Abstract
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 efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.
1. Introduction
Let be the set of all real numbers, be the set of all dimension n real vectors, and a fourth-order real tensor, denoted by , consists of components:
Specifically, 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 coefficients of the following biquadratic homogeneous polynomial optimization problem:
This optimization problem induced by tensor , finds 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 definition.
Definition 1. Let be a partially symmetric real tensor. For , ifwhere and , then the scalar λ is called an M-eigenvalue of the tensor and x and y are called left and right M-eigenvectors of , respectively, which are associated with the M-eigenvalue λ. Denote as the set of all M-eigenvalues of . Then, the M-spectral radius of is denoted byNote that is positive definite if and only if M-eigenvalues of are positive [7]. Hence, effective algorithms for finding M-eigenvalue and the corresponding eigenvector have been implemented [8–16]. Due to the complexity of the tensor eigenvalue problem [17, 18], it is difficult to compute all M-eigenvalues. Thus, some researchers turned to investigating the inclusion sets of M-eigenvalue [19–21]. For example, Che et al. [19] proposed a Gershgorin-type M-inclusion set as follows.
Lemma 1 (Theorem 2.1 of [19]). Suppose be a partially symmetric real tensor. Then,whereUnfortunately, the mentioned inclusion sets always include zero and cannot identify the positive definiteness of .
Example 1. Consider the following partially symmetric tensor defined byFrom Lemma 1, it holds thatBy computation, we can obtain that the corresponding M-eigenvalues are . Hence, is positive definite. However, we could not use to identify the positive definiteness of . 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.
This paper is organized as follows. In Section 2, we establish two M-eigenvalue inclusion intervals for fourth-order 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.
2. M-Eigenvalue Inclusion Intervals for Fourth-Order Partially Symmetric Tensors
In this section, inspired by H-eigenvalue inclusion theorems [22–26] and Z-eigenvalue inclusion intervals [27–31], we establish two M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors. We begin our work by introducing M-identity tensor.
Definition 2. We call an M-identity tensor if its entries arewhere .
Obviously, is a partially symmetric tensor and has the following property:with for all .
Theorem 1. Let be a partially symmetric tensor and be an M-identity tensor. For any , thenwhereFurther, .
Proof. Let be an M-eigenpair of and be an M-identity tensor. From the definition of M-identity tensor and (11), it holdsSetting by , one has From the tth equality of (14), we obtainHence, for any real number , it follows thatTaking modulus in the above equation, one hasTherefore,which implies that . From the arbitrariness of , the conclusion follows.
Theorem 2. Let be a partially symmetric tensor and be an M-identity tensor. For any , thenwhereFurther, .
Proof. Let be an M-eigenpair of and be an M-identity tensor. Set . Since , it holds that . From the tth equation of in (4), for any and any real number , we haveTaking modulus in the above equation and using the triangle inequality giveThus,If , thenwhich shows . Otherwise, for , we obtainThat is,Multiplying (23) with (26) yieldswhich implies that . From the arbitrariness of p, it follows that . Further, . It follows from the arbitrariness of α that .
Remark 1. (i)It is clear that Theorems 1 and 2 reduce to Theorems 2.1 and 2.2 of [19] if one takes , respectively. Consequently, the upper bounds of in Theorems 1 and 2 are smaller than those in Theorems 2.1 and 2.2 of [19].(ii)By using the equation , we can establish some conclusions similar to Theorems 1 and 2.
Corollary 1. Let be a partially symmetric tensor and be an M-identity tensor. For any , then
Proof. For any , without loss of generality, there exists such that , for all . Thus,We now break up the argument into two cases.
Case 1. If , thenHence,Therefore, .
Case 2. If , thenwhich implies thatorThus, .
In summary, and the desired result follows.
The following example exhibits the superiority of the results given in Theorems 1 and 2.
Example 2. Consider the tensor in Example 1.
Set . 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 Theorems 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).
Example 3. All testing partially symmetric tensors are generated with as and other elements are generated randomly in .
The 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).
3. Applications
In this section, based on the inclusion intervals and in Theorems 1 and 2, we propose some sufficient conditions for the positive definiteness and make bound estimations on the M-spectral radius of nonnegative fourth-order partially symmetric tensors.
