Q}T)$$

$$\le {\epsilon }^2\sum _{i=1}^p\sum _{j=2}^{m_i}|({\hat{E}}_{ii}{)}_{j-1,j}|$$

$$\displaystyle\leq\varepsilon^{2}\sqrt{n-p}\Bigg{(}\sum_{i=1}^{p}\sum_{j=2}^{m_{i}}|(\widehat{E}_{ii})_{j-1,j}|^{2}\Bigg{)}^{\frac{1}{2}}$$

Full text

1. Introduction

Let $\mathbb{C}^{m\times n}$ be the set of all $m\times n$ complex matrices, and let $I_{n}$ be the $n\times n$ identity matrix. For any $X\in\mathbb{C}^{m\times n}$, the symbols $X^{\ast}$, $\|X\|_{2}$, and $\|X\|_{F}$ denote the conjugate transpose, the spectral norm, and the Frobenius norm of $X$, respectively. For any $Y\in\mathbb{C}^{n\times n}$, $\operatorname*{tr}(Y)$, $\mathcal{D}(Y)$, $\mathcal{L}(Y)$, and $\mathcal{U}(Y)$ stand for its trace, diagonal part, strictly lower triangular part, and strictly upper triangular part, respectively. Furthermore, we define

[TABLE]

Obviously, $\delta(Y)\leq\|Y\|_{F}$, and $\delta(Y)=\|Y\|_{F}$ if and only if $\operatorname*{tr}(Y)=0$.

Let $A\in\mathbb{C}^{n\times n}$ and $\widetilde{A}=A+E\in\mathbb{C}^{n\times n}$ have the spectra $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$, respectively. For any permutation $\pi$ of $\{1,\ldots,n\}$, we define

[TABLE]

If $A$ and $\widetilde{A}$ are normal matrices, Hoffman and Wielandt [5] proved that there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that

[TABLE]

This is the well-known Hoffman–Wielandt theorem, which reveals the strong stability of the spectrum of a normal matrix. However, the inequality (1.3) may fail when $A$ or $\widetilde{A}$ is non-normal. Over the past decades, various extensions or analogues of the Hoffman–Wielandt theorem have been developed by many researchers; see, e.g., [4, 12, 2, 3, 6, 7, 11, 9, 8, 1, 10, 13].

If $A\in\mathbb{C}^{n\times n}$ is normal and $\widetilde{A}=A+E\in\mathbb{C}^{n\times n}$ is non-normal, Sun [12, Theorem 1.1] showed that

[TABLE]

Recently, Xu and Zhang [13, Theorem 3.6] derived that

[TABLE]

which improved the estimate (1.4) due to $\delta(E)\leq\|E\|_{F}$. Nevertheless, the estimates (1.3)–(1.5) may be invalid for a general matrix $A$. As is well known, for any $A\in\mathbb{C}^{n\times n}$, there is a nonsingular matrix $Q\in\mathbb{C}^{n\times n}$ such that

[TABLE]

where each $J_{i}\in\mathbb{C}^{m_{i}\times m_{i}}$ ($\sum_{i=1}^{p}m_{i}=n$) is a Jordan block. Let

[TABLE]

It was proved by Song [11, Theorem 2.1] that

[TABLE]

In this paper, we establish some new upper bounds for the spectral variation of a general matrix. One of our main results is

[TABLE]

In view of (1.1), $\delta(E_{Q})$ involved in (1.7) is \delta(E_{Q})=\big{(}\|E_{Q}\|_{F}^{2}-\frac{1}{n}|\operatorname*{tr}(E)|^{2}\big{)}^{\frac{1}{2}}. Theoretical analysis shows that the new estimate (1.7) is sharper than (1.6) (see Remark 3.2 for details). Moreover, it is easy to check that (1.7) will reduce to (1.5) if $A$ is a normal matrix. That is, the new estimate (1.7) also generalizes the existing one (1.5).

The rest of this paper is organized as follows. In Section 2, we introduce several auxiliary estimates, which play an important role in our analysis. In Section 3, we present new upper bounds for the spectral variation of a general matrix.

2. Preliminaries

For any square matrix $M$, the first lemma provides an upper bound for $\|\mathcal{L}(M)\|_{F}^{2}+\|\mathcal{U}(M)\|_{F}^{2}$ [13, Lemma 3.1].

