A tighter $Z$-eigenvalue localization set for tensors and its applications

TL;DR

This paper introduces a new, tighter set for localizing Z-eigenvalues of tensors, improving bounds on spectral radii, with theoretical proofs and numerical validation.

Contribution

It presents a novel Z-eigenvalue localization set that is proven to be tighter than existing ones, enhancing spectral radius bounds for certain tensors.

Findings

The new localization set is tighter than previous sets.

A sharper upper bound for the Z-spectral radius is established.

Numerical examples confirm the theoretical improvements.

Abstract

A new $Z$-eigenvalue localization set for tensors is given and proved to be tighter than those presented by Wang \emph{et al}. (Discrete and Continuous Dynamical Systems Series B 22(1): 187-198, 2017) and Zhao (J. Inequal. Appl., to appear, 2017). As an application, 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.

Equations

$$a_{i_1{i}_2\cdots i_m}\in \mathbb{R},$$

$$a_{i_1\cdots i_m}=a_{\pi (i_1\cdots i_m)},\forall \pi \in {\Pi }_m,$$

$$\mathcal{A}{x}^m=\sum _{i_1,i_2,\cdots ,i_m\in N}{a}_{i_1{i}_2\cdots i_m}{x}_{i_1}{x}_{i_2}\cdots x_{i_m}$$

$$\mathcal{A}{x}^{m-1}=\lambda x\text{and}{x}^{T}x=1,$$

$$(\mathcal{A}{x}^{m-1}{)}_i=\sum _{i_2,\cdots ,i_m\in N}{a}_{i{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m}.$$

$$\varrho (\mathcal{A}):=\sup \{|\lambda |:\lambda \in \sigma (\mathcal{A})\}.$$

$$\sigma (\mathcal{A})\subseteq \mathcal{K}(\mathcal{A})=\bigcup _{i\in N}{\mathcal{K}}_i(\mathcal{A}),$$

$${\mathcal{K}}_i(\mathcal{A})=\{z\in \mathbb{C}:|z|\le R_i(\mathcal{A})\},R_i(\mathcal{A})=\sum _{i_2,\cdots ,i_m\in N}|a_{i{i}_2\cdots i_m}|.$$

$$\sigma (\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})=\bigcup _{i\in N}\bigcap _{j\in N,j\ne i}{\mathcal{L}}_{i,j}(\mathcal{A}),$$

$$\mathcal{L}_{i,j}(\mathcal{A})=\left\{z\in{\mathbb{C}}:\big{(}|z|-(R_{i}(\mathcal{A})-|a_{ij\cdots j}|\big{)}|z|\leq|a_{ij\cdots j}|R_{j}(\mathcal{A})\right\}.$$

$$\sigma (\mathcal{A})\subseteq \Psi (\mathcal{A})=\bigcup _{i\in N}\bigcap _{j\in N,j\ne i}{\Psi }_{i,j}(\mathcal{A}),$$

$$\displaystyle\Psi_{i,j}(\mathcal{A})=\left\{z\in\mathbb{C}:\big{(}|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}|z|\leq r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A})\right\},$$

$$r_i^{{\Delta }_j}(\mathcal{A})=\sum _{j\in \{i_2,\cdots ,i_m\}}|a_{i{i}_2\cdots i_m}|,r_i^{{\overline{\Delta }}_j}(\mathcal{A})=\sum _{j\notin \{i_2,\cdots ,i_m\}}|a_{i{i}_2\cdots i_m}|.$$

$$\displaystyle\sigma(\mathcal{A})\subseteq\Omega(\mathcal{A})=\left(\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\hat{\Omega}_{i,j}(\mathcal{A})\right)\bigcup\left(\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\Big{(}\tilde{\Omega}_{i,j}(\mathcal{A})\bigcap\mathcal{K}_{i}(\mathcal{A})\Big{)}\right),$$

