Identifying the Positive Deﬁniteness of Even-Order Weakly Symmetric Tensors via Z -Eigenvalue Inclusion Sets

: The positive deﬁniteness of even-order weakly symmetric tensors plays important roles in asymptotic stability of time-invariant polynomial systems. In this paper, we establish two Brauer-type Z -eigenvalue inclusion sets with parameters by Z -identity tensors, and show that these inclusion sets are sharper than existing results. Based on the new Z -eigenvalue inclusion sets, we propose some sufﬁcient conditions for testing the positive deﬁniteness of even-order weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments are reported to show the efﬁciency of our results.


Introduction
Let A = (a i 1 i 2 ···i m ) ∈ (R n ) ⊗m be an m-th order n dimensional real square tensor and x be a real n-vector and N = {1, 2, · · · , n}. Consider the following real n-vector: (Ax m−1 ) i = ∑ i 2 ,...,i m ∈N a ii 2 ···i m x i 2 . . . x i m .
If there exists a unimodular vector x ∈ R n and a real number λ such that Ax m−1 = λx, then λ is called a Z-eigenvalue of A and x is called a Z-eigenvector of A associated with λ [1,2].
Z-eigenvalue problems of tensors were constantly emerging due to their wide applications in medical resonance [3,4], spectral hypergraph theory [5,6], automatic control [7,8] and machine learning [9]. Some effective algorithms for finding Z-eigenvalues and the corresponding eigenvectors have been proposed [5,[10][11][12][13][14][15]. However, it is difficult to compute all Z-eigenvalues, even the smallest Z-eigenvalue when m and n are large [16,17]. Thus, many researchers turned to investigating Z-eigenvalue inclusion sets [10,[18][19][20][21][22]. Later, Qi et al. [13] investigated Z-eigenvalues to identify the positive definiteness of a degree m with n variables homogeneous polynomials with unit constraint: (1) where A (t) = (a i 1 i 2 ...i t ) ∈ (R n ) ⊗t (t = 2, 4, . . . , 2k). Deng et al. [27] investigated asymptotical stability of time-invariant polynomial systems without constraint via Lyapunov's method. Further, Li et al. [26] established asymptotical stability of systems (2) based on Gershgorintype Z-eigenvalue inclusion sets. So, we want to exactly verify the asymptotical stability of time-invariant polynomial systems use of Brauer-type Z-eigenvalue inclusion sets, which constitutes the second motivation of the paper. This paper is organized as follows. In Section 2, important definitions and preliminary results are recalled. In Section 3, two Brauer-type Z-eigenvalue inclusion sets with parameters are established. In Section 4, some sufficient conditions are proposed for identifying positive definiteness of even-order weakly symmetric tensors and asymptotic stability of time-invariant polynomial systems.

Preliminaries
In this section, we introduce some definitions and important properties related to Z-eigenvalues of a tensor [12,13,26]. Definition 1. Let A be m-order n-dimensional tensors.
(i) We define σ Z (A) as the set of all Z-eigenvalues of A. Assume σ Z (A) = ∅. Then, the Z-spectral radius of A is denoted by where I I m is the permutation group of m indices.
(iii) We say that A is weakly symmetric if the associated homogeneous polynomial Ax m satisfies where ∇ is the differential operator.
Obviously, if tensor A is symmetric, then A is weakly symmetric. However, the converse result may not hold.
Based on variational property of weakly symmetric tensors given in [10], we establish the following result. Lemma 1. Let A ∈ (R n ) ⊗m be a weakly symmetric tensor. Then, f A (x) = Ax m is positive definite if and only if its Z-eigenvalues are positive.
Proof. Since A is a weakly symmetric tensor, we have where ·, · denotes the inner product. Thus, we observe that λ ∈ σ Z (A) if and only if λ is a critical value of Ax m on the standard unit sphere.
Hence, f A (x) > 0 if and only if λ > 0. The conclusion follows. 2 A Z-identity tensor was introduced by [2,12] to propose a shifted power method for computing tensor Z-eigenpairs and investigate a generalization of the characteristic polynomial for symmetric even-order tensors, respectively.

Definition 2.
Assume that m is even. We call I Z a Z-identity tensor if Note that there is no Z-identity tensor for m odd [12]. Meanwhile, Z-identity tensor is not unique in general. For instance, each even tensor in the following is a Z-identity tensor: Case I: (I Z ) iii 2 i 2 ...i k i k = 1, ∀k ∈ N and m = 2k; Case II: (Property 2.4 of [12]): is the standard Kronecker, i.e., To end this section, we introduce the results given in [26].

Sharp Z-Eigenvalue Inclusion Sets for Even Tensors
In this section, we establish Brauer-type Z-eigenvalue inclusion sets and give comparisons among different Z-eigenvalue inclusion sets for even-order tensors. Theorem 1. Let A = (a i 1 i 2 ...i m ) ∈ (R n ) ⊗m and I Z ∈ (R n ) ⊗m be a Z-identity tensor. For any real vector α = (α 1 , . . . , α n ) ∈ R n , then Proof. Let (λ, x) be a Z-eigenpair of A and I Z ∈ (R n ) ⊗m be a Z-identity tensor, i.e., Ax m−1 = λx = λI Z x m−1 and x x = 1. ( Assume Hence, for any real number α t , it holds that Taking modulus in (4) and using the triangle inequality give . Otherwise, |x j | > 0. For any j ∈ N, j = t and any real number α j , we obtain Multiplying (5) with (6) yields for all j, it follows that λ lives in the intersection and hence λ ∈ j∈N,j =t Next, we show L(A, α) ⊆ G(A, α).
..i m ) ∈ (R n ) ⊗m and I Z ∈ (R n ) ⊗m be a Z-identity tensor. For any real vector α = (α 1 , . . . , α n ) ∈ R n , then Proof. For any λ ∈ L(A, α), without loss of generality, there exists t ∈ N with any j = t such that λ ∈ L t,j (A, α), that is, By classifying the index set, we can get an accurate characterization for σ(A). Define Proof. Let (λ, x) be a Z-eigenpair of A and I Z ∈ (R n ) ⊗m be a Z-identity tensor, i.e., The following argument is divided into two cases.
Summing up the above two situations, we draw the conclusion. 2 Now, we show the set N (A, α) is tighter than G(A, α).

