Abstract
A new Z-eigenvalue localization set for tensors is given and proved to be tighter than those in the work of Wang et al. (Discrete Contin. Dyn. Syst., Ser. B 22(1):187-198, 2017). Based on this set, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. 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 \(\{1,2,\ldots ,n\}\). \(\mathbb{C}\) (\(\mathbb{R}\)) denotes the set of all complex (real) numbers. We call \(\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{m}})\) a real tensor of order m dimension n, denoted by \(\mathbb{R}^{[m,n]}\), if
where \(i_{j}\in{N}\) for \(j=1,2,\ldots,m\). \(\mathcal{A}\) is called nonnegative if \(a_{i_{1}i_{2}\cdots i_{m}}\geq0\). \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[m,n]}\) is called symmetric [2] if
where \(\Pi_{m}\) is the permutation group of m indices. \(\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{m}})\in\mathbb{R}^{[m,n]}\) is called weakly symmetric [3] if the associated homogeneous polynomial
satisfies \(\nabla\mathcal{A}x^{m}=m\mathcal{A}x^{m-1}\). It is shown in [3] that a symmetric tensor is necessarily weakly symmetric, but the converse is not true in general.
Given a tensor \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb {R}^{[m,n]}\), if there are \(\lambda\in\mathbb{C}\) and \(x=(x_{1},x_{2}\cdots,x_{n})^{T}\in\mathbb{C}^{n}\backslash\{0\}\) such that
then λ is called an E-eigenvalue of \(\mathcal{A}\) and x an E-eigenvector of \(\mathcal{A}\) associated with λ, where \(\mathcal{A}x^{m-1}\) is an n dimension vector whose ith component is
If λ and x are all real, then λ is called a Z-eigenvalue of \(\mathcal {A}\) and x a Z-eigenvector of \(\mathcal{A}\) associated with λ; for details, see [2, 4].
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\). We define the Z-spectrum of \(\mathcal{A}\), denoted \(\sigma(\mathcal{A})\) to be the set of all Z-eigenvalues of \(\mathcal{A}\). Assume \(\sigma(\mathcal{A})\neq0\), then the Z-spectral radius [3] of \(\mathcal{A}\), denoted \(\varrho (\mathcal{A})\), is defined as
Recently, much literature has focused on locating all Z-eigenvalues of tensors and bounding the Z-spectral radius of nonnegative tensors in [1, 5–10]. It is well known that one can use eigenvalue inclusion sets to obtain the lower and upper bounds of the spectral radius of nonnegative tensors; for details, see [1, 11–14]. Therefore, the main aim of this paper is to give a tighter Z-eigenvalue inclusion set for tensors, and use it to obtain a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors.
In 2017, Wang et al. [1] established the following Gers̆gorin-type Z-eigenvalue inclusion theorem for tensors.
Theorem 1
[1], Theorem 3.1
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\). Then
where
To get a tighter Z-eigenvalue inclusion set than \(\mathcal{K}(\mathcal{A})\), Wang et al. [1] gave the following Brauer-type Z-eigenvalue localization set for tensors.
Theorem 2
[1], Theorem 3.2
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\). Then
where
In this paper, we continue this research on the Z-eigenvalue localization problem for tensors and its applications. We give a new Z-eigenvalue inclusion set for tensors and prove that the new set is tighter than those in Theorem 1 and Theorem 2. As an application of this set, we obtain a new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors, which is sharper than some existing upper bounds.
2 Main results
In this section, we give a new Z-eigenvalue localization set for tensors, and establish the comparison between this set with those in Theorem 1 and Theorem 2. For simplification, we denote
For \(\forall i,j\in N, j\neq i\), let
Then \(R_{i}(\mathcal{A})=r_{i}^{\Delta_{j}}(\mathcal{A})+r_{i}^{\overline{\Delta }_{j}}(\mathcal{A})\).
