Note on regions containing eigenvalues of a matrix

TL;DR

This paper introduces a new eigenvalue inclusion region for matrices by refining existing lphaeta-type regions, providing a tighter bound on where eigenvalues can be located.

Contribution

It proposes a novel eigenvalue inclusion region that is contained within the previous lphaeta-type region, improving eigenvalue localization.

Findings

The new region is strictly contained within the previous lphaeta-type region.

The paper proves the new region's validity and containment.

The approach refines eigenvalue localization methods.

Abstract

By excluding some regions, in which each eigenvalue of a matrix is not contained, from the \alpha\beta-type eigenvalue inclusion region provided by Huang et al.(Electronic Journal of Linear Algebra, 15 (2006) 215-224), a new eigenvalue inclusion region is given. And it is proved that the new region is contained in the \alpha\beta-type eigenvalue inclusion region.

Equations

$$\alpha \bigcup \beta =Nand\alpha \bigcap \beta =\varnothing ,$$

$$r_i^{\alpha }(A):=\sum _{\genfrac{}{}{0ex}{}{j\in \alpha ,}{j\ne i}}|a_{ij}|,c_i^{\alpha }(A):=\sum _{\genfrac{}{}{0ex}{}{j\in \alpha ,}{j\ne i}}|a_{ji}|,$$

$$r_i^{\beta }(A):=\sum _{\genfrac{}{}{0ex}{}{j\in \beta ,}{j\ne i}}|a_{ij}|,c_i^{\beta }(A):=\sum _{\genfrac{}{}{0ex}{}{j\in \beta ,}{j\ne i}}|a_{ji}|.$$

$$r_i(A)=\sum _{j\ne i}|a_{ij}|=r_i^{\alpha }(A)+r_i^{\beta }(A),$$

$$c_i(A)=\sum _{j\ne i}|a_{ji}|=c_i^{\alpha }(A)+c_i^{\beta }(A).$$

$$\lambda \in G^{(\alpha \beta )}(A):=\left(\bigcup _{i\in \alpha }{G}_i^{(\alpha )}(A)\right)\bigcup \left(\bigcup _{j\in \beta }{G}_j^{(\beta )}(A)\right)\bigcup \left(\bigcup _{\genfrac{}{}{0ex}{}{i\in \alpha ,}{j\in \beta }}{G}_{ij}^{(\alpha \beta )}(A)\right),$$

$$G_i^{(\alpha )}(A):=\bigcup _{i\in \alpha }\left\{z\in \mathbb{C}:|a_{ii}-z|\le r_i^{\alpha }(A)\right\},$$

$$G_j^{(\beta )}(A):=\bigcup _{j\in \overline{S}}\left\{z\in \mathbb{C}:|a_{jj}-z|\le r_j^{\beta }(A)\right\},$$

$$G_{ij}^{(\alpha \beta )}(A)$$

$$\left(\bigcap _{i\in \alpha }{G}_i^{(\alpha )}(A)\right)\bigcup \left(\bigcup _{\genfrac{}{}{0ex}{}{i\in \alpha ,}{j\in \beta }}{V}_{ij}^{(\alpha \beta )}(A)\right)$$

$$V_{ij}^{(\alpha \beta )}(A):=\left\{z\in \mathbb{C}:\left(|a_{ii}-z|-r_i^{\alpha }(A)\right)\left(|a_{jj}-z|-r_j^{\beta }(A)\right)\le r_i^{\beta }(A)r_j^{\alpha }(A)\right\},$$

$$A=\left[\begin{array}[]{cccc}1&-0.01&0&7\\ 0.1&0.9&-1.1&0.3\\ -0.11&0.2&0.9&0.2\\ 2&0.4&-0.1&1.1\end{array}\right].$$