$${\hat{\Omega }}_{i,j}(\mathcal{A})=\left\{z\in \mathbb{C}:|z|<r_i^{{\overline{\Delta }}_j}(\mathcal{A}),|z|<r_j^{{\Delta }_j}(\mathcal{A})\right\}$$

$$\displaystyle\tilde{\Omega}_{i,j}(\mathcal{A})=\left\{z\in\mathbb{C}:\big{(}|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}\big{(}|z|-r_{j}^{\Delta_{j}}(\mathcal{A})\big{)}\leq r_{i}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})\right\}.$$

$$\mathcal{A}{x}^{m-1}=\lambda x,\text{and}||x|{|}_2=1.$$

$$\lambda x_t=\sum _{j\in \{i_2,\cdots ,i_m\}}{a}_{t{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m}+\sum _{j\notin \{i_2,\cdots ,i_m\}}{a}_{t{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m}.$$

$$|\lambda ||x_t|$$

$$\displaystyle\big{(}|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}|x_{t}|\leq r_{t}^{\Delta_{j}}(\mathcal{A})|x_{j}|.$$

$$\displaystyle\big{(}|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}\big{(}|\lambda|-r_{j}^{\Delta_{j}}(\mathcal{A})\big{)}\leq 0\leq r_{t}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A}),$$

$$|\lambda ||x_j|$$

$$\displaystyle\big{(}|\lambda|-r_{j}^{\Delta_{j}}(\mathcal{A})\big{)}|x_{j}|\leq r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})|x_{t}|.$$

$$\displaystyle\big{(}|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}\big{(}|\lambda|-r_{j}^{\Delta_{j}}(\mathcal{A})\big{)}\leq r_{t}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A}),$$

$$\Omega (\mathcal{A})\subseteq \Psi (\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})\subseteq \mathcal{K}(\mathcal{A}).$$

$$\big{(}|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}|z|\leq 0\leq r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A}),~{}\forall~{}j\in N,j\neq i,$$

$$|z|\le R_i(\mathcal{A}),$$

$$\displaystyle\big{(}|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}\big{(}|z|-r_{j}^{\Delta_{j}}(\mathcal{A})\big{)}\leq r_{i}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A}).$$

$$\displaystyle\big{(}|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\big{)}|z|\leq 0\leq r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A}),$$

$$|z|\le R_j(\mathcal{A}).$$

Full text

A tighter $Z$-eigenvalue localization set for tensors and its applications

Jianxing Zhao††Corresponding author. E-mail: zjx810204@163.com; zhaojianxing@gzmu.edu.cn (Jianxing Zhao)

*College of Data Science and Information Engineering, Guizhou Minzu University,

Guiyang 550025, P.R. China

Abstract. A new $Z$-eigenvalue localization set for tensors is given and proved to be tighter than those presented by Wang et al. (Discrete and Continuous Dynamical Systems Series B 22(1): 187-198, 2017) and Zhao (J. Inequal. Appl., to appear, 2017). As an application, 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.

Keywords: $Z$-eigenvalue; localization set; nonnegative tensors; spectral radius; weakly symmetric

AMS Subject Classification: 15A18; 15A42; 15A69.

1 Introduction

For a positive integer $n$, $n\geq 2$, $N$ denotes the set $\{1,2,\cdots,n\}$. $\mathbb{C}$ ($\mathbb{R}$) 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

[TABLE]

where $i_{j}\in{N}$ for $j=1,2,\cdots,m$. $\mathcal{A}$ is called nonnegative if $a_{i_{1}i_{2}\cdots i_{m}}\geq 0.$ $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[m,n]}$ is called symmetric [1] if

[TABLE]

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 [2] if the associated homogeneous polynomial

[TABLE]

satisfies $\nabla\mathcal{A}x^{m}=m\mathcal{A}x^{m-1}$. It is shown in [2] 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

[TABLE]

then $\lambda$ is called an $E$-eigenvalue of $\mathcal{A}$ and $x$ an $E$-eigenvector of $\mathcal{A}$ associated with $\lambda$, where $\mathcal{A}x^{m-1}$ is an $n$ dimension vector whose $i$th component is

