ON THE M-EIGENVALUE ESTIMATION OF FOURTH-ORDER PARTIALLY SYMMETRIC TENSORS

. In this article, the M-eigenvalue of fourth-order partially symmetric tensors is estimated by choosing diﬀerent components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors are discussed, which are sharper than existing upper bounds. Finally, numerical examples are reported to verify the obtained results.

c 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 , (1.1) To establish the criteria in identifying the strong ellipticity in elastic mechanics, Qi et al. [17] introduced the following definition [15]. For λ ∈ R, x ∈ R m , y ∈ R n , if x T x = 1, y T y = 1, (1.2) where (C · yxy) i = k∈[m], j,l∈ [n] c ijkl y j x k y l , and (Cxyx·) l = i,k∈ [m],j∈ [n] c ijkl x i y j x k , then the scalar λ is called an M-eigenvalue of the tensor C, and x and y are called left and right M-eigenvectors of C, respectively, which associated with the M-eigenvalue λ.
Based on the M-eigenvalue with the strong ellipticity [13,14,20], Han et al. [10] proposed the strong ellipticity condition to the rank-one positive definiteness of three second-order tensors, three fourth-order tensors, and a sixth-order tensor. Wang et al. [23] presented a practical method to compute the largest M-eigenvalue of a fourth-order partially symmetric tensor. The research in [17] exhibits that the strong ellipticity holds if and only if all M-eigenvalues of the ellipticity tensor is positive.
Another important similar concept in tensor analysis is Z-eigenvalue problem [18,17] which collapses to M-eigenvalue when the underlying tensor is partially symmetric [17]. The Z-eigenvalue plays an important role in best rank-one approximation, which has a wide range of practical applications in statistical data analysis and engineering [15,26]. For this, Chang et al. [1] proposed upper bounds for Z-spectral radius of nonnegative tensors. Moreover, Song et al. [19] improved the upper bounds for Z-spectral radius based on the relationship between the Gelfand formula and the spectral radius. For weakly symmetric and positive tensors, He et al. [11] presented the Ledermann-like upper bound for the largest Z-eigenvalue. For general tensors, Wang et al. [25] established Z-eigenvalue inclusion theorems, and the upper bounds for the largest Z-eigenvalue of a weakly symmetric nonnegative tensor was obtained.
Generally speaking, the study on high order tensors have attracted much attention of researchers, which made tensor analysis an important tool in theoretical physics, continuum mechanics and many other areas of science and engineering [2,3,4,5,6,21,24,28,27,22,12]. Particulary, Wang et al. [25] established Zeigenvalue inclusion theorems, and the upper bounds for the largest Z-eigenvalue of a weakly symmetric nonnegative tensor was obtained. Since Z-eigenvalue is a special kind of M-eigenvalue, the research on Z-eigenvalue motivates us to consider the estimation on M-eigenvalues. This constitutes the main issue considered in this paper. More precisely, in this paper, several M-eigenvalue localization sets for tensors are obtained by choosing different components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative tensors are discussed. Finally, numerical examples are proposed to verify the theoretical results.
The remainder of this paper is organized as follows. In Section 2, we establish some M-eigenvalue inclusion theorems and give comparisons among these eigenvalue inclusion sets. In Section 3, we apply these inclusion theorems to estimate upper bounds of the largest M-eigenvalue for nonnegative tensors.
2. M-eigenvalue inclusion theorems. In this section, we discuss several Meigenvalue inclusion theorems of fourth-order partially symmetric tensors. Furthermore, we establish comparisons among different M-eigenvalue inclusion sets. The M-spectral radius ρ(C) of C is defined as where σ(C) is the M-spectrum of C, which contains all M-eigenvalues of C.
Inspired by the ideas of H-eigenvalue inclusion theorem [18] and Z-eigenvalue inclusion theorems [25], we can establish the following M-eigenvalue inclusion theorems.
Proof. Let λ be an M-eigenvalue of the tensor C with the associated left Meigenvector x ∈ R m and right M-eigenvector y ∈ R n . Then there exists index It follows from (1.2) that c tjkl y j x k y l .
Furthermore, since y T y = 1, one has |y j | ≤ which implies λ ∈ Γ(C) and the desired result holds.
Otherwise, for |x s | > 0, we have c sjkl y j x k y l . |c sjkl |.
The following conclusion exhibits the relationship between σ(C), L(C) and Γ(C).
Proof. By Theorem 2.1 and Theorem 2.2, it is sufficient to prove L(C) ⊆ Γ(C). Without loss of generality, for any λ ∈ L(C), there exists an index t ∈ [m] such that λ ∈ L t,s (C), for all s = t. Thus . We now break up the argument into two cases.
In what follows, let x s denote the component of the left M-eigenvector x with the second largest modulus. Then we can obtain the following technical results for σ(C).
2), we know that the left M-eigenvector x has at least one nonzero component, thus we can assume that x t is a component of x with the largest absolute value and x s is a component of x with the second largest absolute value. It is easy to check that |x t | > 0.
Following the argument of (2.4), we have c sjkl y j x k y l .

