An eigenvalue localization set for tensors and its applications

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.

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

$$a_{i_{1}\cdots i_{m}}\in{\mathbb{C}}(\mathbb{R}), $$

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

$$a_{i_{1}i_{2}\cdots i_{m}}=0, \quad \forall i_{1}\in\mathbb{J}, \forall i_{2},\ldots,i_{m}\notin\mathbb{J}. $$

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

$$\mathcal{A}x^{m-1}=\lambda x^{[m-1]}, $$

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

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

and

$$x^{[m-1]}=\bigl(x_{1}^{m-1},x_{2}^{m-1}, \ldots,x_{n}^{m-1}\bigr)^{T}. $$

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

$$\rho(\mathcal{A})=\max\bigl\{ \vert \lambda \vert :\lambda\in\sigma( \mathcal{A})\bigr\} , $$

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

$$ \delta_{i_{1}\cdots i_{m}}= \textstyle\begin{cases} 1& \mathrm{if}\ i_{1}=\cdots=i_{m}, \\ 0& \mathrm{otherwise}. \end{cases} $$

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

$$\begin{aligned} \sigma(\mathcal{A})\subseteq\Delta(\mathcal{A})=\bigcup _{i,j\in {N},j\neq{i}}\Delta_{i}^{j}(\mathcal{A}), \end{aligned}$$

where

$$\begin{aligned} &\Delta_{i}^{j}(\mathcal{A})= \bigl\{ z\in\mathbb{C}:\bigl\vert (z-a_{i\cdots i}) (z-a_{j\cdots j})-a_{ij\cdots j}a_{ji\cdots i} \bigr\vert \leq \vert z-a_{j\cdots j}\vert r_{i}^{j}( \mathcal{A})+\vert a_{ij\cdots j}\vert r_{j}^{i}( \mathcal {A}) \bigr\} , \\ &r_{i}(\mathcal{A})=\sum_{\delta_{ii_{2}\cdots i_{m}}=0}\vert a_{ii_{2}\cdots i_{m}}\vert ,\qquad r_{i}^{j}(\mathcal{A})=\sum _{\substack{\delta_{ii_{2}\cdots i_{m}}=0,\\ \delta_{ji_{2}\cdots i_{m}}=0}}\vert a_{ii_{2}\cdots i_{m}}\vert =r_{i}(\mathcal{A})-\vert a_{ij\cdots{j}} \vert . \end{aligned}$$

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

$$\begin{aligned} \sigma(\mathcal{A})\subseteq\Delta^{S}(\mathcal{A}) = \biggl(\bigcup _{i\in S,j\in\bar{S}}\Delta_{i}^{j}(\mathcal{A}) \biggr)\cup \biggl(\bigcup_{i\in\bar{S},j\in S}\Delta_{i}^{j}( \mathcal{A}) \biggr). \end{aligned}$$

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

$$\begin{aligned} \min \Bigl\{ \min_{i\in S}\max_{j\in\bar{S}}L_{ij}( \mathcal {A}), \min_{i\in\bar{S}}\max_{j\in S}L_{ij}( \mathcal {A}) \Bigr\} \leq\tau(\mathcal{A})\leq \max \Bigl\{ \max _{i\in S}\min_{j\in\bar{S}}L_{ij}(\mathcal {A}), \max_{i\in\bar{S}}\min_{j\in S}L_{ij}( \mathcal {A}) \Bigr\} , \end{aligned}$$

where

$$\begin{aligned} L_{ij}(\mathcal{A})=\frac{1}{2} \bigl\{ a_{i\cdots i}+a_{j\cdots j}-r_{i}^{j}( \mathcal{A})-\bigl[\bigl(a_{i\cdots i}-a_{j\cdots j}-r_{i}^{j}( \mathcal{A})\bigr)^{2}-4a_{ij\cdots j}r_{j}(\mathcal{A}) \bigr]^{\frac {1}{2}} \bigr\} . \end{aligned}$$

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

$$\begin{aligned} \sigma(\mathcal{A})\subseteq\Delta^{\cap}(\mathcal{A})=\bigcup _{i\in N}\bigcap_{j\in N,j\neq i} \Delta_{i}^{j}(\mathcal{A}). \end{aligned}$$