[TABLE]

If $\lambda$ and $x$ are all real, then $\lambda$ is called a $Z$-eigenvalue of $\mathcal{A}$ and $x$ a $Z$-eigenvector of $\mathcal{A}$ associated with $\lambda$; for details, see [1, 3]. Here, we define the $Z$-spectrum of $\mathcal{A}$, denoted $\sigma(\mathcal{A})$ to be the set of all $Z$-eigenvalues of $\mathcal{A}$. Assume $\sigma(\mathcal{A})\neq 0,$ then the $Z$-spectral radius [2] of $\mathcal{A}$, denoted $\varrho(\mathcal{A})$, is defined as

[TABLE]

Recently, many people have focused on locating all $Z$-eigenvalues of tensors and bounding the $Z$-spectral radius of nonnegative tensors in [2, 4, 5, 6, 7, 8, 9, 11, 10]. In 2017, Wang et al. [4] established the following Ger$\breve{s}$gorin-type $Z$-eigenvalue inclusion theorem for tensors.

Theorem 1**.**

[4, Theorem 3.1]* Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}$. Then*

[TABLE]

where

[TABLE]

To get a tighter $Z$-eigenvalue inclusion set than $\mathcal{K}(\mathcal{A})$, Wang et al. [4] gave the following Brauer-type $Z$-eigenvalue localization set for tensors.

Theorem 2**.**

[4, Theorem 3.2]* Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}$. Then*

[TABLE]

where

[TABLE]

Very recently, Zhao [5] presented another Brauer-type $Z$-eigenvalue localization set for tensors and proved that this set is tighter than those in Theorem 1 and Theorem 2.

Theorem 3**.**

[5, Theorem 3]* Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}$. Then*

[TABLE]

where

[TABLE]

[TABLE]

As we know, one can use eigenvalue inclusion sets to obtain the upper bound of the spectral radius of nonnegative tensors; for details, see [4, 12, 13, 14, 15]. Therefore, the main aim of this paper is to give a new $Z$-eigenvalue inclusion set for tensors and prove that the new set is tighter than those in Theorems 1-3. And as an application, a new upper bound for the $Z$-spectral radius of weakly symmetric nonnegative tensors is obtained and proved to be sharper than some existing upper bounds.

2 Main results

In this section, we give a new Brauer-type $Z$-eigenvalue localization set for tensors, and establish the comparison between the new set with those in Theorems 1-3.

Theorem 4**.**

Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

Let $\lambda$ be a $Z$-eigenvalue of $\mathcal{A}$ with corresponding $Z$-eigenvector $x=(x_{1},\cdots,x_{n})^{T}\in{\mathbb{C}}^{n}\backslash\{0\}$, i.e.,

[TABLE]

Let $|x_{t}|=\max\limits_{i\in N}|x_{i}|$. Obviously, $0<|x_{t}|^{m-1}\leq|x_{t}|\leq 1.$ For $\forall~{}j\in N,j\neq t$, from (1), we have

[TABLE]

Taking modulus in the above equation and using the triangle inequality gives

[TABLE]

i.e.,

[TABLE]

If $|x_{j}|=0$, then $|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\leq 0$ as $|x_{t}|>0$. When $|z|-r_{j}^{\Delta_{j}}(\mathcal{A})\geq 0$, we have

[TABLE]

which implies $\lambda\in\bigcap\limits_{j\in N,j\neq t}\tilde{\Omega}_{t,j}(\mathcal{A})\subseteq\Omega(\mathcal{A})$ from the arbitrariness of $j$. When $|z|-r_{j}^{\Delta_{j}}(\mathcal{A})<0$, from the arbitrariness of $j$, we have $\lambda\in\bigcap\limits_{j\in N,j\neq t}\hat{\Omega}_{t,j}(\mathcal{A})\subseteq\Omega(\mathcal{A})$.

