A new Z-eigenvalue localization set for tensors

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.

Introduction

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

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

where $i_j\in N$ for $j=1,2,\dots ,m$. $\mathcal{A}$ is called nonnegative if $a_{i_1{i}_2\cdots i_m}\ge 0$. $\mathcal{A}=(a_{i_1\cdots i_m})\in {\mathbb{R}}^{[m,n]}$ is called symmetric [2] if

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

where ${\mathrm{\Pi }}_m$ 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

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

satisfies $\mathrm{\nabla }\mathcal{A}{x}^m=m\mathcal{A}{x}^{m-1}$. 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 $x={(x_1,x_2\cdots ,x_n)}^T\in {\mathbb{C}}^n\mathrm{\setminus }\{0\}$ such that

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

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

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

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 $\sigma (\mathcal{A})\ne 0$, then the Z-spectral radius [3] of $\mathcal{A}$, denoted $\varrho (\mathcal{A})$, is defined as

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

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

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

where

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

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

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

where

$${\mathcal{L}}_{i,j}(\mathcal{A})=\{z\in \mathbb{C}:(|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|))|z|\le |a_{ij\cdots j}|R_j(\mathcal{A})\}.$$

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.

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

$$\begin{array}{c}{\mathrm{\Delta }}_j=\{(i_2,i_3,\dots ,i_m):i_k=j\text{ for some }k\in \{2,\dots ,m\},\text{where }j,i_2,\dots ,i_m\in N\},\hfill \\ {\bar{\mathrm{\Delta }}}_j=\{(i_2,i_3,\dots ,i_m):i_k\ne j\text{ for any }k\in \{2,\dots ,m\},\text{where }j,i_2,\dots ,i_m\in N\}.\hfill \end{array}$$

For $\mathrm{\forall }i,j\in N,j\ne i$, let

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

Then $R_i(\mathcal{A})=r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})+r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})$.

Theorem 3

Let $\mathcal{A}=(a_{i_1\cdots i_m})\in {\mathbb{R}}^{[m,n]}$. Then

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

where

$${\mathrm{\Psi }}_{i,j}(\mathcal{A})=\{z\in \mathbb{C}:(|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))|z|\le r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})\}.$$

Proof

Let λ be a Z-eigenvalue of $\mathcal{A}$ with corresponding Z-eigenvector $x={(x_1,\dots ,x_n)}^T\in {\mathbb{C}}^n\mathrm{\setminus }\{0\}$, i.e.,

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

Assume $|x_t|={max}_{i\in N}|x_i|$, then $0<{|x_t|}^{m-1}\le |x_t|\le 1$. For $\mathrm{\forall }j\in N$, $j\ne t$, from (1), we have

$$\lambda x_t=\sum _{(i_2,\dots ,i_m)\in {\mathrm{\Delta }}_j}{a}_{t{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m}+\sum _{(i_2,\dots ,i_m)\in {\bar{\mathrm{\Delta }}}_j}{a}_{t{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m}.$$

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

$$\begin{array}{rcl}|\lambda ||x_t|& \le & \sum _{(i_2,\dots ,i_m)\in {\mathrm{\Delta }}_j}|a_{t{i}_2\cdots i_m}||x_{i_2}|\cdots |x_{i_m}|+\sum _{(i_2,\dots ,i_m)\in {\bar{\mathrm{\Delta }}}_j}|a_{t{i}_2\cdots i_m}||x_{i_2}|\cdots |x_{i_m}|\\ & \le & \sum _{(i_2,\dots ,i_m)\in {\mathrm{\Delta }}_j}|a_{t{i}_2\cdots i_m}||x_j|+\sum _{(i_2,\dots ,i_m)\in {\bar{\mathrm{\Delta }}}_j}|a_{t{i}_2\cdots i_m}||x_t|\\ & =& r_t^{{\mathrm{\Delta }}_j}(\mathcal{A})|x_j|+r_t^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})|x_t|,\end{array}$$

i.e.,

$$(|\lambda |-r_t^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))|x_t|\le r_t^{{\mathrm{\Delta }}_j}(\mathcal{A})|x_j|.$$ 2

If $|x_j|=0$, by $|x_t|>0$, we have $|\lambda |-r_t^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})\le 0$. Then

$$(|\lambda |-r_t^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))|\lambda |\le 0\le r_t^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A}).$$