$$4.8161,-2.6994,0.8917+0.4921\text{i},0.8917-0.4921\text{i},$$

$$G_2^{(\beta )}(A)=\{z\in \mathbb{C}:|z-0.9|\le 0.3\},$$

$$G_4^{(\beta )}(A)=\{z\in \mathbb{C}:|z-1.1|\le 0.4\}.$$

$$\{0.8917+0.4921\text{i},0.8917-0.4921\text{i}\}\subseteq \left(G_2^{(\beta )}(A)\bigcup G_3^{(\beta )}(A)\right)$$

$$\lambda \in E^{(\alpha \beta )}(A)=\left(\bigcup _{i\in \alpha }{G}_i^{(\alpha )}(A)\right)$$

$${\tilde{E}}_{ij}^{(\alpha \beta )}(A)=\left\{z\in \mathbb{C}:(|z-a_{ii}|+r_i^j(A))\left(|z-a_{jj}|+r_j^{\beta }(A)\right)<|a_{ij}|\left(2|a_{ji}|-r_j^{\alpha }(A)\right)\right\},$$

$${\hat{E}}_{ij}^{(\alpha \beta )}(A)=\left\{z\in \mathbb{C}:(|z-a_{jj}|+r_j^i(A))\left(|z-a_{ii}|+r_i^{\alpha }(A)\right)<|a_{ji}|\left(2|a_{ij}|-r_i^{\beta }(A)\right)\right\}.$$

$$\left(G_{ij}^{(\alpha \beta )}(A)\backslash {\tilde{E}}_{ij}^{(\alpha \beta )}(A)\right)\subseteq G_{ij}^{(\alpha \beta )}(A)$$

$$\left(G_{ij}^{(\alpha \beta )}(A)\backslash {\hat{E}}_{ij}^{(\alpha \beta )}(A)\right)\subseteq G_{ij}^{(\alpha \beta )}(A),$$

$$E^{(\alpha \beta )}(A)\subseteq G^{(\alpha \beta )}(A).$$

$$\lambda \in E^{(\alpha \beta )}(A).$$

$$Ax=\lambda x.$$

$$|\lambda -a_{pp}|\le r_p^{\alpha }(A),or|\lambda -a_{qq}|\le r_q^{\beta }(A),$$

$$|a_{pp}-\lambda |>r_p^{\alpha }(A),and|a_{qq}-\lambda |>r_q^{\beta }(A).$$

$$\left(|\lambda -a_{pp}|-r_p^{\alpha }(A)\right)\left(|\lambda -a_{qq}|-r_q^{\beta }(A)\right)\le r_p^{\beta }(A)r_q^{\alpha }(A).$$

$$(\lambda -a_{pp})x_p-\sum _{\genfrac{}{}{0ex}{}{j\in N,}{j\ne p,q}}{a}_{pj}{x}_j=a_{pq}{x}_q,$$

$$|a_{pq}||x_q|$$

$$(\lambda -a_{qq})x_q-\sum _{\genfrac{}{}{0ex}{}{j\in \alpha ,}{j\ne p}}{a}_{qj}{x}_j-\sum _{\genfrac{}{}{0ex}{}{j\in \beta ,}{j\ne q}}{a}_{qj}{x}_j=a_{qp}{x}_p,$$

Full text

††journal: Arkiv

1 Introduction

Let $A=[a_{ij}]\in\mathbb{C}^{n\times n}$be an $n\times n$ complex matrix with $n\geq 2$ , and $\alpha$ and $\beta$ nonempty index sets satisfying

[TABLE]

throughout the paper. Define partial absolute deleted row sums and column sums as follows:

[TABLE]

[TABLE]

If $\alpha=\{i_{0}\}$, then we assume, by convention, that $r_{i_{0}}^{\alpha}(A)=0$. Similarly $r_{i_{0}}^{\beta}(A)=0$ if $\beta=\{i_{0}\}$. Clearly,

[TABLE]

and

[TABLE]

In [9], Huang et al. provided a so-called $\alpha\beta$-type eigenvalue inclusion region by using the partial absolute deleted row sums $r_{i}^{\alpha}(A)$ and $r_{i}^{\beta}(A)$ as follows.

