M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the eciency of the obtained results. As applications, we provide some checkable sucient conditions for the positive deniteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.


Introduction
Let R be the set of all real numbers, R n be the set of all dimension n real vectors, and [n] 1, 2, . . ., n { }a fourthorder real tensor, denoted by (1) n] is called partially symmetric, if its components are invariant under the following permutation of subscripts: ( In fact, the tensor of elastic moduli for elastic materials exactly is partially symmetric [1], and the components of such tensor are regarded as the coe cients of the following biquadratic homogeneous polynomial optimization problem: min f A (x, y) Axyxy i,k∈[m] j,l∈ [n] a ijkl x i y j x k y l s.t.x T x 1, y T y 1, x ∈ R m , y ∈ R n .
is optimization problem induced by tensor A, nds applications in nonlinear elastic materials analysis [2], the ordinary ellipticity and strong ellipticity [1,3], and stability study of nonlinear autonomous systems [4,5].As we know, the strong ellipticity condition is essential in theory of elasticity, which guarantees the existence of solutions of basic boundary-value problems of elastostatics and ensures an elastic material to satisfy some mechanical properties.Qi et al. [6] pointed out that the strong ellipticity condition holds if and only if the optimal value of problem (3) is positive.To establish the criteria in identifying the strong ellipticity in elastic mechanics, Qi et al. [6,7] introduced the following de nition.

De nition 1. Let
where (A • yxy) i �  k∈[m],j,l∈ [n] a ijkl y j x k y l and (Axyx•) l �  i,k∈ [m],j∈ [n] a ijkl x i y j x k , then the scalar λ is called an M-eigenvalue of the tensor A and x and y are called left and right M-eigenvectors of A, respectively, which are associated with the M-eigenvalue λ.Denote σ M (A) as the set of all M-eigenvalues of A. en, the M-spectral radius of A is denoted by Note that f A (x, y) is positive definite if and only if Meigenvalues of A are positive [7].Hence, effective algorithms for finding M-eigenvalue and the corresponding eigenvector have been implemented [8][9][10][11][12][13][14][15][16].Due to the complexity of the tensor eigenvalue problem [17,18], it is difficult to compute all M-eigenvalues.
us, some researchers turned to investigating the inclusion sets of M-eigenvalue [19][20][21].For example, Che et al. [19] proposed a Gershgorin-type Minclusion set as follows. where Unfortunately, the mentioned inclusion sets always include zero and cannot identify the positive definiteness of f A (x, y). [2]defined by

Example 1. Consider the following partially symmetric tensor
From Lemma 1, it holds that By computation, we can obtain that the corresponding M-eigenvalues are 7, 20.Hence, A is positive definite.However, we could not use Γ(A) to identify the positive definiteness of A. To overcome the drawback above, we present new M-eigenvalue inclusion intervals with n parameters, which can be used to identify the positive definiteness of fourth-order partially symmetric tensors. is paper is organized as follows.In Section 2, we establish two M-eigenvalue inclusion intervals for fourthorder partially symmetric tensors.In Section 3, we propose some checkable sufficient conditions of the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.Numerical examples are proposed to verify the efficiency of the obtained results.
Definition 2. We call n] an M-identity tensor if its entries are where n] is a partially symmetric tensor and has the following property: n] be a partially symmetric tensor and I M be an M-identity tensor.For any α � (α where Proof.Let (λ, x, y) be an M-eigenpair of A and I M be an M-identity tensor.From the definition of M-identity tensor and (11), it holds From the tth equality of ( 14), we obtain a tjkl y j x k y l . (15) erefore, which implies that λ ∈ G t (A, α) ⊆ G(A, α).From the arbitrariness of α, the conclusion follows.n] be a partially symmetric tensor and I M be an M-identity tensor.For any α where Proof.Let (λ, x, y) be an M-eigenpair of A and I M be an M-identity tensor.Set From the tth equation of A • yxy � λx in (4), for any p ∈ [m], p ≠ t and any real number α t , we have Taking modulus in the above equation and using the triangle inequality give at is, Multiplying ( 23) with (26) yields which implies that λ ∈ K t,p (A, α).From the arbitrariness of p, it follows that Remark 1.
(i) It is clear that eorems 1 and 2 reduce to eorems 2.1 and 2.2 of [19] if one takes α � 0, respectively.Consequently, the upper bounds of ρ M (A) in eorems 1 and 2 are smaller than those in eorems 2.1 and 2.2 of [19].(ii) By using the equation Axyx � λy, we can establish some conclusions similar to eorems 1 and 2.
be a partially symmetric tensor and I M be an M-identity tensor.For any α Proof.For any λ ∈ K(A, α), without loss of generality, there exists t We now break up the argument into two cases. Hence, which implies that In summary, σ M (A) ⊆ K(A, α) ⊆ G(A, α) and the desired result follows.
e following example exhibits the superiority of the results given in eorems 1 and 2. [2] in Example 1.

Example 2. Consider
Set α � (14, 8.5) T .For this tensor, the bounds via different estimations given in the literature are shown in Table 1.
It is easy to see that the results given in eorems 1 and 2 are sharper than some existing results.
We observe that the suitable parameter α has a great influence on the numerical effects (Table 2).n] are generated with m � n as a ijij � 4i + j and other elements are generated randomly in [− 0.5, 0.5].

Example 3. All testing partially symmetric tensors
e choice of parameter α is derived as follows: For the tensors with different dimensions, the values presented in the table are the average values of 10 examples (Table 3).

Applications
In this section, based on the inclusion intervals G(A, α) and K(A, α) in eorems 1 and 2, we propose some sufficient conditions for the positive definiteness and make bound estimations on the M-spectral radius of nonnegative fourthorder partially symmetric tensors.