Moreover
. Thus, the desired results follow.
From Theorems 2.1 and 2.4, one has the following conclusion on the relationship between σ(C), M(C) and Γ(C).
Theorem 2.5. Suppose C is a partially symmetric tensor as in Theorem 2.4. Then Proof. For any λ ∈ M(C), we break the proof into two cases.
Combining cases 1 and 2 yields that σ(C) ⊆ M(C) ⊆ Γ(C) and the desired result follows.
Following the argument of Theorem 2.4, we can obtain the following M-eigenvalue inclusion theorem. Theorem 2.6. Let C = (c ijkl ) be a partially symmetric tensor with i, k ∈ [m], j, l ∈ [n]. Then From (1.2), for any left M-eigenvector x, without loss of generality, we assume that x t is the component with largest absolute value and x s is the component such c tjkl y j x k y l . Therefore, |c tjtl ||y j x t y l | + |c sjkl |, which is equivalent to . Thus, λ ∈ N t,s (C) ⊆ N (C) and the desired result follows.
Similar to the argument in Theorems 2.3 and 2.5, we can obtain the following conclusion.

3.
Bounds on the largest M-eigenvalue of nonnegative fourth-order partially symmetric tensors. Based on the obtained results in last section, we present some sharp upper bounds estimation on M-spectral radius of nonnegative fourth-order partially symmetric tensors, which improves the corresponding results in [1,19]. To proceed, we first recall some fundamental results on nonnegative tensors. Here, R i (C) and R k i (C) are defined as in Theorems 2.1 and 2.2, respectively. Lemma 3.1. [1] Let A be an m-th order n-dimensional nonnegative tensor. Then Lemma 3.2. [19] Let A be an m-th order n-dimensional nonnegative tensor. Then

Lemma 3.3. [9]
The M-spectral radius of any nonnegative partially symmetric tensor is exactly its greatest M-eigenvalue. Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the M-spectral radius.
With the help of Theorem 2.2, we can present a sharp bound estimation on the largest M-eigenvalue for nonnegative fourth-order partially symmetric tensors.
Theorem 3.1. Suppose the tensor C is a nonnegative fourth-order partially symmetric tensor. Then Proof. By Lemma 3.3, we can assume that ρ(C) is the largest M-eigenvalue of C. It follows from Theorem 2.2 that there exists an index t ∈ [m] such that Since s ∈ [m] is arbitrary, we have Moreover, and the desired result follows.
From Theorem 3.1, the following comparison conclusion can be readily obtained.
Proof. We break the proof into two cases.
In this case, one has which implies the desired result holds.
By Theorem 2.4, we can establish a sharp bound estimation of the largest Meigenvalue for nonnegative fourth-order partially symmetric tensors. Theorem 3.3. Suppose C is a nonnegative fourth-order partially symmetric tensor. Then Proof. Suppose ρ(C) is the largest M-eigenvalue of C. We break the proof into two cases.
Similar to the proof of Theorem 3.2, one has the following conclusion.
Theorem 3.4. Suppose the tensor C is a nonnegative fourth-order partially symmetric tensor. Then From Theorem 2.6, we have the following conclusion, and the proof is omitted. R i (C). Now, we present some running examples [1,17] to illustrate the improvement of the obtained results.

4.
Conclusions. In this article, several M-eigenvalue localization sets for fourthorder partially symmetric tensors were obtained. As an application, several upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors were discussed, which improve the existing corresponding results.