Otherwise, $|x_{j}|>0$. From (1), we can get

[TABLE]

i.e.,

[TABLE]

By (2), it is not difficult to see $\lambda\in\mathcal{K}_{t}(\mathcal{A})$. When $|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\geq 0$ or $|\lambda|-r_{j}^{\Delta_{j}}(\mathcal{A})\geq 0$ holds, multiplying (2) with (3) and noting that $|x_{t}||x_{j}|>0$, we have

[TABLE]

which implies \lambda\in\bigcap\limits_{j\in N,j\neq t}\big{(}\tilde{\Omega}_{t,j}(\mathcal{A})\bigcap\mathcal{K}_{t}(\mathcal{A})\big{)}\subseteq\Omega(\mathcal{A}) from the arbitrariness of $j$. And when $|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})<0$ and $|\lambda|-r_{j}^{\Delta_{j}}(\mathcal{A})<0$ hold, from the arbitrariness of $j$, we have $\lambda\in\bigcap\limits_{j\in N,j\neq t}\hat{\Omega}_{i,j}(\mathcal{A})\subseteq\Omega(\mathcal{A})$. Hence, the conclusion $\sigma(\mathcal{A})\subseteq\Omega(\mathcal{A})$ follows immediately from what we have proved. ∎

Next, a comparison theorem is given for Theorems 1-4.

Theorem 5**.**

Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}$. Then

[TABLE]

Proof.

From Theorem 5 in [5], we have $\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\subseteq\mathcal{K}(\mathcal{A})$. Hence, here only $\Omega(\mathcal{A})\subseteq\Psi(\mathcal{A})$ is proved. Let $z\in\Omega(\mathcal{A})$. Then $z\in\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\hat{\Omega}_{i,j}(\mathcal{A})$ or z\in\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\Big{(}\tilde{\Omega}_{i,j}(\mathcal{A})\bigcap\mathcal{K}_{i}(\mathcal{A})\Big{)}. We next divide the proof into two cases.

Case I: If $z\in\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\hat{\Omega}_{i,j}(\mathcal{A}),$ then there is one index $i\in N$ such that $|z|<r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})$ and $|z|<r_{j}^{\Delta_{j}}(\mathcal{A}),\forall~{}j\in N,j\neq i.$ Then, it is easy to see that

[TABLE]

which implies that $z\in\bigcap\limits_{j\in N,j\neq i}\Psi_{i,j}(\mathcal{A})\subseteq\Psi(\mathcal{A})$. This implies $\Omega(\mathcal{A})\subseteq\Psi(\mathcal{A})$.

Case II: If z\in\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\Big{(}\tilde{\Omega}_{i,j}(\mathcal{A})\bigcap\mathcal{K}_{i}(\mathcal{A})\Big{)}, then there is one index $i\in N$, for any $j\in N,j\neq i$, such that

[TABLE]

and

[TABLE]

(i) If $r_{i}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})=0$, then $|z|\leq r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})$ or $|z|\leq r_{j}^{\Delta_{j}}(\mathcal{A}).$ When $|z|\leq r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})$, we have

[TABLE]

which implies that $z\in\bigcap\limits_{j\in N,j\neq i}\Psi_{i,j}(\mathcal{A})\subseteq\Psi(\mathcal{A})$ from the arbitrariness of $j$. When $|z|\leq r_{j}^{\Delta_{j}}(\mathcal{A}),$ we have

[TABLE]

From (4), we can get

[TABLE]

Multiplying (6) and (7), we have

[TABLE]

which also implies that $z\in\bigcap\limits_{j\in N,j\neq i}\Psi_{i,j}(\mathcal{A})\subseteq\Psi(\mathcal{A})$, consequently, $\Omega(\mathcal{A})\subseteq\Psi(\mathcal{A}).$

(ii) If $r_{i}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})>0$, then dividing both sides by $r_{i}^{\Delta_{j}}(\mathcal{A})r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})$ in (5), we have

[TABLE]

