Exclusion sets in Dashnic–Zusmanovich localization sets

Abstract

A new eigenvalue localization set is given by excluding some proper subsets that do not contain any eigenvalues of matrices from Dashnic–Zusmanovich localization sets. As an application, a sufficient condition for non-singularity of matrices is obtained. In order to locate all eigenvalues of matrices precisely, another set including two positive integers s and k is presented. By adjusting the parameters s and k, one can locate all eigenvalues and judge the non-singularity of matrices accurately.

1 Introduction

Let n be a positive integer, $n\geq 2$ and $N=\{1,2,\ldots ,n\}$. $\mathbb{C}$ (or, respectively, $\mathbb{R}$) denotes the set of all complex (or, respectively, real) numbers, $\mathbb{C}^{n\times n}$ (or, respectively, $\mathbb{R}^{n\times n}$) denotes the set of all ${n\times n}$ complex (or, respectively, real) matrices and I stands for the identity matrix. Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ and $\sigma (A)$ be the set of all eigenvalues of A. Eigenvalue problems of matrices has a wide range of practical applications, such as image restoration [1], linear and multilinear algebra [2], higher order Markov chains [3], etc. In order to locate all eigenvalues of matrices, the authors in [4,5,6,7,8,9,10,11,12,13] found some regions including all eigenvalues of matrices in the complex plane. The first work is due to Geršgorin, who presented such a region called the Geršgorin disk theorem [5], which consists of n disks centered at the diagonal elements of the matrix.

Theorem 1

([5, Geršgorin set])

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$. Then

$$\begin{aligned}& \sigma (A) \subseteq \varGamma (A) = \bigcup_{i\in N} \varGamma _{i}(A), \end{aligned}$$

where

$$\begin{aligned}& \varGamma _{i}(A)= \bigl\{ z\in \mathbb{C}: \vert z-a_{ii} \vert \leq r_{i}(A) \bigr\} \end{aligned}$$

and

$$\begin{aligned}& r_{i}(A)=\sum_{j\in N, j\neq i} \vert a_{ij} \vert . \end{aligned}$$

Although Geršgorin set is concise, its result is not accurate enough. Hence, tighter sets than $\varGamma (A)$ are conjectured till now. The Dashnic–Zusmanovich localization set, which is tighter than Geršgorin set, provided by Dashnic–Zusmanovich (DZ) [14], is described as follows.

Theorem 2

([14, DZ set])

Let $A=[a_{ij}]\in {\mathbb{C}} ^{n\times n}$. Then

$$\begin{aligned}& \sigma (A)\subseteq D(A)=\bigcap_{i\in N}\bigcup _{j\in N, j\neq i} D _{ij}(A), \end{aligned}$$

where

$$\begin{aligned}& D_{ij}(A)=\bigl\{ z\in \mathbb{C}: \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert -r^{i}_{j}(A) \bigr)\leq r _{i}(A) \vert a_{ji} \vert \bigr\} \end{aligned}$$

and

$$\begin{aligned}& r_{j}^{i}(A)=r_{j}(A)- \vert a_{ji} \vert . \end{aligned}$$

It is generally accepted that an eigenvalue localization set is connected with one kind of non-singular matrices [7, 8]. The non-singularity criterions for matrices derived from the Geršgorin set in Theorem 1 and DZ set in Theorem 2 are as follows.

Theorem 3

([7])

If $A=[a_{ij}]\in {\mathbb{C}} ^{n\times n}$ is an SDD matrix, i.e., for each $i\in N$, we have

$$\begin{aligned}& \vert a_{ii} \vert >r_{i}(A), \end{aligned}$$

then it is non-singular.

Theorem 4

([7])

If $A=[a_{ij}]\in {\mathbb{C}} ^{n\times n}$ is a DZ matrix, i.e., there exists an index $i\in N$, for all $j\in N$, $j\neq i$,

$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert -r_{j}^{i}(A)\bigr)>r_{i}(A) \vert a_{ji} \vert , \end{aligned}$$

then it is non-singular.