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.,

$$\begin{aligned} \mathcal{A}x^{m-1}=\lambda x^{[m-1]}. \end{aligned}$$

(1)

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

$$\begin{aligned} \lambda x_{p}^{m-1}=\mathop{\sum _{\delta_{pi_{2}\cdots i_{m}}=0,}}_{\delta _{ji_{2}\cdots i_{m}}=0}a_{pi_{2}\cdots i_{m}}x_{i_{2}}\cdots x_{i_{m}}+a_{p\cdots{p}}x_{p}^{m-1}+a_{pj\cdots{j}}x_{j}^{m-1} \end{aligned}$$

and

$$\begin{aligned} \lambda x_{j}^{m-1}=\mathop{\sum _{\delta_{ji_{2}\cdots i_{m}}=0,}}_{ \delta_{pi_{2}\cdots i_{m}}=0}a_{ji_{2}\cdots i_{m}}x_{i_{2}}\cdots x_{i_{m}}+a_{j\cdots{j}}x_{j}^{m-1}+a_{jp\cdots{p}}x_{p}^{m-1}, \end{aligned}$$

equivalently,

$$\begin{aligned} (\lambda-a_{p\cdots{p}})x_{p}^{m-1}-a_{pj\cdots{j}}x_{j}^{m-1}= \mathop{\sum_{\delta_{pi_{2}\cdots i_{m}}=0,}}_{\delta_{ji_{2}\cdots i_{m}}=0}a_{pi_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}} \end{aligned}$$

(2)

and

$$\begin{aligned} (\lambda-a_{j\cdots{j}})x_{j}^{m-1}-a_{jp\cdots{p}}x_{p}^{m-1}= \mathop{\sum_{\delta_{ji_{2}\cdots i_{m}}=0,}}_{\delta_{pi_{2}\cdots i_{m}}=0}a_{ji_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}}. \end{aligned}$$

(3)

Solving $x_{p}^{m-1}$ from (2) and (3), we get

$$\begin{aligned} &\bigl((\lambda-a_{p\cdots{p}}) (\lambda-a_{j\cdots{j}})-a_{pj\cdots {j}}a_{jp\cdots{p}} \bigr)x_{p}^{m-1}\\ &\quad = (\lambda-a_{j\cdots{j}})\mathop{\sum _{\delta_{pi_{2}\cdots i_{m}}=0,}}_{ \delta_{ji_{2}\cdots i_{m}}=0}a_{pi_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}} +a_{pj\cdots j}\mathop{\sum_{\delta_{ji_{2}\cdots i_{m}}=0,}}_{\delta _{pi_{2}\cdots i_{m}}=0}a_{ji_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}}. \end{aligned}$$

Taking absolute values and using the triangle inequality yields

$$\begin{aligned} &\bigl\vert (\lambda-a_{p\cdots{p}}) (\lambda-a_{j\cdots{j}})-a_{pj\cdots {j}}a_{jp\cdots{p}} \bigr\vert \vert x_{p}\vert ^{m-1}\\ &\quad\leq \vert \lambda-a_{j\cdots j}\vert r_{p}^{j}(\mathcal{A}) \vert x_{p}\vert ^{m-1}+\vert a_{pj\cdots j}\vert r_{j}^{p}(\mathcal{A})\vert x_{p}\vert ^{m-1}. \end{aligned}$$

Furthermore, by $\vert x_{p}\vert >0$, we have

$$\begin{aligned} \bigl\vert (\lambda-a_{p\cdots{p}}) (\lambda-a_{j\cdots{j}})-a_{pj\cdots {j}}a_{jp\cdots{p}} \bigr\vert \leq \vert \lambda-a_{j\cdots j}\vert r_{p}^{j}( \mathcal{A})+\vert a_{pj\cdots j}\vert r_{j}^{p}( \mathcal{A}), \end{aligned}$$

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

$$\begin{aligned} \Delta^{\cap}(\mathcal{A})\subseteq\Delta^{S}(\mathcal{A}) \subseteq\Delta (\mathcal{A}). \end{aligned}$$

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

$$\begin{aligned} \min_{i\in N}\max_{j\neq i}L_{ij}( \mathcal{A})\leq\tau (\mathcal{A})\leq \max_{i\in N}\min _{j\neq i}L_{ij}(\mathcal{A}). \end{aligned}$$

