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

$$a_{i_{1}i_{2}\cdots i_{m}}\in{\mathbb{R}}, $$

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

$$ a_{i_{1}\cdots i_{m}}=a_{\pi(i_{1}\cdots i_{m})},\quad \forall\pi\in\Pi _{m}, $$

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

$$\mathcal{A}x^{m}=\sum_{i_{1},i_{2},\ldots,i_{m}\in N}a_{i_{1}i_{2}\cdots i_{m}}x_{i_{1}}x_{i_{2}} \cdots x_{i_{m}} $$

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

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

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

$$\bigl(\mathcal {A}x^{m-1}\bigr)_{i}=\sum _{i_{2},\ldots,i_{m}\in N} a_{ii_{2}\cdots i_{m}}x_{i_{2}}\cdots x_{i_{m}}. $$

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 \(\sigma(\mathcal{A})\neq0\), then the Z-spectral radius [3] of \(\mathcal{A}\), denoted \(\varrho (\mathcal{A})\), is defined as

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

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

$$ \mathcal{K}_{i}(\mathcal{A})=\bigl\{ z\in{\mathbb{C}}:|z|\leq R_{i}(\mathcal {A})\bigr\} ,\qquad R_{i}(\mathcal{A})=\sum _{i_{2},\ldots, i_{m}\in N}|a_{ii_{2}\cdots i_{m}}|. $$

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

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

where

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

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

$$\begin{aligned}& \Delta_{j}=\bigl\{ (i_{2},i_{3},\ldots, i_{m}): i_{k}=j\mbox{ for some }k\in\{2,\ldots,m\}, \mbox{where }j,i_{2},\ldots, i_{m}\in N\bigr\} , \\& \overline{\Delta}_{j}=\bigl\{ (i_{2},i_{3}, \ldots, i_{m}): i_{k}\neq j\mbox{ for any }k\in\{ 2,\ldots,m \}, \mbox{where }j,i_{2},\ldots, i_{m}\in N\bigr\} . \end{aligned}$$

For \(\forall i,j\in N, j\neq i\), let

$$ r_{i}^{\Delta_{j}}(\mathcal{A})=\sum_{(i_{2},\ldots,i_{m})\in\Delta _{j}}|a_{ii_{2}\cdots i_{m}}|, \qquad r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})=\sum _{(i_{2},\ldots,i_{m})\in \overline{\Delta}_{j}}|a_{ii_{2}\cdots i_{m}}|. $$

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

$$ \sigma(\mathcal{A})\subseteq\Psi(\mathcal{A})=\bigcup _{i\in N}\bigcap_{j\in N, j\neq i} \Psi_{i,j}(\mathcal{A}), $$

where

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

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.,

$$ \mathcal{A}x^{m-1}=\lambda x,\quad \text{and}\quad \|x \|_{2}=1. $$
(1)

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

$$ \lambda x_{t}=\sum_{(i_{2},\ldots, i_{m})\in\Delta_{j}}a_{ti_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}} +\sum_{(i_{2},\ldots, i_{m})\in\overline{\Delta}_{j}}a_{ti_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}}. $$

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

$$\begin{aligned} |\lambda||x_{t}| \leq& \sum_{(i_{2},\ldots, i_{m})\in\Delta _{j}}|a_{ti_{2}\cdots i_{m}}||x_{i_{2}}| \cdots|x_{i_{m}}| +\sum_{(i_{2},\ldots, i_{m})\in\overline{\Delta}_{j}}|a_{ti_{2}\cdots i_{m}}||x_{i_{2}}| \cdots|x_{i_{m}}| \\ \leq& \sum_{(i_{2},\ldots, i_{m})\in\Delta_{j}}|a_{ti_{2}\cdots i_{m}}||x_{j}| +\sum_{(i_{2},\ldots, i_{m})\in\overline{\Delta}_{j}}|a_{ti_{2}\cdots i_{m}}||x_{t}| \\ =&r_{t}^{\Delta_{j}}(\mathcal{A})|x_{j}|+r_{t}^{\overline{\Delta}_{j}}( \mathcal{A})|x_{t}|, \end{aligned}$$

i.e.,

$$ \bigl(|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A}) \bigr)|x_{t}|\leq r_{t}^{\Delta _{j}}(\mathcal{A})|x_{j}|. $$
(2)

If \(|x_{j}|=0\), by \(|x_{t}|>0\), we have \(|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\leq0\). Then

$$ \bigl(|\lambda|-r_{t}^{\overline{\Delta}_{j}}(\mathcal{A})\bigr)|\lambda|\leq0 \leq r_{t}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A}). $$

