Bounds for the M-spectral radius of a fourth-order partially symmetric tensor

M-eigenvalues of fourth-order partially symmetric tensors play an important role in many real fields such as quantum entanglement and nonlinear elastic materials analysis. In this paper, we give two bounds for the maximal absolute value of all the M-eigenvalues (called the M-spectral radius) of a fourth-order partially symmetric tensor and discuss the relation of them. A numerical example is given to explain the proposed results.


Introduction
A fourth-order real tensor A = (a i 1 i 2 i 3 i 4 ) ∈ R m×n×m×n is called partially symmetric [1] if it has the following symmetry: . . , n}. Such a tensor often arises in nonlinear elastic materials analysis [2,3] and entanglement studies in quantum physics [4][5][6]. For this tensor, there are many kinds of eigenvalues such as H-eigenvalues, Z-eigenvalues, and Deigenvalues [7,8]; here we only discuss its M-eigenvalues [1,9]. Definition 1 ([9]) Let A = (a i 1 i 2 i 3 i 4 ) ∈ R m×n×m×n be a partially symmetric tensor, and let λ ∈ R. Suppose that there are real vectors x ∈ R m and y ∈ R n such that where A · yxy ∈ R m and Axyx· ∈ R n with ith components Then λ is called an M-eigenvalue of A with left M-eigenvector x and right M-eigenvector y.
Note that M-eigenvalues of a fourth-order partially symmetric tensor always exist [1]. They have a close relation to many problems in the theory of elasticity and quantum physics [1,9,10]. For example, the largest M-eigenvalue of A = (a i 1 i 2 i 3 i 4 ) ∈ R m×n×m×n , denoted by is the optimum solution of the problem (for details, see [9]) and λ is an M-eigenvalue with the maximal absolute value of A = (a i 1 i 2 i 3 i 4 ) ∈ R m×n×m×n with left M-eigenvector x ∈ R m and right M-eigenvector y ∈ R n , is a partially symmetric best rank-one approximation of A [1], which has wide applications in signal and image processing, wireless communication systems, and independent component analysis [11][12][13][14]. The M-spectral radius of A = (a i 1 i 2 i 3 i 4 ) ∈ R m×n×m×n , denoted by has significant impacts on identifying nonsingular M -tensors, which satisfy the strong ellipticity condition [10]. To our knowledge, there are few results about bounds for the M-spectral radius of a fourth-order partially symmetric tensor. In this paper, we present two bounds for the Mspectral radius and discuss their relation. A numerical example is also given to explain the proposed results.

Two bounds for the M-spectral radius
In this section, we give two bounds for the M-spectral radius of fourth-order partially symmetric tensors and discuss their relation.
be a partially symmetric tensor. Then where Proof Suppose that λ is an M-eigenvalue of A and that x ∈ R m and y ∈ R n are associated left M-eigenvector and right M-eigenvector. Then (1) holds. Let |y k | .
Since x T x = 1 and y T y = 1, we have The pth equation of A · yxy = λx is Taking the absolute values on both sides of (4) and using the triangle inequality give Similarly, by the qth equation of Axyx· = λy we have Multiplying (5) and (6) gives which, together with (3), yields Since (7) C l (A) , and the conclusion follows.

Theorem 2 Let A = (a i 1 i 2 i 3 i 4 ) ∈ R m×n×m×n be a partially symmetric tensor, and let α be any subset of [m] and β be any subset of [n]. Then
where Proof Assume that λ is an M-eigenvalue of A and that x ∈ R m and y ∈ R n are the corresponding left M-eigenvector and right M-eigenvector. Then (1) holds. Let |y k | .
Then (3) holds. The pth equation of A · yxy = λx can be rewritten as By the technique for the inequality in Theorem 1, we obtain from (9) that In addition, by the qth equation of Axyx· = λy we have Multiplying (10) with (11) and using (3) yield Then Note that (13) holds for all M-eigenvalues of A and any α ⊆ [m]. Hence On the other hand, for the qth equation of Axyx· = λy, we have Then By the pth equation of A · yxy = λx we have Multiplying (16) with (17) and using (3), we derive Hence From (14) and (20) we have The proof is completed.
when α = ∅ and β = ∅, we have Therefore, the bound in (8) is tighter than the bound in (2) for the M-spectral radius ρ M (A) of a given tensor A. (8) is tighter than the bound in (2), it is easier to compute the bound in (2) for the M-spectral radius of a given tensor.

Remark 2 Although the bound in
Next, we use a numerical example to show the effectiveness of the bounds in Theorems 1 and 2.

Conclusions
In this paper, we have presented two bounds for the M-spectral radius of a fourth-order partially symmetric tensor and have indicated their relation. To show the effectiveness of the proposed results, a numerical example is also given.