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.,

$$\begin{aligned} \mathcal{A}x^{m-1}=\tau(\mathcal{A}) x^{[m-1]}. \end{aligned}$$

(4)

(I) Let $x_{q}=\min\{x_{i}:i\in N\}$. For any $j\in N, j\neq q$, we have by (4) that

$$\tau(\mathcal{A})x_{q}^{m-1}=\mathop{\sum _{\delta_{qi_{2} \cdots i_{m}}=0,}}_{\delta_{ji_{2} \cdots i_{m}}=0}a_{qi_{2}\cdots i_{m}}x_{i_{2}}\cdots x_{i_{m}}+a_{q\cdots q}x_{q}^{m-1}+a_{qj\cdots j}x_{j}^{m-1} $$

and

$$\tau(\mathcal{A})x_{j}^{m-1}=\mathop{\sum _{\delta_{ji_{2} \cdots i_{m}}=0,}}_{\delta_{qi_{2} \cdots i_{m}}=0}a_{ji_{2}\cdots i_{m}}x_{i_{2}}\cdots x_{i_{m}}+a_{j\cdots j}x_{j}^{m-1}+a_{jq\cdots q}x_{q}^{m-1}, $$

equivalently,

$$\begin{aligned} \bigl(\tau(\mathcal{A})-a_{q\cdots q}\bigr)x_{q}^{m-1}-a_{qj\cdots j}x_{j}^{m-1}= \mathop{\sum_{\delta_{qi_{2} \cdots i_{m}}=0,}}_{\delta_{ji_{2} \cdots i_{m}}=0}a_{qi_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}} \end{aligned}$$

(5)

and

$$\begin{aligned} \bigl(\tau(\mathcal{A})-a_{j\cdots j}\bigr)x_{j}^{m-1}-a_{jq\cdots q}x_{q}^{m-1}= \mathop{\sum_{\delta_{ji_{2} \cdots i_{m}}=0,}}_{\delta_{qi_{2} \cdots i_{m}}=0}a_{ji_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}}. \end{aligned}$$

(6)

Solving $x_{q}^{m-1}$ by (5) and (6), we get

$$\begin{aligned} &\bigl(\bigl(\tau(\mathcal{A})-a_{q\cdots q}\bigr) \bigl(\tau( \mathcal{A})-a_{j\cdots j}\bigr)-a_{qj\cdots j}a_{jq\cdots q} \bigr)x_{q}^{m-1}\\ &\quad=\bigl(\tau(\mathcal{A})-a_{j\cdots j} \bigr)\mathop{\sum_{\delta_{qi_{2} \cdots i_{m}}=0,}}_{\delta_{ji_{2} \cdots i_{m}}=0}a_{qi_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}} +a_{qj\cdots j}\mathop{\sum_{\delta_{ji_{2} \cdots i_{m}}=0,}}_{\delta_{qi_{2} \cdots i_{m}}=0}a_{ji_{2}\cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}}. \end{aligned}$$

From Theorem 2.1 in [9], we have $\tau(\mathcal{A})\leq\min_{i\in N}a_{i\cdots i}$ and

$$\begin{aligned} &\bigl(\bigl(a_{q\cdots q}-\tau(\mathcal{A})\bigr) \bigl(a_{j\cdots j}- \tau(\mathcal{A})\bigr)-a_{qj\cdots j}a_{jq\cdots q}\bigr)x_{q}^{m-1} \\ &\quad=\bigl(a_{j\cdots j}-\tau(\mathcal{A})\bigr)\mathop{\sum _{\delta_{qi_{2} \cdots i_{m}}=0,}}_{\delta_{ji_{2} \cdots i_{m}}=0}\vert a_{qi_{2}\cdots i_{m}}\vert x_{i_{2}}\cdots x_{i_{m}} +\vert a_{qj\cdots j}\vert \mathop{ \sum_{\delta_{ji_{2} \cdots i_{m}}=0,}}_{\delta _{qi_{2} \cdots i_{m}}=0} \vert a_{ji_{2}\cdots i_{m}}\vert x_{i_{2}}\cdots x_{i_{m}}. \end{aligned}$$