By excluding some proper subsets that contain no eigenvalues of matrices from some existing eigenvalue localization sets, the authors in [15,16,17,18] obtained some tighter sets and some sufficient conditions for non-singularity of matrices. Inspired by these effective results, we in this paper first present a new eigenvalue localization set by excluding some proper subsets from DZ set, and we obtain a new sufficient condition for non-singularity of matrices. In order to precisely locate all eigenvalues of matrices, we in Sect. 3 present another set which includes two positive integers s and k, and show by an example that, by adjusting the parameters s and k, one can locate all eigenvalues and judge the non-singularity of matrices accurately.

2 Exclusion sets in Dashnic–Zusmanovich localization sets

In this section, we present a new eigenvalue inclusion set by excluding some proper subsets that contain no eigenvalues of matrices from DZ sets, and, as an application, we provide a sufficient condition to judge the non-singularity of matrices.

Theorem 5

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$. Then

$$\begin{aligned}& \sigma (A)\subseteq \varOmega (A)=\bigcap_{i\in N} \bigcup_{j\in N, j \neq i} \bigl(D_{ij}(A)\setminus \varOmega _{ij}(A) \bigr), \end{aligned}$$

where

$$\begin{aligned}& \varOmega _{ij}(A)=\bigl\{ z\in \mathbb{C}: \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert +r_{j}^{i}(A)\bigr)< \bigl(2 \vert a _{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert \bigr\} . \end{aligned}$$

Proof

Let λ be an eigenvalue of A and $x=(x_{1},\ldots ,x_{n})^{T} \in \mathbb{C}^{n}\setminus \{0\}$ be its eigenvector. Then

$$\begin{aligned}& Ax=\lambda x. \end{aligned}$$

(1)

Let $|x_{p}|=\max _{j\in N}|x_{j}|$. Obviously, $|x_{p}|>0$. For any $i\in N$, $i\neq p$, the pth formula of (1) can be written as

$$\begin{aligned}& (\lambda -a_{pp})x_{p}=\sum _{s\neq p,i}a_{ps}x_{s}+a_{pi}x_{i}. \end{aligned}$$

(2)

Taking the absolute value of (2) and using the triangle inequality, we have

$$ \vert \lambda -a_{pp} \vert \vert x_{p} \vert \leq \sum_{s\neq p,i} \vert a_{ps} \vert \vert x_{s} \vert + \vert a_{pi} \vert \vert x _{i} \vert \leq r_{p}^{i}(A) \vert x_{p} \vert + \vert a_{pi} \vert \vert x_{i} \vert , $$

i.e.,

$$ \bigl( \vert \lambda -a_{pp} \vert -r_{p}^{i}(A)\bigr) \vert x_{p} \vert \leq \vert a_{pi} \vert \vert x_{i} \vert . $$

(3)

If $|x_{i}|>0$, then from the ith formula of (1), i.e.,

$$\begin{aligned}& (\lambda -a_{ii})x_{i}=\sum _{t\neq i}a_{it}x_{t}, \end{aligned}$$

(4)

we have

$$\begin{aligned}& \vert \lambda -a_{ii} \vert \vert x_{i} \vert \leq r_{i}(A) \vert x_{p} \vert . \end{aligned}$$

(5)

Multiplying (3) and (5) and noting that $|x_{p}||x_{i}|>0$, we have

$$\begin{aligned}& \vert \lambda -a_{ii} \vert \bigl( \vert \lambda -a_{pp} \vert -r_{p}^{i}(A)\bigr) \leq r_{i}(A) \vert a_{pi} \vert . \end{aligned}$$

(6)