From (4), we can get (7) and furthermore $\frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})}{r_{i}^{\Delta_{j}}(\mathcal{A})}\leq 1.$ When $\frac{|z|-r_{j}^{\Delta_{j}}(\mathcal{A})}{r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})}\leq 1$, then (6) holds. Multiplying (6) and (7), we can get (8), which implies that $z\in\bigcap\limits_{j\in N,j\neq i}\Psi_{i,j}(\mathcal{A})\subseteq\Psi(\mathcal{A})$, consequently, $\Omega(\mathcal{A})\subseteq\Psi(\mathcal{A}).$

And when $\frac{|z|-r_{j}^{\Delta_{j}}(\mathcal{A})}{r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})}>1$, we can obtain $|z|>R_{j}(\mathcal{A}).$ Let $a=|z|,b=r_{j}^{\Delta_{j}}(\mathcal{A})-|a_{jj\cdots j}|,c=|a_{jj\cdots j}|$ and $d=r_{j}^{\overline{\Delta}_{j}}(\mathcal{A})$. By Lemma 2.3 in [12], we have

[TABLE]

If $|z|>r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})$, by (9) and (10), we have

[TABLE]

equivalently,

[TABLE]

which implies that $z\in\bigcap\limits_{j\in N,j\neq i}\Psi_{i,j}(\mathcal{A})\subseteq\Psi(\mathcal{A})$ from the arbitrariness of $j$. If $|z|\leq r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})$, we have

[TABLE]

This also leads to $z\in\bigcap\limits_{j\in N,j\neq i}\Psi_{i,j}(\mathcal{A})\subseteq\Psi(\mathcal{A})$, consequently, $\Omega(\mathcal{A})\subseteq\Psi(\mathcal{A}).$ The conclusion follows from Case I and Case II. ∎

Remark 1**.**

Theorem 5 shows that the set $\Omega(\mathcal{A})$ in Theorem 4 is tighter than $\mathcal{K}(\mathcal{A})$ in Theorem 1, $\mathcal{L}(\mathcal{A})$ in Theorem 2 and $\Psi(\mathcal{A})$ in Theorem 3, that is, $\Omega(\mathcal{A})$ can capture all $Z$-eigenvalues of $\mathcal{A}$ more precisely than $\mathcal{K}(\mathcal{A})$, $\mathcal{L}(\mathcal{A})$ and $\Psi(\mathcal{A})$.***

Now, an example is given to verify the fact in Remark 2.

Example 1**.**

Let $\mathcal{A}=(a_{ijkl})\in{\mathbb{R}}^{[4,2]}$ be a symmetric tensor defined by

[TABLE]

By computation, we get that all the $Z$-eigenvalues of $\mathcal{A}$ are $-0.2044,-0.2044,5.0000$ and $5.0000$. By Theorem 1, we have

[TABLE]

By Theorem 2, we have

[TABLE]

By Theorem 3, we have

[TABLE]

By Theorem 4, we have

[TABLE]

The $Z$-eigenvalue inclusion sets $\mathcal{K}(\mathcal{A})$, $\mathcal{L}(\mathcal{A})$, $\Psi(\mathcal{A})$, $\Omega(\mathcal{A})$ and the exact $Z$-eigenvalues are drawn in Figure 1, where $\mathcal{K}(\mathcal{A})$, $\mathcal{L}(\mathcal{A})$, $\Psi(\mathcal{A})$ and $\Omega(\mathcal{A})$ are represented by black dashed boundary, green solid boundary, blue point line boundary and red solid boundary, respectively. The exact eigenvalues are plotted by black “$+$”. It is easy to see $\sigma(\mathcal{A})\subseteq\Omega(\mathcal{A})\subset\Psi(\mathcal{A})\subset\mathcal{L}(\mathcal{A})\subset\mathcal{K}(\mathcal{A})$, that is, $\Omega(\mathcal{A})$ can capture all $Z$-eigenvalues of $\mathcal{A}$ more precisely than $\Psi(\mathcal{A})$, $\mathcal{L}(\mathcal{A})$ and $\mathcal{K}(\mathcal{A})$.