Hence,

$$\begin{aligned} &\bigl(\bigl(a_{q\cdots q}-\tau(\mathcal{A})\bigr) \bigl(a_{j\cdots j}- \tau(\mathcal{A})\bigr)-\vert a_{qj\cdots j}\vert \vert a_{jq\cdots q} \vert \bigr)x_{q}^{m-1} \\ &\quad \geq\bigl(a_{j\cdots j}-\tau( \mathcal{A})\bigr)\mathop{\sum_{\delta_{qi_{2} \cdots i_{m}}=0,}}_{\delta_{ji_{2} \cdots i_{m}}=0} \vert a_{qi_{2}\cdots i_{m}}\vert x_{q}^{m-1} +\vert a_{qj\cdots j}\vert \mathop{\sum_{\delta_{ji_{2} \cdots i_{m}}=0,}}_{\delta_{qi_{2} \cdots i_{m}}=0} \vert a_{ji_{2}\cdots i_{m}}\vert x_{q}^{m-1}. \end{aligned}$$

From $x_{q}>0$, we have

$$\begin{aligned} &\bigl(a_{q\cdots q}-\tau(\mathcal{A})\bigr) \bigl(a_{j\cdots j}-\tau( \mathcal{A})\bigr)-\vert a_{qj\cdots j}\vert \vert a_{jq\cdots q}\vert \\ &\quad \geq\bigl(a_{j\cdots j}-\tau(\mathcal{A})\bigr)\mathop{\sum _{\delta_{qi_{2} \cdots i_{m}}=0,}}_{\delta_{ji_{2} \cdots i_{m}}=0}\vert a_{qi_{2}\cdots i_{m}}\vert +\vert a_{qj\cdots j}\vert \mathop{\sum_{\delta_{ji_{2} \cdots i_{m}}=0,}}_{\delta_{qi_{2} \cdots i_{m}}=0} \vert a_{ji_{2}\cdots i_{m}}\vert \\ &\quad=\bigl(a_{j\cdots j}-\tau(\mathcal{A})\bigr)r_{q}^{j}( \mathcal{A})+\vert a_{qj\cdots j}\vert r_{j}^{q}( \mathcal{A}), \end{aligned}$$

equivalently,

$$\bigl(a_{q\cdots q}-\tau(\mathcal{A})\bigr) \bigl(a_{j\cdots j}-\tau( \mathcal{A})\bigr)-\bigl(a_{j\cdots j}-\tau(\mathcal{A})\bigr)r_{q}^{j}( \mathcal{A})-\vert a_{qj\cdots j}\vert r_{j}(\mathcal{A})\geq0, $$

that is,

$$\tau(\mathcal{A})^{2}-\bigl(a_{q\cdots q}+a_{j\cdots j}-r_{q}^{j}( \mathcal{A})\bigr)\tau(\mathcal{A})+a_{q\cdots q}a_{j\cdots j}-a_{j\cdots j}r_{q}^{j}( \mathcal{A})+a_{qj\cdots j}r_{j}(\mathcal{A})\geq0. $$

Solving for $\tau(\mathcal{A})$ gives

$$\tau(\mathcal{A})\leq\frac{1}{2} \bigl\{ a_{q\cdots q}+a_{j\cdots j}-r_{q}^{j}( \mathcal{A})- \bigl[\bigl(a_{q\cdots q}-a_{j\cdots j}-r_{q}^{j}( \mathcal{A})\bigr)^{2}-4a_{qj\cdots j}r_{j}(\mathcal{A}) \bigr]^{\frac {1}{2}} \bigr\} =L_{qj}(\mathcal{A}). $$

For the arbitrariness of j, we have $\tau(\mathcal{A})\leq\min_{j\neq q}L_{qj}(\mathcal{A})$. Furthermore, we have

$$\begin{aligned} \tau(\mathcal{A})\leq\max_{i\in N}\min_{j\neq i}L_{ij}( \mathcal{A}). \end{aligned}$$

(II) Let $x_{p}=\max\{x_{i}:i\in N\}$. Similar to (I), we have

$$\begin{aligned} \tau(\mathcal{A})\geq\min_{i\in N}\max_{j\neq i}L_{ij}( \mathcal{A}). \end{aligned}$$