If $|x_{i}|=0$ in (3), then $|\lambda -a_{pp}|-r _{p}^{i}(A)\leq 0$ as $|x_{p}|>0$, which implies that (6) also holds. Therefore, $\lambda \in D_{ip}(A)$. On the other hand, from (2) and (4), we have

$$\begin{aligned}& a_{pi}x_{i}=(\lambda -a_{pp})x_{p}- \sum_{s\neq p,i}a_{ps}x_{s} \end{aligned}$$

and

$$\begin{aligned}& a_{ip}x_{p}=(\lambda -a_{ii})x_{i}- \sum_{t\neq i,p}a_{it}x_{t}, \end{aligned}$$

which leads to

$$\begin{aligned} \vert a_{pi} \vert \vert x_{i} \vert \leq & \vert \lambda -a_{pp} \vert \vert x_{p} \vert +\sum_{s\neq p,i} \vert a _{ps} \vert \vert x_{s} \vert \leq \vert \lambda -a_{pp} \vert \vert x_{p} \vert +\sum _{s\neq p,i} \vert a_{ps} \vert \vert x _{p} \vert \\ =&\bigl( \vert \lambda -a_{pp} \vert +r_{p}^{i}(A)\bigr) \vert x_{p} \vert \end{aligned}$$

(7)

and

$$\begin{aligned}& \vert a_{ip} \vert \vert x_{p} \vert \leq \vert \lambda -a_{ii} \vert \vert x_{i} \vert + \sum_{t\neq i,p} \vert a_{it} \vert \vert x _{t} \vert \leq \vert \lambda -a_{ii} \vert \vert x_{i} \vert +r_{i}^{p}(A)) \vert x_{p} \vert , \end{aligned}$$

i.e.,

$$\begin{aligned}& \bigl( \vert a_{ip} \vert -r_{i}^{p}(A) \bigr) \vert x_{p} \vert \leq \vert \lambda -a_{ii} \vert \vert x_{i} \vert . \end{aligned}$$

(8)

By (7) and (8), we have

$$\begin{aligned}& \vert a_{pi} \vert \bigl(2 \vert a_{ip} \vert -r_{i}(A)\bigr)\leq \vert \lambda -a_{ii} \vert \bigl( \vert \lambda -a_{pp} \vert +r _{p}^{i}(A) \bigr), \end{aligned}$$

which implies that $\lambda \notin \varOmega _{ip}(A)$. Hence, $\lambda \in (D_{ip}(A)\setminus \varOmega _{ip}(A) )$.

For some certain $i\in N$, $i\neq p$, since we do not know which p is appropriate to λ, we can only conclude that

$$\begin{aligned}& \lambda \in \bigcup_{p\in N, i \neq p} \bigl(D_{ip}(A) \setminus \varOmega _{ip}(A) \bigr). \end{aligned}$$

Furthermore, by the arbitrariness of i, we have

$$\begin{aligned}& \lambda \in \bigcap_{i\in N}\bigcup _{j\in N, j\neq i} \bigl(D_{ij}(A) \setminus \varOmega _{ij}(A) \bigr). \end{aligned}$$

The conclusion follows. □

Now, a comparison theorem for Theorems 1, 2 and 5 is obtained.

Theorem 6

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$. Then

$$\begin{aligned}& \varOmega (A)\subseteq D(A)\subseteq \varGamma (A). \end{aligned}$$

Proof

It is showed in Theorem 7 of [19] that $D(A)\subseteq \varGamma (A)$. For any $i,j\in N$, $j\neq i$, by $(D_{ij}(A)\setminus \varOmega _{ij}(A)) \subseteq D_{ij}(A)$, obviously,

$$\begin{aligned}& \bigcap_{i\in N}\bigcup_{j\in N, i\neq j} \bigl(D_{ij}(A)\setminus \varOmega _{ij}(A)\bigr)\subseteq \bigcap_{i\in N}\bigcup_{j\in N, i\neq j}D_{ij}(A). \end{aligned}$$

Hence, $\varOmega (A)\subseteq D(A)$ holds. □

Next, based on the fact that $\det (A)=0$ if and only if $0\in \sigma (A)$ for a matrix A, we can obtain the following condition for judging the non-singularity of matrices.