Theorem 3
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\). Then
where
Proof
Let λ be a Z-eigenvalue of \(\mathcal{A}\) with corresponding Z-eigenvector \(x=(x_{1},\ldots,x_{n})^{T}\in{\mathbb{C}}^{n}\backslash \{0\}\), i.e.,
Assume \(|x_{t}|=\max_{i \in N}|x_{i}|\), then \(0<|x_{t}|^{m-1}\leq|x_{t}|\leq1\). For \(\forall j\in N\), \(j\neq t\), from (1), we have
Taking the modulus in the above equation and using the triangle inequality give
i.e.,
If \(|x_{j}|=0\), by \(|x_{t}|>0\), we have \(|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\leq0\). Then
Obviously, \(\lambda\in\Psi_{t,j}(\mathcal{A})\). Otherwise, \(|x_{j}|>0\). From (1), we have
Multiplying (2) with (3) and noting that \(|x_{t}||x_{j}|>0\), we have
which implies that \(\lambda\in\Psi_{t,j}(\mathcal{A})\). From the arbitrariness of j, we have \(\lambda\in\bigcap_{j\in N, j\neq t}\Psi_{t,j}(\mathcal{A})\). Furthermore, we have \(\lambda\in\bigcup_{i\in N}\bigcap_{j\in N, j\neq i}\Psi _{i,j}(\mathcal{A})\). □
Next, a comparison theorem is given for Theorem 1, Theorem 2 and Theorem 3.
Theorem 4
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\). Then
Proof
By Corollary 3.1 in [1], \(\mathcal{L}(\mathcal{A})\subseteq \mathcal{K}(\mathcal{A})\) holds. Here, we only prove \(\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\). Let \(z\in\Psi(\mathcal{A})\). Then there exists \(i\in N\), such that \(z\in\Psi_{i,j}(\mathcal{A})\), \(\forall j\in N\), \(j\neq i\), that is,
Next, we divide our subject in two cases to prove \(\Psi(\mathcal {A})\subseteq\mathcal{L}(\mathcal{A})\).
Case I: If \(r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A})=0\), then we have
which implies that \(z\in\bigcap_{j\in N, j\neq i}\mathcal{L}_{i,j}(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})\) from the arbitrariness of j, consequently, \(\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\).
Case II: If \(r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A})>0\), then dividing both sides by \(r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal {A})\) in (4), we have
which implies
or
Let \(a=|z|\), \(b=r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\), \(c=r_{i}^{\Delta _{j}}(\mathcal{A})-|a_{ij\cdots j}|\) and \(d=|a_{ij\cdots j}|\). When (6) holds and \(d=|a_{ij\cdots j}|>0\), from Lemma 2.2 in [11], we have
Furthermore, from (5) and (8), we have
equivalently,
which implies that \(z\in\bigcap_{j\in N, j\neq i}\mathcal{L}_{i,j}(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})\) from the arbitrariness of j, consequently, \(\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\). When (6) holds and \(d=|a_{ij\cdots j}|=0\), we have
and furthermore
This also implies \(\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\).
On the other hand, when (7) holds, we only prove \(\Psi (\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\) under the case that
From (9), we have \(\frac{a}{b+c+d}=\frac{|z|}{R_{i}(\mathcal{A})}>1\). When (7) holds and \(|a_{ji\cdots i}|>0\), by Lemma 2.3 in [11], we have
By (7), Lemma 2.2 in [11] and similar to the proof of (8), we have
Multiplying (10) and (11), we have
equivalently,
This implies \(z\in\bigcap_{i\in N, i\neq j}\mathcal{L}_{j,i}(\mathcal {A})\subseteq\mathcal{L}(\mathcal{A})\) and \(\Psi(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})\) from the arbitrariness of i. When (7) holds and \(|a_{ji\cdots i}|=0\), we can obtain
and
This also implies \(\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\). The conclusion follows from Case I and Case II. □
Remark 1
Theorem 4 shows that the set \(\Psi(\mathcal{A})\) in Theorem 3 is tighter than \(\mathcal{K}(\mathcal{A})\) in Theorem 1 and \(\mathcal{L}(\mathcal{A})\) in Theorem 2, that is, \(\Psi(\mathcal{A})\) can capture all Z-eigenvalues of \(\mathcal{A}\) more precisely than \(\mathcal{K}(\mathcal{A})\) and \(\mathcal{L}(\mathcal{A})\).
