An eigenvalue localization set for tensors and its applications

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 M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}$\end{document}-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,…,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N=\{1,2,\ldots,n\}$\end{document}, we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}$\end{document}-tensors. Finally, numerical examples are given to verify the theoretical results.


Introduction
For a positive integer n, n ≥ , N denotes the set {, , . . . , n}. C (respectively, R) denotes the set of all complex (respectively, real) numbers. We call A = (a i  ···i m ) a complex (real) tensor of order m dimension n, denoted by C [m,n] (R [m,n] ), if where i j ∈ N for j = , , . . . , m. A is called reducible if there exists a nonempty proper index subset J ⊂ N such that then λ is called an eigenvalue of A and x an eigenvector of A associated with λ, where Ax m- is an n dimension vector whose ith component is  Recently, many people have focused on locating eigenvalues of tensors and using obtained eigenvalue inclusion theorems to determine the positive definiteness of an evenorder real symmetric tensor or to give the lower and upper bounds for the spectral radius of nonnegative tensors and the minimum eigenvalue of M-tensors. For details, see [, , -].
In , Li et al. [] proposed the following Brauer-type eigenvalue localization set for tensors.
To reduce computations, Huang et al. [] presented an S-type eigenvalue localization set by breaking N into disjoint subsets S andS, whereS is the complement of S in N .
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  and  without considering the selection of S. And then we use this set to obtain new lower and upper bounds for the minimum eigenvalue of M-tensors and prove that new bounds are sharper than those in Theorem .

Main results
Now, we give a new eigenvalue inclusion set for tensors and establish the comparison between this set with those in Theorems  and .
Next, a comparison theorem is given for Theorems ,  and . Remark  Theorem  shows that the set ∩ (A) in Theorem  is tighter than those in Theorems  and , that is, ∩ (A) can capture all eigenvalues of A more precisely than (A) and S (A).

Theorem  Let
In the following, we give new lower and upper bounds for the minimum eigenvalue of M-tensors.
Proof Let x = (x  , x  , . . . , x n ) T be an associated positive eigenvector of A corresponding to τ (A), i.e., (  ) (I) Let x q = min{x i : i ∈ N}. For any j ∈ N, j = q, we have by () that Solving x m- q by () and (), we get Hence, From x q > , we have equivalently, Remark  Theorem  shows that the bounds in Theorem  are shaper than those in Theorem , Theorem . of [] and Theorem  of [] without considering the selection of S, which is also the advantage of our results.

Numerical examples
In this section, two numerical examples are given to verify the theoretical results. ] be an irreducible M-tensor with elements defined as follows:

Conclusions
In this paper, we give a new eigenvalue inclusion set for tensors and prove that this set is tighter than those in [, ]. As an application, we obtain new lower and upper bounds for the minimum eigenvalue of M-tensors and prove that the new bounds are sharper than those in [, , ]. Compared with the results in [], 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 M-tensors.