Positive Definiteness of Fourth-Order Partially Symmetric Tensors
be a partially symmetric tensor and I M be an M-identity tensor.For i ∈ [m], if there exists positive real vector α � (α 1 , . . ., α m ) T such that then A is positive definite and f A (x, y) defined in ( 3) is positive definite.
Proof.Suppose on the contrary that λ ≤ 0. From eorem 1, there exists i On the other hand, by α i 0 > 0 and λ ≤ 0, we have which contradicts (35).Hence, λ > 0. Since A is partially symmetric and all M-eigenvalues are positive, A is positive definite and f A (x, y) defined in (3) is positive definite.□ be a partially symmetric tensor and I M be an M-identity tensor.For i ∈ [m], if there exist a positive real vector α � (α 1 , . . ., α m ) T and k ≠ v such that then A is positive definite and f A (x, y) defined in ( 3) is positive definite.

Complexity
Proof.Suppose on the contrary that λ ≤ 0. From eorem 2, there exist Further, it follows from α i > 0 and λ ≤ 0 that which contradicts (38).Hence, λ > 0. Since A is partially symmetric and all M-eigenvalues are positive, A is positive definite and f A (x, y) defined in (3) is positive definite.e following example show eorems 3 and 4 can judge the positive definiteness of fourth-order partially symmetric tensors. [2]defined by
Set α � (8, 4) T .According to eorem 3, we have Hence, A satisfies all conditions of eorem 3, which implies that A is positive definite.According to eorem 4, it holds (43) Hence, A satisfies all conditions of eorem 4, which implies that A is positive definite.e following example reveals that eorem 4 can judge the positive definiteness of partially symmetric tensors more accurately than eorem 3. [2] defined by a ijkl � a 1111 � 10, a 1212 � 8, a 1122 � a 1221 � 0.5;

Example 5. Consider the partially symmetric tensor
By eorem 7 of [7], we observe that the minimum Meigenvalue and corresponding with left and right M-eigenvectors are (λ, x, y) � (2.5774, (0.2724, 0.9622), (− 0.0452, 0.9990)).Hence, A is positive definite.For any positive real number α 2 , we have which implies that the condition of eorem 3 is not satisfied.us, eorem 3 is not suitable to check the positive definiteness of A. However, taking α � (10, 5) T , from eorem 4, we have which can show the positive definiteness of A.

Bound Estimations on the M-Spectral Radius.
Based on eorems 1 and 2, we present sharp bound estimations on M-spectral radius of fourth-order partially symmetric nonnegative tensors, which improves the corresponding results in [19,20].For M-eigenvalues and associated left and right M-eigenvectors of fourth-order partially symmetric tensors, Qi and Luo [7] provided several related results.
Lemma 2 (Theorem 1 of [7]).M-eigenvalues always exist.If x and y are left and right M-eigenvectors of A, associated with an M-eigenvalue λ, then λ � Axyxy.
be a partially symmetric nonnegative tensor.e M-spectral radius of A is For the medium-sized tensors, we show the validity of the estimations given by our theorems.

Complexity
exactly its largest M-eigenvalue.Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the Mspectral radius.
Proof.Assume that λ * is the largest M-eigenvalue of A. It is clear that λ * ≤ ρ M (A).In the following, we show λ * ≥ ρ M (A).It follows from Lemma 2 that there exist left and right M-eigenvectors (x * , y * ) of λ * such that Obviously, λ * ≥ 0. Next, we show (52) Setting a feasible solution of (52) we have (55) From ( 52) and (55), the conclusion follows.
be a partially symmetric nonnegative tensor and I M be an M-identity tensor.For real vector α  6 Complexity Furthermore, n] be a partially symmetric nonnegative tensor and I M be an M-identity tensor.For real vector α Proof In the following, we use Example 1 of [20] to show the superiority of our results.□ Example 6.Consider the partially symmetric tensor A � (a ijkl ) ∈ R [2] ]×[ [2] ]×[ [2] ]×[ [2] defined by  4.
It is easy to see that the result given in eorem 6 is sharper than some existing results.

Conclusions
In this paper, we introduced M-identity tensor to establish sharp M-eigenvalue inclusion intervals.Further, we proposed some sufficient conditions for the positive definiteness of four-order partially symmetric tensors.
e given experiments show the validity of the obtained results.It is worth noting that suitable parameter α has a great influence on the numerical effects and positive definiteness.erefore, how to select the suitable parameter α is our further research.

)
Proof.Without loss of generality, we assume that ρ(A) � λ * is the largest M-eigenvalue of A by Lemma 3. It follows from eorem 1 that there exists t ∈ N such that ρ M (A) − α t         ≤ R t A, α t .(57)SinceA is nonnegative andα i ≤ max i∈[m],j∈[n] a ijij  , from Lemma 4 and (57), we deduce

Table 3 References
Noting that A is nonnegative andα i ≤ max i∈[m],j∈[n] a ijij , from Lemma 4 and (61), we have whereΔ t,v (A) � (α t − α v + [(R t (A, α t ) − R v t (A, α t ))]) 2 + 4(R v t (A, α t )R v (A, α v )).Since v ∈ [m] is chosen arbitrarily, it holds ρ M (A) ≤ min . Without loss of generality, we assume that ρ M (A) � λ * is the largest M-eigenvalue of A by Lemma 3. It follows from eorem 2 that there exists t ∈ N such thatλ − α i         − R i A, α i  − R v i A, α i     λ − α v         ≤ R v i A, α i R v A, α v , ∀v ≠ t, (61) M (A) − α t − R i A, α i  − R v i A, α i    ρ M (A) − α v  ≤ R v i A, α i R v A, α v , ∀v ≠ t. t + α v + R t A, α t  − R M (A) ≤ max i∈[m]min v≠i,v∈[m]