Corollary 2.
Let A = (a i 1 i 2 ...i m ) ∈ (R n ) ⊗m and I Z ∈ (R n ) ⊗m be a Z-identity tensor. For any real vector α = (α 1 , . . . , α n ) ∈ R n , then Proof. For any λ ∈ N (A, α), we break the proof into two parts. Case 1. There exist t, s ∈ N with s = t such that λ ∈ N t,s (A, α), that is, Otherwise, |a ts...s − α t (I Z ) ts...s |r Θ t s (A, α s ) > 0. Then, (11) entails Next, we give a numerical comparison among Theorems 1 and 2 and existing results. By simple computation, all Z-eigenvalues of A are 5.0000 and 10.0000. Taking positive vector α = [10,7] and I Z as follows: We compute Table 1 to show the comparisons different methods with our results. Numerical results show that Theorems 1 and 2 are tighter than existing results. In the following, setting α 1 = [10,7] , α 2 = [9, 5] and α 3 = [9, 5.5] , we obtain inclusion sets by different theorems. Consequently, the parameter α has a great influence on the numerical effects. The parameter α has a great influence on the numerical effects from Table 2.

Positive Definiteness of even Order Weakly Symmetric Tensors and Asymptotic Stability of Polynomial Systems
In this section, based on the inclusion sets L(A, α) and N (A, α) in Theorems 1 and 2, we propose some sufficient conditions for the positive definiteness of weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems.

Positive Definiteness of even Order Weakly Symmetric Tensors
Li et al. [26] proposed the following theorem to test the positive definiteness of polynomial systems via Gershgorin-type Z-eigenvalue inclusion sets. [26]). Let A = (a i 1 i 2 ...i m ) ∈ (R n ) ⊗m and λ be a Z-eigenvalue of A. If there exists a positive real vector α = (α 1 , . . . , α n ) such that (1) is positive definite.

Theorem 3.
Let λ be a Z-eigenvalue of A = (a i 1 i 2 ...i m ) ∈ (R n ) ⊗m and I Z ∈ (R n ) ⊗m be a Zidentity tensor. If there exists a positive real vector α = (α 1 , . . . , α n ) and i, j ∈ N with j = i such that then λ > 0. Further, if A is weakly symmetric, then A is positive definite and f A (x) defined in (1) is positive definite.
Proof. Suppose on the contrary that λ ≤ 0. From Theorem 1, there exists t ∈ N with any j = t such that λ ∈ L t,j (A, α), i.e., Further, it follows from α t > 0 and λ ≤ 0 that which contradicts (12). Thus, λ > 0. When A is a weakly symmetric tensor and all Zeigenvalues are positive, we obtain that A is positive definite and f A (x) defined in (1) is positive definite by Lemma 1. and then λ > 0. Further, if A is weakly symmetric, then A is positive definite and f A (x) defined in (1) is positive definite.
Proof. Suppose on the contrary that λ ≤ 0. The following argument is divided into two cases. Case 1. There exist t, s ∈ N with s = t such that λ ∈ N t,s (A, α), i.e., Further, it follows from α s > 0, α t > 0 and λ ≤ 0 that which contradicts (13). Thus, λ > 0. Case 2. There exist t, s ∈ N with s = t such that λ ∈ H t,s (A, α), i.e., Further, it follows from α s > 0, α t > 0 and λ ≤ 0 that which contradicts (14). Thus, λ > 0. From the above two cases, when A is a weakly symmetric tensor and all Z-eigenvalues are positive, we obtain that A is positive definite and f A (x) defined in (1) is positive definite from Lemma 1.
2 Remark 1. Compared with Theorem 3.2 of [26], our conclusions can more accurately determine the positive definiteness for even order weakly symmetric tensors.
The following example reveals that Theorems 3 and 4 can judge the positive definiteness of weakly symmetric tensors. Firstly, we can verify that A is not a symmetric tensor, but a weakly symmetric tensor. By computations, we obtain that the minimum Z-eigenvalue and corresponding with the Z-eigenvector are (λ,x) = (1.3543, (0.2300, 0.9732)). Hence, A is positive definite.

Example 2. Let
Taking the Z-identity tensor I Z as Case I or Case II in Definition 2, we cannot find positive real number α 2 such that which shows that Theorem 3.2 of [26] fails to check the positive definiteness of weakly symmetric tensor A.
Proof. Following the proof of Theorem 3.3 of [27], we find that the equilibrium point of ∑ in (2) is asymptotically stable via Lyapunov's method. 2 The following example shows the validity of Theorem 5.
Hence, −A (4) is positive definite. Further, it follows from Theorem 5 that the equilibrium point of ∑ is asymptotically stable.

Conclusions
In this paper, we established new Brauer-type Z-eigenvalue inclusion sets for evenorder tensors by Z-identity tensor and proposed some sufficient conditions for the positive definiteness of weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments showed its validity. It is remarkable that suitable parameter α has a great influence on the numerical effect and positive definiteness. Therefore, how to select the suitable parameter α is our further research.

Conflicts of Interest:
The authors declare no conflicts of interest.