Abstract
A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al. (Linear Algebra Appl. 481:36-53, 2015) and Huang et al. (J. Inequal. Appl. 2016:254, 2016). As an application of this set, new bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al., the advantage of our results is that, without considering the selection of nonempty proper subsets S of \(N=\{1,2,\ldots,n\}\), we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors. Finally, numerical examples are given to verify the theoretical results.
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1 Introduction
For a positive integer n, \(n\geq2\), N denotes the set \(\{1,2,\ldots ,n\}\). \(\mathbb{C}\) (respectively, \(\mathbb{R}\)) denotes the set of all complex (respectively, real) numbers. We call \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\) a complex (real) tensor of order m dimension n, denoted by \(\mathbb{C}^{[m,n]}(\mathbb {R}^{[m,n]})\), if
where \(i_{j}\in{N}\) for \(j=1,2,\ldots,m\). \(\mathcal{A}\) is called reducible if there exists a nonempty proper index subset \(\mathbb{J}\subset N\) such that
If \(\mathcal{A}\) is not reducible, then we call \(\mathcal{A}\) irreducible [3].
Given a tensor \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb {C}^{[m,n]}\), if there are \(\lambda\in\mathbb{C}\) and \(x=(x_{1},x_{2},\ldots,x_{n})^{T}\in\mathbb{C}\backslash\{0\}\) such that
then λ is called an eigenvalue of \(\mathcal{A}\) and x an eigenvector of \(\mathcal{A}\) associated with λ, where \(\mathcal{A}x^{m-1}\) is an n dimension vector whose ith component is
and
If λ and x are all real, then λ is called an H-eigenvalue of \(\mathcal {A}\) and x an H-eigenvector of \(\mathcal{A}\) associated with λ; see [4, 5]. Moreover, the spectral radius \(\rho(\mathcal{A})\) of \(\mathcal{A}\) is defined as
where \(\sigma(\mathcal{A})\) is the spectrum of \(\mathcal{A}\), that is, \(\sigma(\mathcal{A})=\{\lambda:\lambda \mbox{ is an eigenvalue of } \mathcal{A}\}\); see [3, 6].
A real tensor \(\mathcal{A}\) is called an \(\mathcal{M}\)-tensor if there exist a nonnegative tensor \(\mathcal{B}\) and a positive number \(\alpha>\rho(\mathcal{B})\) such that \(\mathcal {A}=\alpha\mathcal{I}-\mathcal{B}\), where \(\mathcal{I}\) is called the unit tensor with its entries
Denote by \(\tau(\mathcal{A})\) the minimal value of the real part of all eigenvalues of an \(\mathcal{M}\)-tensor \(\mathcal{A}\). Then \(\tau (\mathcal{A})>0\) is an eigenvalue of \(\mathcal{A}\) with a nonnegative eigenvector. If \(\mathcal{A}\) is irreducible, then \(\tau(\mathcal{A})\) is the unique eigenvalue with a positive eigenvector [7–9].
Recently, many people have focused on locating eigenvalues of tensors and using obtained eigenvalue inclusion theorems to determine the positive definiteness of an even-order real symmetric tensor or to give the lower and upper bounds for the spectral radius of nonnegative tensors and the minimum eigenvalue of \(\mathcal {M}\)-tensors. For details, see [1, 2, 10–14].
In 2015, Li et al. [1] proposed the following Brauer-type eigenvalue localization set for tensors.
Theorem 1
[1], Theorem 6
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{C}}^{[m,n]}\). Then
where
To reduce computations, Huang et al. [2] presented an S-type eigenvalue localization set by breaking N into disjoint subsets S and S̄, where S̄ is the complement of S in N.
Theorem 2
[2], Theorem 3.1
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{C}}^{[m,n]}\), S be a nonempty proper subset of N, S̄ be the complement of S in N. Then
Based on Theorem 2, Huang et al. [2] obtained the following lower and upper bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors.
Theorem 3
[2], Theorem 3.6
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[m, n]}\) be an \(\mathcal{M}\)-tensor, S be a nonempty proper subset of N, S̄ be the complement of S in N. Then
where
The main aim of this paper is to give a new eigenvalue inclusion set for tensors and prove that this set is tighter than those in Theorems 1 and 2 without considering the selection of S. And then we use this set to obtain new lower and upper bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors and prove that new bounds are sharper than those in Theorem 3.
2 Main results
Now, we give a new eigenvalue inclusion set for tensors and establish the comparison between this set with those in Theorems 1 and 2.
Theorem 4
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{C}}^{[m,n]}\). Then
Proof
For any \(\lambda\in\sigma(\mathcal{A})\), let \(x=(x_{1},\ldots ,x_{n})^{T}\in{\mathbb{C}}^{n}\backslash\{0\}\) be an associated eigenvector, i.e.,
Let \(\vert x_{p}\vert =\max\{\vert x_{i}\vert :i \in N\}\). Then \(\vert x_{p}\vert >0\). For any \(j\in N, j\neq p\), then from (1) we have
and
equivalently,
and
Solving \(x_{p}^{m-1}\) from (2) and (3), we get
Taking absolute values and using the triangle inequality yields
Furthermore, by \(\vert x_{p}\vert >0\), we have
which implies that \(\lambda\in\Delta_{p}^{j}(\mathcal{A})\). From the arbitrariness of j, we have \(\lambda\in\bigcap_{j\in N, j\neq p}\Delta_{p}^{j}(\mathcal{A})\). Furthermore, we have \(\lambda\in\bigcup_{i\in N}\bigcap_{j\in N, j\neq i}\Delta _{i}^{j}(\mathcal{A})\). The conclusion follows. □
Next, a comparison theorem is given for Theorems 1, 2 and 4.