Corollary 1

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$. If there exists $i\in N$, for any $j\in N$, $j\neq i$, either

$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert -r_{j}^{i}(A) \bigr)>r_{i}(A) \vert a_{ji} \vert , \end{aligned}$$

(9)

or

$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert +r_{j}^{i}(A)\bigr)< \bigl(2 \vert a_{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert , \end{aligned}$$

(10)

then A is non-singular.

Proof

Let $0\in \sigma (A)$. By Theorem 5, we have $0\in \varOmega (A)$, i.e., for each $i\in N$, there exists $j\in N$, $j \neq i$, such that

$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert -r_{j}^{i}(A)\bigr)\leq r_{i}(A) \vert a_{ji} \vert \end{aligned}$$

and

$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert +r_{j}^{i}(A)\bigr)\geq \bigl(2 \vert a_{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert . \end{aligned}$$

This contradicts (9) and (10). Consequently, $0\notin \sigma (A)$, that is, A is non-singular. □

Remark 1

(i) Let i and j be any two elements of N, and $i\neq j$. If $(2|a_{ij}|-r_{i}(A))|a_{ji}|>0$, then, by $|a_{ij}|\leq r_{i}(A)$, we have $0<2|a_{ij}|-r_{i}(A)\leq r_{i}(A)$ and

$$\begin{aligned} \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert -r^{i}_{j}(A)\bigr) \leq & \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert +r ^{i}_{j}(A) \bigr) \\ < &\bigl(2 \vert a_{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert \\ \leq & r_{i}(A) \vert a_{ji} \vert , \end{aligned}$$

which implies that

$$\begin{aligned}& \varOmega _{ij}(A)\subseteq D_{ij}(A) \quad \mbox{and}\quad \bigl(D_{ij}(A)\setminus \varOmega _{ij}(A) \bigr)\subseteq D_{ij}(A). \end{aligned}$$

(11)

That is to say, $\varOmega _{ij}(A)$ is well defined. If $(2|a_{ij}|-r _{i}(A))|a_{ji}|\leq 0$, then $\varOmega _{ij}(A)=\emptyset$. Obviously, (11) also holds. Here, $\varOmega _{ij}(A)$ is called the exclusion set of $D_{ij}(A)$.

(ii) As has been shown in [7, 8], the wider the class of non-singular matrices is, the tighter eigenvalue localization set it will lead to. Obviously, this conclusion, in turn, holds true. By Corollary 1, one can conclude that the conditions of Corollary 1 for judging the non-singularity of matrices are weaker than those in Theorems 3 and 4.

Next, an example is given to show that $\varOmega (A)$ can catch all eigenvalues of a matrix A more precisely than $\varGamma (A)$ and $D(A)$, and that Theorems 3 and 4 cannot be used to judge the non-singularity of A in some cases, but Corollary 1 works better.

Example 1

Let

$$A=\begin{bmatrix} 11&4+\mathbf{{i}}&0&15-\mathbf{{i}} \\ -2-\mathbf{{i}}&10&5-\mathbf{{i}}&0 \\ 0&6&12+\mathbf{{i}}&4 \\ 16&2&0&11 \end{bmatrix}. $$

By computations, all eigenvalues of $\mathcal{A}$ are $26.4293-0.7552\mathbf{{i}}$, $-4.4930+0.6450\mathbf{{i}}$, $16.0853+0.0217\mathbf{{i}}$, $5.9784+1.0885\mathbf{{i}}$. Next, the eigenvalue location and the determination of non-singularity for A are considered.

(I) Eigenvalue inclusion sets for A.

