Two S-type Z-eigenvalue inclusion sets for tensors

In this paper, we present two S-type Z-eigenvalue inclusion sets involved with a nonempty proper subset S of N for general tensors. It is shown that the new sets are tighter than those provided by Wang et al. (Discrete Contin. Dyn. Syst., Ser. B 22(1):187-198, 2017). Furthermore, we obtain upper bounds for the spectral radius of weakly symmetric nonnegative tensors, which are sharper than existing results.


Introduction
Let C (R) be the set of all complex (real) numbers and N = {, , . . . , n}. A real m-order n-dimensional tensor A consists of n m elements: A is called nonnegative (positive) if a i  i  ...i m ≥  (a i  i  ...i m > ).
The following two definitions of eigenpairs were introduced by Qi [] and Lim [], respectively.
Definition  Let A be a tensor with order m and dimension n. If there exist a nonzero vector x = [x  , x  , . . . , x n ] T ∈ C n and a number λ ∈ C satisfying the equation Definition  Let A be a tensor with order m and dimension n. We say that (λ, x) ∈ C ×

(C n \ {}) is an E-eigenpair of A if
Ax m- = λx and x T x = .
(λ, x) is called a Z-eigenpair if they are real.
As we know, the Z-eigenpair for nonnegative tensors plays an important role in some applications such as high order Markov chains [, ] and best rank-one approximations in statistical data analysis [, ]. Some effective algorithms for finding Z-eigenvalue and the corresponding eigenvector of tensors have been implemented [, ]. Generally, we cannot judge that Z-eigenvalues generated by the above algorithms are the largest Z-eigenvalues. Therefore, the following definitions were introduced and used by Qi [] and by Chang [] for studying important characterizations of the largest Z-eigenvalue of a tensor.

Definition  ([]) Let
A be a tensor with order m and dimension n. We define σ (A) the Zspectrum of A by the set of all Z-eigenvalues of A. Assume σ (A) = ∅. Then the Z-spectral radius of A is denoted as [] introduced an S-partition method and established S-type H-eigenvalue localization sets, which may reduce computations. Therefore, we want to use the S-partition method and propose S-type Z-eigenvalue inclusion sets for general tensors.

Definition  ([]) Let
The remainder of this paper is organized as follows. In Section , we establish S-type Zeigenvalue inclusion sets for general tensors by breaking N into a disjoint subset S and its complement, which is proved to be tighter than the sets in []. In Section , as applications of the above results, we propose some new bounds on the Z-spectral radius of a weakly symmetric tensor and show that they are tighter than the existing bounds in [, , -, ] by Example .

S-Type Z-eigenvalue inclusion sets
In this section, we give S-type Z-eigenvalue inclusion sets of the tensor A by dividing N into disjoint subsets S andS, whereS is the complement of S in N . Furthermore, we establish comparisons among different Z-eigenvalue inclusion sets.
In what follows, we introduce a lemma for a general tensor.

Lemma  (Theorem . of []) Let
A be a tensor with order m and dimension n ≥ . Then all Z-eigenvalues of A are located in the union of the following sets: By using the partition technique in [], we present the following notations. Let A be an mth order n-dimensional tensor and S be a nonempty proper subset of N . Set Theorem  Let A be a tensor with order m and dimension n ≥  and S be a nonempty proper subset of N . Then all Z-eigenvalues of A are located in the union of the following sets: Proof Let λ be a Z-eigenvalue of A with corresponding eigenvector x, i.e., Then at least one of |x t | and |x s | is nonzero. We next divide the proof into three parts.
Noting that |x t | m- ≤ |x t | ≤ , |x s | m- ≤ |x s | ≤  and taking modulus in the above equation, one has Dividing both sides by |x s | in (), we get On the other hand, by (), we obtain Dividing both sides by |x t | in the above inequality and from |x s | m- ≤ |x s |, one has Similar to the proof of (i), we can get that (iii) If x t x s = , without loss of generality, let |x t | =  and |x s | = . It follows from () that For any i ∈ S, we have . The result follows from (i), (ii) and (iii).

Corollary  Let
A be a tensor with order m and dimension n ≥ , and S be a nonempty proper subset of N . Then Proof Let z be a point of K(A). Two cases are discussed as follows: (ii) There exist s ∈S and t ∈ S such that z ∈ G¯S s,t (A), i.e., . So, the result holds.
Based on an exact characterization of (), another S-type Z-eigenvalue localization set involved with a proper subset S of N is given below.

Theorem  Let
A be a tensor with order m and dimension n ≥  and S be a nonempty proper subset of N . Then Proof Let λ be a Z-eigenvalue of A with corresponding eigenvector x. Let |x t | = max i∈S |x i | and |x s | = max i∈S |x i |. Similar to the proof of Theorem , we also divide the proof into three cases as follows.
(i) If x t x s =  and |x s | ≥ |x t |, then |x s | = max{|x i | : i ∈ N}. By an exact characterization of (), one has  (ii) Suppose that z ∈ G¯S(A), then there exist s ∈S and t ∈ S such that z ∈ G¯S s,t (A). Similar to the proof of (i), the conclusion holds.
(II) Let z ∈ S (A), then z ∈ i∈S,j∈S . We also divide the proof into two parts.
(i) Suppose that z ∈ i∈S,j∈S (ii) Suppose that z ∈ i∈S,j∈S ( And it follows from Theorem  that Proof According to Lemma , we assume that ρ(A) = λ * is the largest Z-eigenvalue of A. From Theorem , we get For the case that ρ(A) ∈ i∈S,j∈S G S i,j (A), there exist t ∈ S, s ∈S such that Solving ρ(A) in inequality (), we obtain For another case that ρ(A) ∈ i∈S,j∈S G¯S i,j (A), we also get It follows from () and () that the upper bound holds.
On the basis of Theorem , we obtain another sharp bound of the largest Z-eigenvalue for a weakly symmetric nonnegative tensor. Proof Similar to the proof of Theorem , according to Lemma  and Theorem , the conclusion holds.

Theorem  Suppose that an m-order n-dimensional nonnegative tensor A is weakly symmetric and S is a nonempty proper subset of N . Then
Remark  For a weakly symmetric nonnegative tensor A, as shown in the proofs of Theorem  and Theorem , it is not hard to obtain that u S ≤ max i∈N R i (A) and v S ≤ max i∈N R i (A).
Next, we take the following example to show the efficiency of our new upper bounds.

Conclusions
In this paper, we consider the Z-eigenvalue for general tensors and obtain two new S-type Z-eigenvalue inclusion sets. According to the above results, we present upper bounds on the spectral radius of weakly symmetric nonnegative tensors and show that the results are sharper than the upper bounds provided by [, , -, ] in Example .