Obviously, \(\lambda\in\Psi_{t,j}(\mathcal{A})\). Otherwise, \(|x_{j}|>0\). From (1), we have

$$ |\lambda||x_{j}|\leq\sum_{i_{2},\ldots, i_{m}\in N}|a_{ji_{2}\cdots i_{m}}||x_{i_{2}}| \cdots|x_{i_{m}}| \leq\sum_{i_{2},\ldots, i_{m}\in N}|a_{ji_{2}\cdots i_{m}}||x_{t}|^{m-1} \leq R_{j}(\mathcal{A})|x_{t}|. $$
(3)

Multiplying (2) with (3) and noting that \(|x_{t}||x_{j}|>0\), we have

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

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

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

Proof

By Corollary 3.1 in [1], \(\mathcal{L}(\mathcal{A})\subseteq \mathcal{K}(\mathcal{A})\) holds. Here, we only prove \(\Psi(\mathcal{A})\subseteq\mathcal{L}(\mathcal{A})\). 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,

$$ \bigl(|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\bigr)|z|\leq r_{i}^{\Delta _{j}}(\mathcal{A})R_{j}(\mathcal{A}), \quad \forall j\in N, j\neq i. $$
(4)

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

$$ \bigl(|z|-\bigl(R_{i}(\mathcal{A})-|a_{ij\cdots j}|\bigr) \bigr)|z| \leq \bigl(|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})\bigr)|z| \leq r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}(\mathcal{A})=0 \leq|a_{ij\cdots j}|R_{j}(\mathcal{A}), $$

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

$$ \frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})}{r_{i}^{\Delta _{j}}(\mathcal{A})} \frac{|z|}{R_{j}(\mathcal{A})}\leq1, $$
(5)

which implies

$$ \frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})}{r_{i}^{\Delta _{j}}(\mathcal{A})}\leq1, $$
(6)

or

$$ \frac{|z|}{R_{j}(\mathcal{A})}\leq1. $$
(7)

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

$$ \frac{|z|-(R_{i}(\mathcal{A})-|a_{ij\cdots j}|)}{|a_{ij\cdots j}|}=\frac {a-(b+c)}{d} \leq\frac{a-b}{c+d}= \frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal {A})}{r_{i}^{\Delta_{j}}(\mathcal{A})}. $$
(8)

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

$$ \frac{|z|-(R_{i}(\mathcal{A})-|a_{ij\cdots j}|)}{|a_{ij\cdots j}|}\frac {|z|}{R_{j}(\mathcal{A})} \leq\frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})}{r_{i}^{\Delta _{j}}(\mathcal{A})}\frac{|z|}{R_{j}(\mathcal{A})}\leq1, $$

equivalently,

$$ \bigl(|z|-\bigl(R_{i}(\mathcal{A})-|a_{ij\cdots j}|\bigr) \bigr)|z| \leq|a_{ij\cdots j}|R_{j}(\mathcal{A}), $$

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

$$ |z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})-r_{i}^{\Delta_{j}}( \mathcal {A})\leq0,\quad \textit{i.e.},\quad |z|-\bigl(R_{i}( \mathcal{A})-|a_{ij\cdots j}|\bigr)\leq0, $$

and furthermore

$$ \bigl(|z|-\bigl(R_{i}(\mathcal{A})-|a_{ij\cdots j}|\bigr) \bigr)|z| \leq0=|a_{ij\cdots j}|R_{j}(\mathcal{A}). $$

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

$$ \frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})}{r_{i}^{\Delta _{j}}(\mathcal{A})}>1. $$
(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

$$ \frac{|z|}{R_{i}(\mathcal{A})}=\frac{a}{b+c+d} \leq\frac{a-b}{c+d}= \frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal {A})}{r_{i}^{\Delta_{j}}(\mathcal{A})}. $$
(10)

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

$$ \frac{|z|-(R_{j}(\mathcal{A})-|a_{ji\cdots i}|)}{|a_{ji\cdots i}|} \leq \frac{|z|}{R_{j}(\mathcal{A})}. $$
(11)

Multiplying (10) and (11), we have

$$ \frac{|z|-(R_{j}(\mathcal{A})-|a_{ji\cdots i}|)}{|a_{ji\cdots i}|}\frac {|z|}{R_{i}(\mathcal{A})} \leq \frac{|z|-r_{i}^{\overline{\Delta}_{j}}(\mathcal{A})}{r_{i}^{\Delta _{j}}(\mathcal{A})}\frac{|z|}{R_{j}(\mathcal{A})} \leq1; $$

equivalently,

$$ \bigl(|z|-\bigl(R_{j}(\mathcal{A})-|a_{ji\cdots i}|\bigr) \bigr)|z| \leq|a_{ji\cdots i}|R_{i}(\mathcal{A}). $$

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

$$ |z|-R_{j}(\mathcal{A})\leq0,\quad \textit{i.e.},\quad |z|- \bigl(R_{j}(\mathcal{A})-|a_{ji\cdots i}|\bigr)\leq0 $$

and

$$ \bigl(|z|-\bigl(R_{j}(\mathcal{A})-|a_{ji\cdots i}|\bigr) \bigr)|z| \leq0=|a_{ji\cdots i}|R_{i}(\mathcal{A}). $$