From Theorem 1, we have

$$\begin{aligned}& \varGamma (A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \leq 19.1564 \bigr\} . \end{aligned}$$

From Theorem 2, we have

$$\begin{aligned}& D(A)=D_{24}(A)= \bigl\{ z\in \mathbb{C}: \vert z-10 \vert \bigl( \vert z-11 \vert -16 \bigr)\leq 15.0662 \bigr\} . \end{aligned}$$

From Theorem 5, we have

$$\begin{aligned}& \varOmega (A)= \bigl[D_{12}(A)\cup D_{13}(A) \cup \bigl(D_{24}(A)\setminus \varOmega _{41}(A)\bigr) \cup \bigl(D_{43}(A)\cap \varOmega _{14}(A)\bigr)\bigr], \end{aligned}$$

where

$$\begin{aligned}& D_{12}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-10 \vert -5.0990 \bigr)\leq 42.8356 \bigr\} , \\& D_{13}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-12-\mathbf{{i}} \vert -10 \bigr)\leq 0 \bigr\} , \\& D_{43}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-12-\mathbf{{i}} \vert -18 \bigr)\leq 72 \bigr\} , \\& \varOmega _{14}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-11 \vert +2 \bigr)\leq 175.568 \bigr\} , \\& \varOmega _{41}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-11 \vert +4.1231 \bigr)\leq 210.4662 \bigr\} . \end{aligned}$$

The eigenvalue localization sets $\varGamma (A)$, $D(A)$ and $\varOmega (A)$ are drawn in Fig. 1, respectively, as red boundary, black boundary and yellow zones, and all eigenvalues are plotted as red asterisks. It is obvious that

$$\begin{aligned} \sigma (A)\subseteq \varOmega (A)\subseteq D(A)\subset \varGamma (A), \end{aligned}$$

another way of stating it is, $\varOmega (A)$ can capture all eigenvalues of A more precisely than $D(A)$ and $\varGamma (A)$.

Figure 1

Comparisons of $\varOmega (A)$, $D(A)$ and $\varGamma (A)$

(II) The determination for non-singularity of A.

From Fig. 1, one can see that $0\in \varGamma (A)$ and $0\in D(A)$, but $0\notin \varOmega (A)$, that is, the sets $\varGamma (A)$ and $D(A)$ cannot be used to judge the non-singularity of A. However, by $0\notin \varOmega (A)$, one can conclude that A is non-singular.

Furthermore, as

$$\begin{aligned} \vert a_{11} \vert =11< 19.1564= r_{1}(A) \end{aligned}$$

and

$$\begin{aligned}& \vert a_{11} \vert \bigl( \vert a_{44} \vert -r_{4}^{1}(A) \bigr)=99.0000 < 306.5024=r_{1}(A) \vert a _{41} \vert , \\& \vert a_{22} \vert \bigl( \vert a_{44} \vert -r_{4}^{2}(A) \bigr)=-50.0000 < 14.6702=r_{2}(A) \vert a _{42} \vert , \\& \vert a_{33} \vert \bigl( \vert a_{44} \vert -r_{4}^{3}(A) \bigr)=-84.2912 < 0=r_{3}(A) \vert a _{43} \vert , \\& \vert a_{44} \vert \bigl( \vert a_{33} \vert -r_{3}^{4}(A) \bigr)= 66.4575 < 72.0000 =r _{4}(A) \vert a_{34} \vert , \end{aligned}$$

we know that the conditions in Theorems 3 and 4 do not hold, that is, Theorems 3 and 4 do not work. However, by

$$\begin{aligned}& \vert a_{11} \vert \bigl( \vert a_{22} \vert -r_{2}^{1}(A) \bigr)=53.9108 > 42.8350=r_{1}(A) \vert a _{21} \vert , \\& \vert a_{11} \vert \bigl( \vert a_{33} \vert -r_{3}^{1}(A) \bigr)=22.4575 > 0= r_{1}(A) \vert a _{31} \vert , \\& \vert a_{11} \vert \bigl( \vert a_{44} \vert +r_{4}^{1}(A) \bigr)=143.0000 < 174.5631=\bigl(2 \vert a _{14} \vert - r_{1}(A)\bigr) \vert a_{41} \vert , \end{aligned}$$

we know that there exists an index $i=1$, for $j=2,3,4$, either (9) or (10) holds. Then we can conclude the non-singularity of A by Corollary 1.

