A new Z-eigenvalue localization set for tensors.

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 ≥ , N denotes the set {, , . . . , n}. C (R) denotes the set of all complex (real) numbers. We call A = (a i  i  ···i m ) a real tensor of order m dimension n, denoted by R [m,n] , if It is shown in [] that a symmetric tensor is necessarily weakly symmetric, but the converse is not true in general.
Given a tensor A = (a i  ···i m ) ∈ R [m,n] , if there are λ ∈ C and x = (x  , x  · · · , x n ) T ∈ C n \{} such that Ax m- = λx and x T x = , then λ is called an E-eigenvalue of A and x an E-eigenvector of A associated with λ, where Ax m- is an n dimension vector whose ith component is If λ and x are all real, then λ is called a Z-eigenvalue of A and x a Z-eigenvector of A associated with λ; for details, see [, ]. n] . We define the Z-spectrum of A, denoted σ (A) to be the set of all Z-eigenvalues of A. Assume σ (A) = , then the Z-spectral radius [] of A, denoted (A), is defined as Recently, much literature has focused on locating all Z-eigenvalues of tensors and bounding the Z-spectral radius of nonnegative tensors in [, -]. 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 [, -]. 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 , Wang et al.
[] established the following Gersgorin-type Z-eigenvalue inclusion theorem for tensors.
To get a tighter Z-eigenvalue inclusion set than K(A), Wang et al.
[] gave the following Brauer-type Z-eigenvalue localization set for tensors.
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  and Theorem . 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  and Theorem . For simplification, we denote Taking the modulus in the above equation and using the triangle inequality give i.e., Obviously, λ ∈ t,j (A). Otherwise, |x j | > . From (), we have Multiplying () with () and noting that |x t ||x j | > , we have which implies that λ ∈ t,j (A). From the arbitrariness of j, we have λ ∈ j∈N,j =t t,j (A). Furthermore, we have λ ∈ i∈N j∈N,j =i i,j (A).
Next, a comparison theorem is given for Theorem , Theorem  and Theorem . .
(   ) By (), Lemma . in [] and similar to the proof of (), we have Multiplying () and (), we have This implies z ∈ i∈N,i =j L j,i (A) ⊆ L(A) and (A) ⊆ L(A) from the arbitrariness of i. When () holds and |a ji···i | = , we can obtain This also implies (A) ⊆ L(A). The conclusion follows from Case I and Case II.

Remark  Theorem  shows that the set (A) in Theorem  is tighter than K(A) in
Theorem  and L(A) in Theorem , that is, (A) can capture all Z-eigenvalues of A more precisely than K(A) and L(A).

Now, we give an example to show that (A) is tighter than K(A) and L(A).
Example  Let A = (a ijkl ) ∈ R [,] be a symmetric tensor defined by a  = , a  = , and a ijkl =  elsewhere.
By computation, we see that all the Z-eigenvalues of A are -.,  and .. By Theorem , we have By Theorem , we have By Theorem , we have The

A new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors
As an application of the results in Section , we in this section give a new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors.
Finally, we show that the upper bound in Theorem  is sharper than those in [, -, ] by the following example. It is not difficult to verify that A is a weakly symmetric nonnegative tensor.