Obviously, $\lambda \in {\mathrm{\Psi }}_{t,j}(\mathcal{A})$. Otherwise, $|x_j|>0$. From (1), we have

$$|\lambda ||x_j|\le \sum _{i_2,\dots ,i_m\in N}|a_{j{i}_2\cdots i_m}||x_{i_2}|\cdots |x_{i_m}|\le \sum _{i_2,\dots ,i_m\in N}|a_{j{i}_2\cdots i_m}|{|x_t|}^{m-1}\le R_j(\mathcal{A})|x_t|.$$ 3

Multiplying (2) with (3) and noting that $|x_t||x_j|>0$, we have

$$(|\lambda |-r_t^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))|\lambda |\le r_t^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A}),$$

which implies that $\lambda \in {\mathrm{\Psi }}_{t,j}(\mathcal{A})$. From the arbitrariness of j, we have $\lambda \in {\bigcap }_{j\in N,j\ne t}{\mathrm{\Psi }}_{t,j}(\mathcal{A})$. Furthermore, we have $\lambda \in {\bigcup }_{i\in N}{\bigcap }_{j\in N,j\ne i}{\mathrm{\Psi }}_{i,j}(\mathcal{A})$. □

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

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

Proof

By Corollary 3.1 in [1], $\mathcal{L}(\mathcal{A})\subseteq \mathcal{K}(\mathcal{A})$ holds. Here, we only prove $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$. Let $z\in \mathrm{\Psi }(\mathcal{A})$. Then there exists $i\in N$, such that $z\in {\mathrm{\Psi }}_{i,j}(\mathcal{A})$, $\mathrm{\forall }j\in N$, $j\ne i$, that is,

$$(|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))|z|\le r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A}),\mathrm{\forall }j\in N,j\ne i.$$ 4

Next, we divide our subject in two cases to prove $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$.

Case I: If $r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})=0$, then we have

$$(|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|))|z|\le (|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))|z|\le r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})=0\le |a_{ij\cdots j}|R_j(\mathcal{A}),$$

which implies that $z\in {\bigcap }_{j\in N,j\ne i}{\mathcal{L}}_{i,j}(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$ from the arbitrariness of j, consequently, $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$.

Case II: If $r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})>0$, then dividing both sides by $r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})$ in (4), we have

$$\frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}\frac{|z|}{R_j(\mathcal{A})}\le 1,$$ 5

which implies

$$\frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}\le 1,$$ 6

or

$$\frac{|z|}{R_j(\mathcal{A})}\le 1.$$ 7

Let $a=|z|$, $b=r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})$, $c=r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})-|a_{ij\cdots j}|$ and $d=|a_{ij\cdots j}|$. When (6) holds and $d=|a_{ij\cdots j}|>0$, from Lemma 2.2 in [11], we have

$$\frac{|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|)}{|a_{ij\cdots j}|}=\frac{a-(b+c)}{d}\le \frac{a-b}{c+d}=\frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}.$$ 8

Furthermore, from (5) and (8), we have

$$\frac{|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|)}{|a_{ij\cdots j}|}\frac{|z|}{R_j(\mathcal{A})}\le \frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}\frac{|z|}{R_j(\mathcal{A})}\le 1,$$

equivalently,

$$(|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|))|z|\le |a_{ij\cdots j}|R_j(\mathcal{A}),$$

which implies that $z\in {\bigcap }_{j\in N,j\ne i}{\mathcal{L}}_{i,j}(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$ from the arbitrariness of j, consequently, $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$. When (6) holds and $d=|a_{ij\cdots j}|=0$, we have

$$|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})-r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})\le 0,\mathit{\text{i.e.}},|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|)\le 0,$$

and furthermore

$$(|z|-(R_i(\mathcal{A})-|a_{ij\cdots j}|))|z|\le 0=|a_{ij\cdots j}|R_j(\mathcal{A}).$$

This also implies $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$.