3 An eigenvalue localization set with parameters

In this section, an eigenvalue localization set with parameters and its applications is considered.

Theorem 7

Let $A=sI-B\in \mathbb{C}^{n\times n}$ and $s\in \mathbb{C}$. Given an arbitrary positive integer k, then

$$\begin{aligned} \sigma (A)\subseteq \varOmega _{k}^{s}(A)=\bigcap _{i\in N} \bigcup_{j\in N , j\neq i} \bigl(D^{s}_{ij}\bigl(B^{k}\bigr)\setminus \varOmega ^{s} _{ij}\bigl(B^{k}\bigr) \bigr), \end{aligned}$$

where

$$\begin{aligned} D^{s}_{ij}\bigl(B^{k}\bigr)= \bigl\{ z\in \mathbb{C}: \bigl\vert (s-z)^{k}-\bigl(B^{k} \bigr)_{ii} \bigr\vert \bigl( \bigl\vert (s-z)^{k}- \bigl(B ^{k}\bigr)_{jj} \bigr\vert -r_{j}^{i} \bigl(B^{k}\bigr)\bigr)\leq r_{i}\bigl(B^{k} \bigr) \bigl\vert \bigl(B^{k}\bigr)_{ji} \bigr\vert \bigr\} \end{aligned}$$

and

$$\begin{aligned} \varOmega ^{s}_{ij}\bigl(B^{k}\bigr) =& \bigl\{ z\in \mathbb{C}: \bigl\vert (s-z)^{k}-\bigl(B^{k} \bigr)_{ii} \bigr\vert \bigl( \bigl\vert (s-z)^{k}- \bigl(B ^{k}\bigr)_{jj} \bigr\vert +r_{j}^{i} \bigl(B^{k}\bigr)\bigr) \\ &{}< \bigl(2 \bigl\vert \bigl(B^{k}\bigr)_{ij} \bigr\vert -r_{i}\bigl(B^{k}\bigr)\bigr) \bigl\vert \bigl(B^{k}\bigr)_{ji} \bigr\vert \bigr\} . \end{aligned}$$

Proof

Let $\lambda \in \sigma (A)$. Given an arbitrary positive integer k, suppose that $\lambda \notin \varOmega _{k}^{s}(A)$. Then there exists $i\in N$, for all $j\in N$, $j\neq i$, $\lambda \notin (D^{s}_{ij}(B ^{k})\setminus \varOmega ^{s}_{ij}(B^{k}) )$, that is, $\lambda \notin D^{s}_{ij}(B^{k})$ or $\lambda \in \varOmega ^{s}_{ij}(B^{k})$, i.e.,

$$\begin{aligned} \bigl\vert (s-\lambda )^{k}-\bigl(B^{k} \bigr)_{ii} \bigr\vert \bigl( \bigl\vert (s-\lambda )^{k}-\bigl(B^{k}\bigr)_{jj} \bigr\vert -r_{j} ^{i}\bigl(B^{k}\bigr)\bigr)> r_{i}\bigl(B^{k}\bigr) \bigl\vert \bigl(B^{k}\bigr)_{ji} \bigr\vert , \end{aligned}$$

or

$$\begin{aligned} \bigl\vert (s-\lambda )^{k}-\bigl(B^{k} \bigr)_{ii} \bigr\vert \bigl( \bigl\vert (s-\lambda )^{k}-\bigl(B^{k}\bigr)_{jj} \bigr\vert +r_{j} ^{i}\bigl(B^{k}\bigr)\bigr)< \bigl(2 \bigl\vert \bigl(B^{k}\bigr)_{ij} \bigr\vert -r_{i}\bigl(B^{k}\bigr)\bigr) \bigl\vert \bigl(B^{k}\bigr)_{ji} \bigr\vert . \end{aligned}$$

By Corollary 1, $(s-\lambda )^{k}I-B^{k}$ is non-singular, which implies that $(s-\lambda )^{k}$ is not an eigenvalue of $B^{k}$.