Lemma 2.1**.**

Let $M$ be a square matrix. Then

[TABLE]

where $\delta(\cdot)$ is defined by (1.1).

The following lemma gives an upper bound for the spectral variation of a normal matrix [13, Theorem 3.6], which plays a key role in the subsequent analysis.

Lemma 2.2**.**

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 $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$. Then there exists a permutation $\pi$ of $\{1,\dots,n\}$ such that

[TABLE]

where $\delta(\cdot)$ is defined by (1.1).

For any $A\in\mathbb{C}^{n\times n}$, it can be factorized as

[TABLE]

where $Q\in\mathbb{C}^{n\times n}$ is nonsingular, and each $J_{i}\in\mathbb{C}^{m_{i}\times m_{i}}$ $(\sum_{i=1}^{p}m_{i}=n)$ is a Jordan block with the form

[TABLE]

Let $0<\varepsilon\leq 1$ be a parameter, and let

[TABLE]

where T_{i}=\operatorname*{diag}\big{(}1,\varepsilon,\ldots,\varepsilon^{m_{i}-1}\big{)} for all $i=1,\ldots,p$. Then

[TABLE]

where \Lambda=\operatorname*{diag}\big{(}\lambda_{1}I_{m_{1}},\ldots,\lambda_{p}I_{m_{p}}\big{)}, and \Omega=\operatorname*{diag}\big{(}\Omega_{1},\dots,\Omega_{p}\big{)} with

[TABLE]

We are now in a position to present the fundamental estimate of this paper.

Lemma 2.3**.**

Let $A\in\mathbb{C}^{n\times n}$ be factorized as in (2.3), and let $\widetilde{A}=A+E\in\mathbb{C}^{n\times n}$ be a perturbed matrix. Let

[TABLE]

where T_{i}=\operatorname*{diag}\big{(}1,\varepsilon,\ldots,\varepsilon^{m_{i}-1}\big{)} with $0<\varepsilon\leq 1$. Then, it holds that

[TABLE]

where

[TABLE]

with $m=\max\limits_{1\leq i\leq p}m_{i}$ and $E_{Q}=Q^{-1}EQ$.

Proof.

From (2.4), we have

[TABLE]

which yields

[TABLE]

In what follows, we establish the upper bounds for $\|T^{-1}E_{Q}T\|_{F}^{2}$, $\operatorname*{Re\,tr}(\Omega^{\ast}T^{-1}E_{Q}T)$, and $\|\Omega\|_{F}^{2}$.

(i) Partitioning $E_{Q}$ into the block form $E_{Q}=(\widehat{E}_{ij})_{p\times p}$ with $\widehat{E}_{ij}\in\mathbb{C}^{m_{i}\times m_{j}}$, we have

[TABLE]

Hence,

[TABLE]

Note that

[TABLE]

Thus,

[TABLE]

(ii) It is easy to see that

[TABLE]

Due to $(T_{i}^{-1}\widehat{E}_{ii}T_{i})_{j-1,j}=\varepsilon(\widehat{E}_{ii})_{j-1,j}$, it follows that

[TABLE]

Since $\|\mathcal{U}(E_{Q})\|_{F}\leq\delta(E_{Q})$ (see (2.1)), we obtain

[TABLE]

(iii) In addition, we have

[TABLE]

Combining (2.6)–(2.9), we can arrive at the estimate (2.5). ∎

3. Main results

In light of (2.2) and (2.5), we can derive the following estimate.

Theorem 3.1**.**

Let $A\in\mathbb{C}^{n\times n}$ have the factorization (2.3), and let $\widetilde{A}=A+E$, where $E\in\mathbb{C}^{n\times n}$ is a perturbation. Let $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$ be the spectra of $A$ and $\widetilde{A}$, respectively. Then there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that

[TABLE]

where $m=\max\limits_{1\leq i\leq p}m_{i}$ and $E_{Q}=Q^{-1}EQ$.

Proof.