Now, we give an example to show that \(\Psi(\mathcal{A})\) is tighter than \(\mathcal{K}(\mathcal{A})\) and \(\mathcal{L}(\mathcal{A})\).
Example 1
Let \(\mathcal{A}=(a_{ijkl})\in{\mathbb{R}}^{[4,2]}\) be a symmetric tensor defined by
By computation, we see that all the Z-eigenvalues of \(\mathcal{A}\) are −0.5000, 0 and 2.7000. By Theorem 1, we have
By Theorem 2, we have
By Theorem 3, we have
The Z-eigenvalue inclusion sets \(\mathcal{K}(\mathcal{A})\), \(\mathcal {L}(\mathcal{A})\), \(\Psi(\mathcal{A})\) and the exact Z-eigenvalues are drawn in Figure 1, where \(\mathcal{K}(\mathcal{A})\) and \(\mathcal{L}(\mathcal{A})\) are represented by blue dashed boundary, \(\Psi(\mathcal{A})\) is represented by red solid boundary and the exact eigenvalues are plotted by ‘+’, respectively. It is easy to see \(\sigma(\mathcal{A})\subseteq\Psi(\mathcal {A})\subset\mathcal{L}(\mathcal{A})\subseteq\mathcal{K}(\mathcal{A})\), that is, \(\Psi(\mathcal{A})\) can capture all Z-eigenvalues of \(\mathcal{A}\) more precisely than \(\mathcal{L}(\mathcal{A})\) and \(\mathcal{K}(\mathcal{A})\).
3 A new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors
As an application of the results in Section 2, we in this section give a new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors.
Theorem 5
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\) be a weakly symmetric nonnegative tensor. Then
where
Proof
From Lemma 4.4 in [1], we know that \(\varrho(\mathcal{A})\) is the largest Z-eigenvalue of \(\mathcal{A}\). It follows from Theorem 3 that there exists \(i\in N\) such that
Solving \(\varrho(\mathcal{A})\) in (12) gives
From the arbitrariness of j, we have \(\varrho(\mathcal{A})\leq\min_{j\in N, j\neq i}\Phi_{i,j}(\mathcal{A})\). Furthermore, \(\varrho(\mathcal{A})\leq\max_{i\in N}\min_{j\in N, j\neq i}\Phi _{i,j}(\mathcal{A})\). □
By Theorem 4, Theorem 4.5 and Corollary 4.1 in [1], the following comparison theorem can be derived easily.
Theorem 6
Let \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}\) be a weakly symmetric nonnegative tensor. Then the upper bound in Theorem 5 is sharper than those in Theorem 4.5 of [1] and Corollary 4.5 of [5], that is,
Finally, we show that the upper bound in Theorem 5 is sharper than those in [1, 5–8, 10] by the following example.
Example 2
Let \(\mathcal{A}=(a_{ijk})\in{\mathbb{R}}^{[3,3]}\) with the entries defined as follows:
It is not difficult to verify that \(\mathcal{A}\) is a weakly symmetric nonnegative tensor. By both Corollary 4.5 of [5] and Theorem 3.3 of [6], we have
By Theorem 3.5 of [7], we have
By Theorem 4.6 of [1], we have
By both Theorem 4.5 of [1] and Theorem 6 of [8], we have
By Theorem 4.7 of [1], we have
By Theorem 2.9 of [10], we have
By Theorem 5, we obtain
which shows that the upper bound in Theorem 5 is sharper.
4 Conclusions
In this paper, we present a new Z-eigenvalue localization set \(\Psi (\mathcal{A})\) and prove that this set is tighter than those in [1]. As an application, we obtain a new upper bound \(\max_{i\in N}\min_{j\in N, j\neq i}\Phi_{i,j}(\mathcal{A})\) for the Z-spectral radius of weakly symmetric nonnegative tensors, and we show that this bound is sharper than those in [1, 5–8, 10] in some cases by a numerical example.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 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. A new Z-eigenvalue localization set for tensors. J Inequal Appl 2017, 85 (2017). https://doi.org/10.1186/s13660-017-1363-6
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DOI: https://doi.org/10.1186/s13660-017-1363-6