On the other hand, let x be an eigenvector corresponding to λ, i.e., $Ax=\lambda x$, which leads to that $Bx=(s-\lambda )x$. Furthermore, we have $B^{k}x=(s-\lambda )^{k}x$, which implies that $(s -\lambda )^{k}$ is an eigenvalue of $B^{k}$. This is a contradiction. Hence, $\lambda \in \varOmega _{k}^{s}(A)$. □

By Theorem 7, the following general method for non-singularity of matrices is obtained.

Corollary 2

Let $A=sI-B\in \mathbb{C}^{n\times n}$ and $s\in \mathbb{C}$. Given an arbitrary positive integer k, if there exists an index i, for all $j\in N$, $j\neq i$, either

$$\begin{aligned} \bigl( \bigl\vert s^{k}-\bigl(B^{k} \bigr)_{ii} \bigr\vert -r_{i}^{j} \bigl(B^{k}\bigr)\bigr) \bigl\vert s^{k}- \bigl(B^{k}\bigr)_{jj} \bigr\vert > \bigl\vert \bigl(B^{k}\bigr)_{ij} \bigr\vert r _{j} \bigl(B^{k}\bigr), \end{aligned}$$

(12)

or

$$\begin{aligned} \bigl( \bigl\vert s^{k}-\bigl(B^{k} \bigr)_{ii} \bigr\vert +r_{i}^{j} \bigl(B^{k}\bigr)\bigr) \bigl\vert s^{k}- \bigl(B^{k}\bigr)_{jj} \bigr\vert < \bigl\vert \bigl(B^{k}\bigr)_{ij} \bigr\vert \bigl(2 \bigl\vert \bigl(B ^{k}\bigr)_{ji} \bigr\vert -r_{j} \bigl(B^{k}\bigr)\bigr), \end{aligned}$$

(13)

then A is non-singular.

Remark 2

Taking $k=1$ and arbitrary complex number s in Theorem 7 and Corollary 2, then they degenerate, respectively, into Theorem 5 and Corollary 1. Hence, Theorem 7 and Corollary 2 can be viewed as generalizations of Theorem 5 and Corollary 1. Furthermore, by selecting appropriate parameters s and k, one may locate all eigenvalues and judge the non-singularity of matrices precisely.

Example 2

Consider again the matrix A in Example 1. Taking $s=11$ and $k=2$, the sets $\varOmega (A)$ and $\varOmega _{2}^{11}(A)$ are drawn in Fig. 2, respectively, as yellow zones and blue zones. All eigenvalues are plotted as red asterisks. It can be seen from Fig. 2 that the set $\varOmega _{2}^{11}(A)$ can be used to more precisely locate all eigenvalues of A and judge the non-singularity of A.

Figure 2

Comparisons of $\varOmega (A)$ and $\varOmega _{2}^{11}(A)$

4 Conclusions

In this paper, we present a new eigenvalue inclusion set $\varOmega (A)$ by excluding some proper subsets that contain no eigenvalues of matrices from Dashnic–Zusmanovich localization sets $D(A)$, and we prove that $\varOmega (A)$ is tighter than $D(A)$ for a matrix A. After that, by the set $\varOmega (A)$, we obtain a sufficient condition for judging the non-singularity of matrices. To catch all eigenvalues of matrices precisely, we put forward another eigenvalue inclusion set $\varOmega _{k}^{s}(A)$ including two parameters s and k. By selecting these two positive integer s and k appropriately, one can locate all eigenvalues of matrices and judge the non-singularity of matrices precisely. However, how to choose s and k to make $\varOmega _{k}^{s}(A)$ works better? This question at present is far from being solved.