On the other hand, when (7) holds, we only prove $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$ under the case that

$$\frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}>1.$$ 9

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

$$\frac{|z|}{R_i(\mathcal{A})}=\frac{a}{b+c+d}\le \frac{a-b}{c+d}=\frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}.$$ 10

By (7), Lemma 2.2 in [11] and similar to the proof of (8), we have

$$\frac{|z|-(R_j(\mathcal{A})-|a_{ji\cdots i}|)}{|a_{ji\cdots i}|}\le \frac{|z|}{R_j(\mathcal{A})}.$$ 11

Multiplying (10) and (11), we have

$$\frac{|z|-(R_j(\mathcal{A})-|a_{ji\cdots i}|)}{|a_{ji\cdots i}|}\frac{|z|}{R_i(\mathcal{A})}\le \frac{|z|-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})}{r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})}\frac{|z|}{R_j(\mathcal{A})}\le 1;$$

equivalently,

$$(|z|-(R_j(\mathcal{A})-|a_{ji\cdots i}|))|z|\le |a_{ji\cdots i}|R_i(\mathcal{A}).$$

This implies $z\in {\bigcap }_{i\in N,i\ne j}{\mathcal{L}}_{j,i}(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$ and $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$ from the arbitrariness of i. When (7) holds and $|a_{ji\cdots i}|=0$, we can obtain

$$|z|-R_j(\mathcal{A})\le 0,\mathit{\text{i.e.}},|z|-(R_j(\mathcal{A})-|a_{ji\cdots i}|)\le 0$$

and

$$(|z|-(R_j(\mathcal{A})-|a_{ji\cdots i}|))|z|\le 0=|a_{ji\cdots i}|R_i(\mathcal{A}).$$

This also implies $\mathrm{\Psi }(\mathcal{A})\subseteq \mathcal{L}(\mathcal{A})$. The conclusion follows from Case I and Case II. □

Remark 1

Theorem 4 shows that the set $\mathrm{\Psi }(\mathcal{A})$ in Theorem 3 is tighter than $\mathcal{K}(\mathcal{A})$ in Theorem 1 and $\mathcal{L}(\mathcal{A})$ in Theorem 2, that is, $\mathrm{\Psi }(\mathcal{A})$ 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 $\mathrm{\Psi }(\mathcal{A})$ 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

$$a_{1222}=1,a_{2222}=2,\text{and}{a}_{ijkl}=0\text{elsewhere}.$$

By computation, we see that all the Z-eigenvalues of $\mathcal{A}$ are −0.5000, 0 and 2.7000. By Theorem 1, we have

$$\begin{array}{rcl}\mathcal{K}(\mathcal{A})& =& {\mathcal{K}}_1(\mathcal{A})\cup {\mathcal{K}}_2(\mathcal{A})=\{z\in \mathbb{C}:|z|\le 1\}\cup \{z\in \mathbb{C}:|z|\le 5\}\\ & =& \{z\in \mathbb{C}:|z|\le 5\}.\end{array}$$

By Theorem 2, we have

$$\begin{array}{rcl}\mathcal{L}(\mathcal{A})& =& {\mathcal{L}}_{1,2}(\mathcal{A})\cup {\mathcal{L}}_{2,1}(\mathcal{A})=\{z\in \mathbb{C}:|z|\le 2.2361\}\cup \{z\in \mathbb{C}:|z|\le 5\}\\ & =& \{z\in \mathbb{C}:|z|\le 5\}.\end{array}$$

By Theorem 3, we have

$$\begin{array}{rcl}\mathrm{\Psi }(\mathcal{A})& =& {\mathrm{\Psi }}_{1,2}(\mathcal{A})\cup {\mathrm{\Psi }}_{2,1}(\mathcal{A})=\{z\in \mathbb{C}:|z|\le 2.2361\}\cup \{z\in \mathbb{C}:|z|\le 3\}\\ & =& \{z\in \mathbb{C}:|z|\le 3\}.\end{array}$$

The Z-eigenvalue inclusion sets $\mathcal{K}(\mathcal{A})$, $\mathcal{L}(\mathcal{A})$, $\mathrm{\Psi }(\mathcal{A})$ 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, $\mathrm{\Psi }(\mathcal{A})$ is represented by red solid boundary and the exact eigenvalues are plotted by ‘+’, respectively. It is easy to see $\sigma (\mathcal{A})\subseteq \mathrm{\Psi }(\mathcal{A})\subset \mathcal{L}(\mathcal{A})\subseteq \mathcal{K}(\mathcal{A})$, that is, $\mathrm{\Psi }(\mathcal{A})$ can capture all Z-eigenvalues of $\mathcal{A}$ more precisely than $\mathcal{L}(\mathcal{A})$ and $\mathcal{K}(\mathcal{A})$.