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

$$\begin{aligned} \min_{i\in N}\max_{j\neq i}L_{ij}( \mathcal{A})\leq\tau (\mathcal{A})\leq \max_{i\in N}\min _{j\neq i}L_{ij}(\mathcal{A}). \end{aligned}$$

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

$$\begin{aligned} \min_{i\in N}R_{i}(\mathcal{A})&\leq \min _{j\neq i}L_{ij}(\mathcal{A})\leq \min \Bigl\{ \min _{i\in S}\max_{j\in\bar{S}}L_{ij}(\mathcal {A}), \min_{i\in\bar{S}}\max_{j\in S}L_{ij}( \mathcal {A}) \Bigr\} \leq \min_{i\in N}\max_{j\neq i}L_{ij}( \mathcal{A}) \\ &\leq \max_{i\in N}\min_{j\neq i}L_{ij}( \mathcal{A}) \leq \max \Bigl\{ \max_{i\in S}\min _{j\in\bar{S}}L_{ij}(\mathcal {A}), \max_{i\in\bar{S}} \min_{j\in S}L_{ij}(\mathcal {A}) \Bigr\} , \end{aligned}$$

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:

$$\begin{aligned} &\mathcal{A}(:,:,1)= \begin{pmatrix} 62&-3&-4&-2\\ -4&-2&-2&-1\\ -3&-1&-3&-3\\ -3&-3&-2&-2 \end{pmatrix} ,\qquad \mathcal{A}(:,:,2)= \begin{pmatrix} 0&-4&-3&-3\\ -1&28&-2&-2\\ -1&-2&-2&-4\\ -2&-2&-3&-1 \end{pmatrix}, \\ &\mathcal{A}(:,:,3)= \begin{pmatrix} -2&-1&-2&-1\\ -1&-1&-1&-2\\ -2&-4&63&-4\\ -4&-4&-2&-2 \end{pmatrix},\qquad \mathcal{A}(:,:,4)= \begin{pmatrix} -4&-2&-2&-1\\ -1&-2&-3&-1\\ -2&-3&-3&-2\\ -2&-2&-4&61 \end{pmatrix}. \end{aligned}$$

By Theorem 2.1 in [9], we have

$$2=\min_{i\in N}R_{i}(\mathcal{A})\leq\tau(\mathcal{A}) \leq\min\Bigl\{ \max_{i\in N}R_{i}(\mathcal{A}),\min _{i\in N}a_{i\cdots i}\Bigr\} =28. $$

By Theorem 4 in [13], we have

$$\tau(\mathcal{A})\geq\min_{j\neq i}L_{ij}( \mathcal{A})=2.3521. $$

By Theorem 3, we have

$$\begin{aligned} &\mbox{if }S=\{1\},\bar{S}=\{2,3,4\},\quad 3.6685\leq\tau(\mathcal{A})\leq 24.2948;\\ &\mbox{if }S=\{2\},\bar{S}=\{1,3,4\},\quad 3.6685\leq\tau(\mathcal{A})\leq 19.7199;\\ &\mbox{if }S=\{3\},\bar{S}=\{1,2,4\},\quad 2.3569\leq\tau(\mathcal{A})\leq 27.7850;\\ &\mbox{if }S=\{4\},\bar{S}=\{1,2,3\},\quad 2.3521\leq\tau(\mathcal{A})\leq 27.8536;\\ &\mbox{if }S=\{1,2\},\bar{S}=\{3,4\},\quad 2.3569\leq\tau(\mathcal{A})\leq 27.7850;\\ &\mbox{if }S=\{1,3\},\bar{S}=\{2,4\},\quad 3.6685\leq\tau(\mathcal{A})\leq 23.0477;\\ &\mbox{if }S=\{1,4\},\bar{S}=\{2,3\},\quad 3.6685\leq\tau(\mathcal{A})\leq 23.9488. \end{aligned}$$

By Theorem 7, we have

$$3.6685\leq\tau(\mathcal{A})\leq19.7199. $$

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:

$$a_{1111}=6,\qquad a_{1222}=-1,\qquad a_{2111}=-2,\qquad a_{2222}=5, $$

other $a_{ijkl}=0$. By Theorem 7, we have

$$4\leq\tau(\mathcal{A})\leq4. $$

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.