Theorem 1**.**

[9, Theorem 2.1]** Let $A=[a_{ij}]\in\mathbb{C}^{n\times n}$ and $\lambda$ be an eigenvalue of $A$. Then

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

The region $G^{(\alpha\beta)}(A)$ in Theorem 1 is equivalent to

[TABLE]

provided by Farid in [6], where

[TABLE]

for details, see Remark 2.4, [6]. It seems that each $G_{j}^{(\beta)}(A)$, $j\in\beta$ dose not work, which is illustrated by the matrix

[TABLE]

All eigenvalues of $A$ are

[TABLE]

where $\textbf{i}^{2}=-1$. Take $\alpha=\{1,3\}$ and $\beta=\{2,4\}$, then

[TABLE]

and

[TABLE]

It is easy to see that all eigenvalues are not contained in $G_{2}^{(\beta)}(A)\bigcup G_{4}^{(\beta)}(A)$, see Figure 1. Hence $G_{2}^{(\beta)}(A)\bigcup G_{4}^{(\beta)}(A)$ can be excluded from $G^{(\alpha\beta)}(A)$ for the matrix $A$. On the other hand, by taking $\beta=\{2,3\}$, Figure 2 shows that

[TABLE]

where $G_{2}^{(\beta)}(A)=\{z\in\mathbb{C}:|z-0.9|\leq 1.1\}$ and $G_{3}^{(\beta)}(A)=\{z\in\mathbb{C}:|z-0.9|\leq 0.2\}.$ This means that each $G_{j}^{(\beta)}(A)$ for $j=2,3$ cannot be excluded from $G^{(\alpha\beta)}(A)$ for the this case. This motivates us to find another regions in which any eigenvalue of a matrix $A$ is not contained to exclude them. In this paper, a new inclusion region by excluding some such regions from $G^{(\alpha\beta)}(A)$ is given. This provides a sufficient condition for the non-singularity of a matrix. Numerical examples are also given to verify the corresponding results.

2 Main results

In this section, some regions in which any eigenvalue of a matrix is not contained, are excluded from $G^{(\alpha\beta)}(A)$.

Theorem 2**.**

Let $A=[a_{ij}]\in\mathbb{C}^{n\times n}$ and $\lambda$ be an eigenvalue of $A$. Then

[TABLE]

where $G_{i}^{(\alpha)}(A)$, $G_{j}^{(\beta)}(A)$ and $G^{(\alpha\beta)}_{ij}(A)$ are defined in Theorem 1,

[TABLE]

and

[TABLE]

Furthermore, $E^{(\alpha\beta)}(A)\subseteq G^{(\alpha\beta)}(A)$.

Proof.

Since

[TABLE]

and

[TABLE]

hold for each $i\in\alpha$ and $j\in\beta$, it implies that

[TABLE]

Hence, we only need to prove

[TABLE]

Suppose that $x=[x_{1},x_{2},\ldots,x_{n}]^{T}$ is an eigenvector of $A$ corresponding to the eigenvalue $\lambda$, that is,

[TABLE]

Let $|x_{p}|=\max\limits_{i\in\alpha}\{|x_{i}|\}$ and $|x_{q}|=\max\limits_{j\in\beta}\{|x_{j}|\}$. Obviously, at least one of $|x_{p}|$ and $|x_{q}|$ is nonzero. Note that if

[TABLE]

then $\lambda\in\left(\bigcup\limits_{i\in\alpha}G_{i}^{(\alpha)}(A)\right)\bigcup\left(\bigcup\limits_{j\in\beta}G_{j}^{(\beta)}(A)\right)\subseteq E^{(\alpha\beta)}(A)$. Hence, next we only prove that (1) holds for the case that

[TABLE]

(I) Suppose that $|x_{p}|\geq|x_{q}|$. Similarly to the proof of Theorem 2.1 in [9], we can get that

[TABLE]

Furthermore, consider the $p$-th equation of (2) and rewrite it into

[TABLE]

Taking absolute values on both sides of (5) and using the triangle inequality give