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

$$a_{1222}=1,\qquad a_{2222}=2, \quad \mbox{and}\quad a_{ijkl}=0\quad \mbox{elsewhere}. $$

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

$$\begin{aligned} \mathcal{K}(\mathcal{A}) =&\mathcal{K}_{1}(\mathcal{A})\cup\mathcal {K}_{2}(\mathcal{A}) =\bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq1 \bigr\} \cup\bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq5\bigr\} \\ =&\bigl\{ z\in{ \mathbb{C}}: \vert z \vert \leq5\bigr\} . \end{aligned}$$

By Theorem 2, we have

$$\begin{aligned} \mathcal{L}(\mathcal{A}) =&\mathcal{L}_{1,2}(\mathcal{A})\cup\mathcal {L}_{2,1}(\mathcal{A}) =\bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq2.2361\bigr\} \cup\bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq 5\bigr\} \\ =& \bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq5\bigr\} . \end{aligned}$$

By Theorem 3, we have

$$\begin{aligned} \Psi(\mathcal{A}) =&\Psi_{1,2}(\mathcal{A})\cup\Psi_{2,1}( \mathcal{A}) =\bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq2.2361\bigr\} \cup \bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq 3\bigr\} \\ =&\bigl\{ z\in{\mathbb{C}}: \vert z \vert \leq3\bigr\} . \end{aligned}$$

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 \(\mathcal{K}(\mathcal{A})\).

Figure 1
figure 1

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

Full size image

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

$$ \varrho(\mathcal{A})\leq\max_{i\in N}\min_{j\in N, j\neq i} \Phi _{i,j}(\mathcal{A}), $$

where

$$ \Phi_{i,j}(\mathcal{A})=\frac{1}{2} \Bigl\{ r_{i}^{\overline{\Delta}_{j}}( \mathcal{A})+\sqrt{\bigl(r_{i}^{\overline{\Delta }_{j}}(\mathcal{A}) \bigr)^{2}+4r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}( \mathcal{A})} \Bigr\} . $$

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

$$ \bigl(\varrho(\mathcal{A})-r_{i}^{\overline{\Delta}_{j}}( \mathcal{A}) \bigr)\varrho(\mathcal{A})\leq r_{i}^{\Delta_{j}}( \mathcal{A})R_{j}(\mathcal{A}),\quad \forall j\in N, j\neq i. $$
(12)

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

$$ \varrho(\mathcal{A})\leq\frac{1}{2} \Bigl\{ r_{i}^{\overline{\Delta}_{j}}( \mathcal{A})+\sqrt{\bigl(r_{i}^{\overline{\Delta }_{j}}(\mathcal{A}) \bigr)^{2}+4r_{i}^{\Delta_{j}}(\mathcal{A})R_{j}( \mathcal{A})} \Bigr\} =\Phi_{i,j}(\mathcal{A}). $$

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,

$$\begin{aligned} \varrho(\mathcal{A}) \leq&\max_{i\in N}\min_{j\in N, j\neq i} \Phi _{i,j}(\mathcal{A}) \\ \leq&\max_{i\in N}\min_{j\in N, j\neq i}\frac{1}{2} \bigl\{ R_{i}(\mathcal{A})-a_{ij\cdots j}+\sqrt{ \bigl(R_{i}(\mathcal{A})-a_{ij\cdots j}\bigr)^{2}+4a_{ij\cdots j}R_{j}( \mathcal{A})} \bigr\} \\ \leq&\max_{i\in N}R_{i}(\mathcal{A}). \end{aligned}$$

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:

$$\begin{aligned}& \mathcal{A}(:,:,1)=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 3&3&0\\ 3&2&2.5\\ 0.5&2.5&0 \end{array}\displaystyle \right ),\qquad \mathcal{A}(:,:,2)=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 3&2&2\\ 2&0&3\\ 2.5&3&1 \end{array}\displaystyle \right ), \\& \mathcal{A}(:,:,3)=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1&3&0\\ 2.5&3&1\\ 0&1&0 \end{array}\displaystyle \right ). \end{aligned}$$

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

$$\varrho(\mathcal{A})\leq19. $$

By Theorem 3.5 of [7], we have

$$\varrho(\mathcal{A})\leq18.6788. $$

By Theorem 4.6 of [1], we have

$$\varrho(\mathcal{A})\leq18.6603. $$

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

$$\varrho(\mathcal{A})\leq18.5656. $$

By Theorem 4.7 of [1], we have

$$\varrho(\mathcal{A})\leq18.3417. $$

By Theorem 2.9 of [10], we have

$$\varrho(\mathcal{A})\leq17.2063. $$

By Theorem 5, we obtain

$$\varrho(\mathcal{A})\leq15.2580, $$

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.