3 A sharper upper bound for the $Z$-spectral radius of weakly symmetric nonnegative tensors

As the $Z$-spectral radius of weakly symmetric nonnegative tensors plays a fundamental role in the symmetric best rank-one approximation [10, 16], recently, many people focus on bounding the $Z$-spectral radius of weakly symmetric nonnegative tensors. As an application of the set in Theorem 4, we in this section give a sharper upper bound for the $Z$-spectral radius of weakly symmetric nonnegative tensors.

Theorem 6**.**

Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in{\mathbb{R}}^{[m,n]}$ be a weakly symmetric nonnegative tensor. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

By Lemma 4.4 in [4], we know that $\varrho(\mathcal{A})$ is a $Z$-eigenvalue of $\mathcal{A}$. By Theorem 4, we have

[TABLE]

If $\varrho(\mathcal{A})\in\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\hat{\Omega}_{i,j}(\mathcal{A}),$ then there is one index $i\in N$ such that

[TABLE]

Then we have \varrho(\mathcal{A})\leq\min\limits_{j\in N,j\neq i}\min\big{\{}r_{i}^{\overline{\Delta}_{j}}(\mathcal{A}),r_{j}^{\Delta_{j}}(\mathcal{A})\big{\}}. Furthermore, we have

[TABLE]

If \varrho(\mathcal{A})\in\bigcup\limits_{i\in N}\bigcap\limits_{j\in N,j\neq i}\Big{(}\tilde{\Omega}_{i,j}(\mathcal{A})\bigcap\mathcal{K}_{i}(\mathcal{A})\Big{)}, then there is one index $i\in N$, for any $j\in N,j\neq i$, such that

[TABLE]

and

[TABLE]

Solving $\varrho(\mathcal{A})$ in above inequality gives

[TABLE]

Combining (11) and (12), and by the arbitrariness of $j$, we have

[TABLE]

The conclusion follows from what we have proved. ∎

By Corollary 4.1 of [4], Theorem 6 of [5] and Theorem 5, the following comparison theorem can be derived easily.

Theorem 7**.**

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 6 is smaller than those in Theorem 5 of [5], Theorem 4.5 of [4] and Corollary 4.5 of [6], that is,

[TABLE]

Finally, we show that the upper bound in Theorem 6 is smaller than those in [4, 5, 6, 7, 8, 9, 10] by the following example.

Example 2**.**

Let $\mathcal{A}=(a_{ijk})\in{\mathbb{R}}^{[3,3]}$ be a weakly symmetric nonnegative tensor with entries defined as follows:*

[TABLE]

By Corollary 4.5 of [6] and Theorem 3.3 of [7], we both have

[TABLE]

By Theorem 3.5 of [8], we have

[TABLE]

By Theorem 4.6 of [4], we have

[TABLE]

By Theorem 4.5 of [4] and Theorem 6 of [9], we both have

[TABLE]

By Theorem 4.7 of [4], we have

[TABLE]

By Theorem 2.9 of [10], we have

[TABLE]

By Theorem 5 of [5], we obtain

[TABLE]

By Theorem 6, we obtain

[TABLE]

This example shows that the bound in Theorem 6 is the smallest.**

Remark 2**.**

From Example 1, it is not difficult to see that the upper bound in Theorem 6 could reach the true value of $\varrho(\mathcal{A})$ in some cases.***

4 Conclusion

In this paper, we present a new $Z$-eigenvalue localization set $\Omega(\mathcal{A})$ and prove that this set is tighter than those in [4, 5]. As an application, we obtain a new upper bound $\Omega_{max}(\mathcal{A})$ for the $Z$-spectral radius of weakly symmetric nonnegative tensors, and show that this bound is sharper than those in [4, 5, 6, 7, 8, 9, 10] in some cases by a numerical example.

Acknowledgments

This work is supported by National Natural Science Foundations of China (Grant No.11501141), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073) and Natural Science Programs of Education Department of Guizhou Province (Grant No.[2016]066).