Observe that $\Lambda$ is a normal matrix with spectrum $\{\lambda_{i}\}_{i=1}^{n}$, and the spectrum of $T^{-1}Q^{-1}\widetilde{A}QT$ is $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$. Applying Lemma 2.2 to $\Lambda$ and $T^{-1}Q^{-1}\widetilde{A}QT$ yields

[TABLE]

where we have used the estimate (2.5). Take

[TABLE]

Direct calculations yield

[TABLE]

Thus, the estimate (3.1) is valid. ∎

Remark 3.2*.*

If $\|E_{Q}\|_{F}<1$, then (3.1) reads

[TABLE]

Due to

[TABLE]

it follows that

[TABLE]

On the other hand, if $\|E_{Q}\|_{F}\geq 1$, then (3.1) reads

[TABLE]

Then

[TABLE]

Hence, the estimate (3.1) is sharper than (1.6).

The next two estimates are based on the different constraints for $E_{Q}$.

Theorem 3.3**.**

Under the assumptions of Theorem 3.1, it holds that

[TABLE]

Proof.

Take

[TABLE]

Direct computation yields

[TABLE]

Similarly to Theorem 3.1, one can show that the estimate (3.2) holds. ∎

Theorem 3.4**.**

Under the assumptions of Theorem 3.1, it holds that

[TABLE]

where

[TABLE]

Proof.

We first note that $A$ is diagonalizable if and only if $n=p$ (or $m=1$).

(i) If $A$ is diagonalizable, then $T=I_{n}$, $n=p$, and $m=1$. In this case, the estimate (2.5) reduces to

[TABLE]

An application of Lemma 2.2 yields

[TABLE]

(ii) If $A$ cannot be diagonalized, then $n>p$ and $m>1$. Direct calculation yields

[TABLE]

Here, $\mathscr{V}^{\prime}(\varepsilon)$ denotes the derivative of $\mathscr{V}(\varepsilon)$ with respect to $\varepsilon$. It is easy to check that

[TABLE]

Take

[TABLE]

Direct computation yields

[TABLE]

The rest of this proof is similar to Theorem 3.1. ∎

Remark 3.5*.*

If $A$ is diagonalizable, the condition ${\rm C}_{2}$ will be satisfied. From (3.3), we have

[TABLE]

which coincides with (3.4). That is, (3.3) has contained the diagonalizable case.

Remark 3.6*.*

In particular, if $A$ is normal, then $Q$ can be chosen as a unitary matrix. In this case, the estimates (3.1)–(3.3) all reduce to

[TABLE]

which is exactly (2.2).

Example 3.7**.**

Let

[TABLE]

where $a\in\mathbb{R}$, $b\in\mathbb{R}$, and $\mathbf{i}=\sqrt{-1}$. In this case,

[TABLE]

The upper bounds in (1.6), (3.1), (3.2), and (3.3) are listed below.

Table 1 displays that the new upper bounds in (3.1)–(3.3) are smaller than that in (1.6).

Under the assumptions of Lemma 2.2, if the original matrix is Hermitian, then the following estimate (see [13, Theorem 4.2]) holds:

[TABLE]

In what follows, we consider a special case that the eigenvalues of $A$ are all real. In such a case, we can derive more accurate estimates for $\mathbb{D}_{2}$ based on (3.5), which are presented in the following three theorems.

Theorem 3.8**.**

Let $A\in\mathbb{C}^{n\times n}$ be factorized as in (2.3), 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 the eigenvalues $\{\lambda_{i}\}_{i=1}^{n}$ of $A$ are all real, then there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that

[TABLE]

Theorem 3.9**.**

Under the assumptions of Theorem 3.8, it holds that

[TABLE]

Theorem 3.10**.**

Under the assumptions of Theorem 3.8, it holds that

[TABLE]

where ${\rm C}_{1}$ and ${\rm C}_{2}$ are given in Theorem 3.4.

Example 3.11**.**

Let

[TABLE]

where $\lambda\in\mathbb{R}$. In this example, it holds that

[TABLE]

The upper bounds in (1.6), (3.6), (3.7), and (3.8) are listed below.

From Table 2, one can see that the new estimates (3.6)–(3.8) are sharper than (1.6).

Remark 3.12*.*

Define

[TABLE]

Using

[TABLE]

one can derive some deductive estimates for $\mathbb{D}_{2}$. Furthermore, using the relation $\mathbb{D}_{\infty}\leq\mathbb{D}_{2}$, one can readily obtain the corresponding estimates for $\mathbb{D}_{\infty}$.