Figure 1.

Comparisons of $\mathcal{K}\mathbf{(}\mathcal{A}\mathbf{)}$ , $\mathcal{L}\mathbf{(}\mathcal{A}\mathbf{)}$ and $\mathbf{\Psi }\mathbf{(}\mathcal{A}\mathbf{)}$ .

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

$$\varrho (\mathcal{A})\le \underset{i\in N}{max}\underset{j\in N,j\ne i}{min}{\mathrm{\Phi }}_{i,j}(\mathcal{A}),$$

where

$${\mathrm{\Phi }}_{i,j}(\mathcal{A})=\frac{1}{2}\{r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})+\sqrt{{(r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))}^2+4r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})}\}.$$

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

$$(\varrho (\mathcal{A})-r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))\varrho (\mathcal{A})\le r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A}),\mathrm{\forall }j\in N,j\ne i.$$ 12

Solving $\varrho (\mathcal{A})$ in (12) gives

$$\varrho (\mathcal{A})\le \frac{1}{2}\{r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A})+\sqrt{{(r_i^{{\bar{\mathrm{\Delta }}}_j}(\mathcal{A}))}^2+4r_i^{{\mathrm{\Delta }}_j}(\mathcal{A})R_j(\mathcal{A})}\}={\mathrm{\Phi }}_{i,j}(\mathcal{A}).$$

From the arbitrariness of j, we have $\varrho (\mathcal{A})\le {min}_{j\in N,j\ne i}{\mathrm{\Phi }}_{i,j}(\mathcal{A})$. Furthermore, $\varrho (\mathcal{A})\le {max}_{i\in N}{min}_{j\in N,j\ne i}{\mathrm{\Phi }}_{i,j}(\mathcal{A})$. □

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,

$$\begin{array}{rcl}\varrho (\mathcal{A})& \le & \underset{i\in N}{max}\underset{j\in N,j\ne i}{min}{\mathrm{\Phi }}_{i,j}(\mathcal{A})\\ & \le & \underset{i\in N}{max}\underset{j\in N,j\ne i}{min}\frac{1}{2}\{R_i(\mathcal{A})-a_{ij\cdots j}+\sqrt{{(R_i(\mathcal{A})-a_{ij\cdots j})}^2+4a_{ij\cdots j}{R}_j(\mathcal{A})}\}\\ & \le & \underset{i\in N}{max}{R}_i(\mathcal{A}).\end{array}$$

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:

$$\begin{array}{c}\mathcal{A}(:,:,1)=\left(\begin{array}{ccc}3& 3& 0\\ 3& 2& 2.5\\ 0.5& 2.5& 0\end{array}\right),\mathcal{A}(:,:,2)=\left(\begin{array}{ccc}3& 2& 2\\ 2& 0& 3\\ 2.5& 3& 1\end{array}\right),\hfill \\ \mathcal{A}(:,:,3)=\left(\begin{array}{ccc}1& 3& 0\\ 2.5& 3& 1\\ 0& 1& 0\end{array}\right).\hfill \end{array}$$

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

$$\varrho (\mathcal{A})\le 19.$$

By Theorem 3.5 of [7], we have

$$\varrho (\mathcal{A})\le 18.6788.$$

By Theorem 4.6 of [1], we have

$$\varrho (\mathcal{A})\le 18.6603.$$

By both Theorem 4.5 of [1] and Theorem 6 of [8], we have

$$\varrho (\mathcal{A})\le 18.5656.$$

By Theorem 4.7 of [1], we have

$$\varrho (\mathcal{A})\le 18.3417.$$

By Theorem 2.9 of [10], we have

$$\varrho (\mathcal{A})\le 17.2063.$$

By Theorem 5, we obtain

$$\varrho (\mathcal{A})\le 15.2580,$$

which shows that the upper bound in Theorem 5 is sharper.

Conclusions

In this paper, we present a new Z-eigenvalue localization set $\mathrm{\Psi }(\mathcal{A})$ and prove that this set is tighter than those in [1]. As an application, we obtain a new upper bound ${max}_{i\in N}{min}_{j\in N,j\ne i}{\mathrm{\Phi }}_{i,j}(\mathcal{A})$ 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.