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.
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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
where
and
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
where
and
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
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 \(j\in N\), \(j\neq i\),
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
where
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
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
Taking the absolute value of (2) and using the triangle inequality, we have
i.e.,
If \(|x_{i}|>0\), then from the ith formula of (1), i.e.,
we have
Multiplying (3) and (5) and noting that \(|x_{p}||x_{i}|>0\), we have
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
and
which leads to
and
i.e.,
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
Furthermore, by the arbitrariness of i, we have
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
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,
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
or
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
and
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
which implies that
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
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
From Theorem 2, we have
From Theorem 5, we have
where
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
another way of stating it is, \(\varOmega (A)\) can capture all eigenvalues of A more precisely than \(D(A)\) 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
and
we know that the conditions in Theorems 3 and 4 do not hold, that is, Theorems 3 and 4 do not work. However, by
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
where
and
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.,
or
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
or
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.
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.
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Acknowledgements
The authors are grateful to the referees for their useful and constructive suggestions.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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This work is supported by Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066), National Natural Science Foundation of China (Grant No. 11501141), and Natural Science Foundation of Guizhou Minzu University.
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Zhao, J., She, L. Exclusion sets in Dashnic–Zusmanovich localization sets. J Inequal Appl 2019, 228 (2019). https://doi.org/10.1186/s13660-019-2179-3
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DOI: https://doi.org/10.1186/s13660-019-2179-3