Theorem 5
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{C}}^{[m,n]}\), S be a nonempty proper subset of N. Then
Proof
By Theorem 3.2 in [2], \(\Delta^{S}(\mathcal{A})\subseteq\Delta(\mathcal{A})\). Here, only \(\Delta^{\cap}(\mathcal{A})\subseteq\Delta^{S}(\mathcal{A})\) is proved. Let \(z\in\Delta^{\cap}(\mathcal{A})\), then there exists some \(i_{0}\in N\) such that \(z\in\Delta_{i_{0}}^{j}(\mathcal {A}),\forall j\in N, j\neq i_{0}\). Let S̄ be the complement of S in N. If \(i_{0}\in S\), then taking \(j\in\bar{S}\), obviously, \(z\in\bigcup_{i_{0}\in S,j\in\bar{S}}\Delta_{i_{0}}^{j}(\mathcal{A})\subseteq \Delta^{S}(\mathcal{A})\). If \(i_{0}\in\bar{S}\), then taking \(j\in S\), obviously, \(z\in\bigcup_{i_{0}\in\bar{S},j\in S}\Delta_{i_{0}}^{j}(\mathcal {A})\subseteq\Delta^{S}(\mathcal{A})\). The conclusion follows. □
Remark 1
Theorem 5 shows that the set \(\Delta^{\cap}(\mathcal{A})\) in Theorem 4 is tighter than those in Theorems 1 and 2, that is, \(\Delta^{\cap}(\mathcal{A})\) can capture all eigenvalues of \(\mathcal {A}\) more precisely than \(\Delta(\mathcal{A})\) and \(\Delta^{S}(\mathcal{A})\).
In the following, we give new lower and upper bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors.
Theorem 6
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[m, n]}\) be an irreducible \(\mathcal{M}\)-tensor. Then
Proof
Let \(x=(x_{1},x_{2},\ldots,x_{n})^{T}\) be an associated positive eigenvector of \(\mathcal{A}\) corresponding to \(\tau(\mathcal{A})\), i.e.,
(I) Let \(x_{q}=\min\{x_{i}:i\in N\}\). For any \(j\in N, j\neq q\), we have by (4) that
and
equivalently,
and
Solving \(x_{q}^{m-1}\) by (5) and (6), we get
From Theorem 2.1 in [9], we have \(\tau(\mathcal{A})\leq\min_{i\in N}a_{i\cdots i}\) and
Hence,
From \(x_{q}>0\), we have
equivalently,
that is,
Solving for \(\tau(\mathcal{A})\) gives
For the arbitrariness of j, we have \(\tau(\mathcal{A})\leq\min_{j\neq q}L_{qj}(\mathcal{A})\). Furthermore, we have
(II) Let \(x_{p}=\max\{x_{i}:i\in N\}\). Similar to (I), we have
The conclusion follows from (I) and (II). □
Similar to the proof of Theorem 3.6 in [2], we can extend the results of Theorem 6 to a more general case.
Theorem 7
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[m, n]}\) be an \(\mathcal{M}\)-tensor. Then
By Theorems 3, 6 and 7 in [13], the following comparison theorem is obtained easily.
Theorem 8
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[m, n]}\) be an \(\mathcal{M}\)-tensor, S be a nonempty proper subset of N, S̄ be the complement of S in N. Then
where \(R_{i}(\mathcal{A})=\sum_{i_{2},\ldots,i_{m}\in N}a_{ii_{2}\cdots i_{m}}\).
Remark 2
Theorem 8 shows that the bounds in Theorem 7 are shaper than those in Theorem 3, Theorem 2.1 of [9] and Theorem 4 of [13] without considering the selection of S, which is also the advantage of our results.
3 Numerical examples
In this section, two numerical examples are given to verify the theoretical results.
Example 1
Let \(\mathcal{A}=(a_{ijk})\in\mathbb{R}^{[3, 4]}\) be an irreducible \(\mathcal{M}\)-tensor with elements defined as follows:
By Theorem 2.1 in [9], we have
By Theorem 4 in [13], we have
By Theorem 3, we have
By Theorem 7, we have
In fact, \(\tau(\mathcal{A})=14.4049\). Hence, this example verifies Theorem 8 and Remark 2, that is, the bounds in Theorem 7 are sharper than those in Theorem 3, Theorem 2.1 of [9] and Theorem 4 of [13] without considering the selection of S.
Example 2
Let \(\mathcal{A}=(a_{ijkl})\in\mathbb{R}^{[4, 2]}\) be an \(\mathcal {M}\)-tensor with elements defined as follows:
other \(a_{ijkl}=0\). By Theorem 7, we have
In fact, \(\tau(\mathcal{A})=4\).
4 Conclusions
In this paper, we give a new eigenvalue inclusion set for tensors and prove that this set is tighter than those in [1, 2]. As an application, we obtain new lower and upper bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors and prove that the new bounds are sharper than those in [2, 9, 13]. Compared with the results in [2], the advantage of our results is that, without considering the selection of S, we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of \(\mathcal{M}\)-tensors.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11361074, 11501141), the Foundation of Guizhou Science and Technology Department (Grant No. [2015]2073) and the Natural Science Programs of Education Department of Guizhou Province (Grant No. [2016]066).
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Zhao, J., Sang, C. An eigenvalue localization set for tensors and its applications. J Inequal Appl 2017, 59 (2017). https://doi.org/10.1186/s13660-017-1331-1
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DOI: https://doi.org/10.1186/s13660-017-1331-1