Z -EIGENVALUE INCLUSION THEOREMS FOR TENSORS

. In this paper, we establish Z -eigenvalue inclusion theorems for general tensors, which reveal some crucial diﬀerences between Z -eigenvalues and H -eigenvalues. As an application, we obtain upper bounds for the largest Z -eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds.

(iii) [3] We say that A is weakly symmetric if the associated homogeneous polynomial Ax m satisfies ∇Ax m = mAx m−1 .
Qi [14] investigated the distribution of H-eigenvalue and established the Gersgorin H-eigenvalue inclusion theorem of real supersymmetric tensors. These results can be generalized to general tensors [11,20]. [11,20]) Let A be a complex tensor of order m and dimension n. The following example shows that Lemma 2.2 cannot be generalized to Z-eigenvalues of a general tensor. ].
3. Z-eigenvalue inclusion theorems. In this section, we establish Z-eigenvalue inclusion theorems of tensor A, which is completely different from H-eigenvalue inclusion theorems. Furthermore, we establish comparisons among different Zeigenvalue inclusion sets.
Theorem 3.1. Let A be a tensor with order m and dimension n ≥ 2. 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., where x = [x 1 , x 2 , . . . , x n ] T . Since x is an eigenvector and ||x|| 2 = 1, it has at least one nonzero component. Define x t as a component of x with the largest absolute value, i.e., |x t | ≥ |x j | for all j ∈ N. It follows from the definition of Z-eigenvalues that

GANG WANG, GUANGLU ZHOU AND LOUIS CACCETTA
Taking the modulus in (2) and dividing both sides by |x t |, we get Remark 1. When m = 2, exact eigenvalue inclusion theorems can be found in [18].
Theorem 3.2. Let A be a tensor with order m and dimension n ≥ 2. 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. Similar to the proof of Theorem 3.1, we define x t as a component of x with the largest modulus such that |x t | ≥ |x j | for all j ∈ N. Note that Dividing both sides by |x t | in (3) and from |x s | m−1 ≤ |x s |, we have If |x s | = 0, then (|λ| − (R t (A) − |a ts...s |) ≤ 0, and it is obvious that λ ∈ L t,s (A) ⊆ L(A). Otherwise, |x s | > 0. Moreover, similar to (3), we obtain Dividing both sides by |x s | in (5) and from |x t | ≤ 1, one has Multiplying (4) with (6), we see which implies λ ∈ L t,s (A). From the arbitrariness of s, we have λ ∈ j∈N,j =t L t,j (A). Furthermore, λ ∈ i∈N j∈N,j =t L t,j (A). Proof. For any λ ∈ L(A), without loss of generality, there exists t ∈ N such that λ ∈ L t,s (A), that is, If |a ts...s |R s (A) = 0, then In the following theorem, choosing x s as a component of x with the second largest modulus, we obtain some sharp results for σ(A). Theorem 3.3. Let A be a tensor with order m and dimension n ≥ 2. 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., Ax m−1 = λx. Since x is an eigenvector, it has at least one nonzero component.
Obviously, |x t | > 0. Similar to the characterization of (4), we get

GANG WANG, GUANGLU ZHOU AND LOUIS CACCETTA
Dividing both sides by |x s | in (9), one has  Proof. For any λ ∈ M(A), we divide the proof into two parts.
Theorem 3.4. Let A be a tensor with order m and dimension n ≥ 2. Then, all Z-eigenvalues of A are located in the union of the following sets: Furthermore, If |x s | = 0, then (|λ| −   According to Theorem 3.1, we get According to Theorem 3.2, one has It follows from Theorem 3.3 that According to Theorem 3.4, we have 4. Bounds on the largest Z-eigenvalue of weakly symmetric nonnegative tensors. In this section, we give some sharp upper bounds for a weakly symmetric nonnegative tensor, which generalize the results of [3,7,17]. We start this section with some fundamental results of nonnegative tensors [3,7,17].    [7]) Suppose that an m-order n-dimensional nonnegative tensor A is weakly symmetric and positive, and x is an eigenvector associated with the largest Z-eigenvalue ρ(A) of A. Then, Based on the assumption that A is weakly symmetric, Chang et al. [3] established the equivalent relation between the largest Z-eigenvalue and Z-spectral radius of nonnegative tensors. We shall devote to finding sharper upper bounds of the largest Z-eigenvalue for a weakly symmetric nonnegative tensor.
Theorem 4.5. Suppose that an m-order n-dimensional nonnegative tensor A is weakly symmetric. Then, Proof. Using Lemma 4.4, without loss of generality, we assume that ρ(A) = λ * is the largest Z-eigenvalue of A . It follows from Theorem 3.2 that there exists t ∈ N such that Since s ∈ N is chosen arbitrarily, it holds Furthermore,

Remark 2. It is necessary that
whereū is defined in Theorem 4.5.

GANG WANG, GUANGLU ZHOU AND LOUIS CACCETTA
Proof. We divide the proof into two parts. Furthermore, Thus, (14) It follows from (13) and (14) that this corollary holds.
Based on Theorem 3.3, we obtain some sharp bounds of the largest Z-eigenvalue for a weakly symmetric nonnegative tensor. Theorem 4.6. Suppose that an m-order n-dimensional nonnegative tensor A is weakly symmetric. Then,  Suppose that an m-order n-dimensional nonnegative tensor A is weakly symmetric. Then, wherev is defined in Theorem 4.6.
Proof. Similar to the proof of Corollary 4.1, the conclusion holds.
From Theorem 3.4 and Corollary 3.3, we obtain some sharp bounds of the largest Z-eigenvalue for a weakly symmetric nonnegative tensor.

5.
Conclusion. In this paper, we establish Z-eigenvalue inclusion theorems for general tensors and show that Gersgorin-type H-eigenvalue inclusion theorem do not apply to Z-eigenvalue of a general tensor. Moreover, using these Z-eigenvalue inclusion theorems, we obtain some new upper bounds for the largest Z-eigenvalue of a weakly symmetric nonnegative tensor and show that these upper bounds are sharper than the bounds in [3,7,17].