[TABLE]

On the other hand, consider the $q$-th equation of (2) and rewrite it into

[TABLE]

Taking absolute values on both sides of (7) and using the triangle inequality give

[TABLE]

Hence

[TABLE]

If $|x_{q}|>0$, then multiplying (6) with (8) gives

[TABLE]

If $|x_{q}|=0$, then by (8) we have

[TABLE]

and hence (9) also holds. From (4) and (9), it follows that

[TABLE]

where $\overline{\widetilde{E}^{(\alpha\beta)}_{ij}(A)}$ is the complementary of the set $\widetilde{E}^{(\alpha\beta)}_{ij}(A)$.

(II) Suppose that $|x_{q}|\geq|x_{p}|$. Similarly to (I), we can easily obtain that (4) holds, and

[TABLE]

Hence

[TABLE]

From (I) and (II) the conclusion follows.∎

Remark 1**.**

Theorem 2 shows that for any eigenvalue $\lambda$ of a matrix $A,$

[TABLE]

And hence $\widetilde{E}^{(\alpha\beta)}_{ij}(A)$ (or $\widehat{E}^{(\alpha\beta)}_{ij}(A)$) can be excluded from the set $G^{(\alpha\beta)}(A)$. Consider again the matrix $A$ in the introduction section, and take $\alpha=\{1,2\}$ and $\beta=\{3,4\}$. From Figure 3, it is not difficult to see that all eigenvalues

[TABLE]

but are not contained in $\widetilde{E}^{(\alpha\beta)}_{ij}(A)$ (or $\widehat{E}^{(\alpha\beta)}_{ij}(A)$). On the other hand, it should be pointed out here that

[TABLE]

holds in some cases. Also consider the matrix $A$ above, and take $\alpha=\{1,4\}$ and $\beta=\{2,3\}$, in which case (11) holds.

It is well-known that an eigenvalue inclusion region in the complex can give a sufficient condition of the non-singularity of a matrix [4]. Hence, by Theorem 2 we can get easily the following result.

Corollary 1**.**

Let $A=[a_{ij}]\in\mathbb{C}^{n\times n}$. If for each $i\in\alpha$, $j\in\beta$, the following holds:

(I)

[TABLE]

(II)

[TABLE]

(III)

[TABLE]

or

[TABLE]

(IV)

[TABLE]

or

[TABLE]

then $A$ is nonsingular.

3 Conclusions

In this paper, a new eigenvalue inclusion region $E^{(\alpha\beta)}(A)$ is given by excluding some regions $\widetilde{E}^{(\alpha\beta)}_{ij}(A)$ and $\widehat{E}^{(\alpha\beta)}_{ij}(A)$ from the $\alpha\beta$-type eigenvalue inclusion region $G^{(\alpha\beta)}(A).$ As shown in Remark 1, $E^{(\alpha\beta)}(A)=G^{(\alpha\beta)}(A)$ holds in some cases, hence besides $\widetilde{E}^{(\alpha\beta)}_{ij}(A)$ and $\widehat{E}^{(\alpha\beta)}_{ij}(A)$, it is very interesting to find other regions which do not contain any eigenvalues of a matrix to exclude them. Furthermore, for the well-known eigenvalue inclusion regions, such as, regions in [1, 2, 3, 4, 5, 7, 10, 11, 12], we can try to find some regions like $\widetilde{E}^{(\alpha\beta)}_{ij}(A)$ and $\widehat{E}^{(\alpha\beta)}_{ij}(A)$, and exclude them from the corresponding existing eigenvalue inclusion regions to give new regions which capture all eigenvalues of a matrix more precisely.

Acknowledgements

This paper is dedicated to Professor *** on the occasion of his 60th birthday. This work is supported in part by National Natural Science Foundations of China (11601473 and 11361074), the National Natural Science Foundation of Zhejiang Province (LY14A010007, LQ14G010002), Ningbo Natural Science Foundation (2015A610173), and CAS ”Light of West China” Program.