3.1. Positive Definiteness of Fourth-Order Partially Symmetric Tensors
Theorem 3. Let be a partially symmetric tensor and be an M-identity tensor. For , if there exists positive real vector such thatthen is positive definite and defined in (3) is positive definite.
Proof. Suppose on the contrary that . From Theorem 1, there exists such that , i.e.,On the other hand, by and , we havewhich contradicts (35). Hence, . Since is partially symmetric and all M-eigenvalues are positive, is positive definite and defined in (3) is positive definite.
Theorem 4. Let be a partially symmetric tensor and be an M-identity tensor. For , if there exist a positive real vector and such thatthen is positive definite and defined in (3) is positive definite.
Proof. Suppose on the contrary that . From Theorem 2, there exist such that , i.e.,Further, it follows from and thatwhich contradicts (38). Hence, . Since is partially symmetric and all M-eigenvalues are positive, is positive definite and defined in (3) is positive definite.
The following example show Theorems 3 and 4 can judge the positive definiteness of fourth-order partially symmetric tensors.
Example 4. Consider the partially symmetric tensor defined byFrom the calculation method provided in Theorem 7 of [7], we obtain that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are . Hence, is positive definite.
Set . According to Theorem 3, we haveHence, satisfies all conditions of Theorem 3, which implies that is positive definite.
According to Theorem 4, it holdsHence, satisfies all conditions of Theorem 4, which implies that is positive definite.
The following example reveals that Theorem 4 can judge the positive definiteness of partially symmetric tensors more accurately than Theorem 3.
Example 5. Consider the partially symmetric tensor defined byBy Theorem 7 of [7], we observe that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are . Hence, is positive definite. For any positive real number , we havewhich implies that the condition of Theorem 3 is not satisfied. Thus, Theorem 3 is not suitable to check the positive definiteness of . However, taking , from Theorem 4, we havewhich can show the positive definiteness of .
3.2. Bound Estimations on the M-Spectral Radius
Based on Theorems 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 , associated with an M-eigenvalue λ, then .
Lemma 3. Let be a partially symmetric nonnegative tensor. The M-spectral radius of is exactly its largest M-eigenvalue. Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the M-spectral radius.
Proof. Assume that is the largest M-eigenvalue of . It is clear that . In the following, we show . It follows from Lemma 2 that there exist left and right M-eigenvectors of such thatObviously, . Next, we show is a M-eigenpair of . Since is nonnegative, and , we obtainwhich implies . Consequently, is a nonnegative M-eigenpair of . Meanwhile, let be a corresponding M-eigenvector of with . Since is nonnegative and is largest value of , we havewhich showsThus, .
Lemma 4. Let be a partially symmetric tensor. If is nonnegative, then
Proof. Without loss of generality, we assume that is the largest M-eigenvalue of by Lemma 3. It follows from Lemma 2 thatLet Setting a feasible solution of (52)we havewhich implies .
Meanwhile, taking a feasible solution from (52), we obtainFrom (52) and (55), the conclusion follows.
Theorem 5. Let be a partially symmetric nonnegative tensor and be an M-identity tensor. For real vector with , then
Proof. Without loss of generality, we assume that is the largest M-eigenvalue of by Lemma 3. It follows from Theorem 1 that there exists such thatSince is nonnegative and , from Lemma 4 and (57), we deduceFurthermore,
Theorem 6. Let be a partially symmetric nonnegative tensor and be an M-identity tensor. For real vector with , thenwhere .
Proof. Without loss of generality, we assume that is the largest M-eigenvalue of by Lemma 3. It follows from Theorem 2 that there exists such thatNoting that is nonnegative and , from Lemma 4 and (61), we haveSolving (63) for giveswhere . Since is chosen arbitrarily, it holdsFurthermore,In the following, we use Example 1 of [20] to show the superiority of our results.
Example 6. Consider the partially symmetric tensor defined byIn fact, . From Lemma 4, we compute . Set . For this tensor, the bounds via different estimations given in the literature are shown in Table 4.
It is easy to see that the result given in Theorem 6 is sharper than some existing results.
4. 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. The 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. Therefore, how to select the suitable parameter α is our further research.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this manuscript.
Acknowledgments
This work was supported by the Natural Science Foundation of China (11671228 and 11801430).