New Sufﬁcient Condition for the Positive Deﬁniteness of Fourth Order Tensors

: In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z -spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufﬁcient condition for the positive deﬁniteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efﬁciency of our results.


Introduction
Let A = (a i 1 i 2 ···i m ) ∈ R [m,n] be an m-th order n dimensional real square tensor, x be a real n-vector. Then, let N = {1, 2, . . . , n}, we define the following real n-vector: If there exists a real vector x and a real number λ such that Ax m−1 = λx [m−1] , then λ is called H-eigenvalue of A and x is called H-eigenvector of A associated with λ. If there exists a real vector x and a real number λ such that then λ is called Z-eigenvalue of A and x is called Z-eigenvector of A associated with λ [1,2].
An mth-degree homogeneous polynomial form of n variables is positive definite, i.e., f (x) > 0, if and only if the real symmetric tensor A is positive definite [2]. When m is even, an eigenvalue method is given to verify the positive definiteness of A.
Theorem 1 ([2]). Let A be an even-order real symmetric tensor. Then (1) A is positive definite if and only if all of its H-eigenvalues are positive; (2) A is positive definite if and only if all of its Z-eigenvalues are positive.
From Theorem 1, we can verify the positive definiteness of A by the H-eigenvalues or the Z-eigenvalues of A. But when m and n are large, it is difficult to compute all the H-eigenvalues (or Z-eigenvalues) or the smallest H-eigenvalue (or Z-eigenvalue) of an order m dimension n real tensor A. Based on the Geršhgorin-type theorem for H-eigenvalues, which is introduced in [2], Li et al. provided some sufficient conditions for the positive definiteness of an even-order real symmetric tensor [3], and some improved results are obtained in [4][5][6][7][8].
The following useful theorem is given in [7].

Theorem 2 ([7]
). Let A be an even-order real symmetric tensor with all positive diagonal entries.
(1) If A is strictly diagonally dominant, then A is positive definite; (2) If A is quasi-doubly strictly diagonally dominant, then A is positive definite.
Positive definiteness of fourth order tensors has important applications in signal processing, automatic control, and magnetic resonance imaging [9][10][11][12]. Recently, in order to preserve positive definiteness for a fourth order tensor, a ternary quartics approach is proposed in [13]. Extending the Riemannian framework from 2nd order tensors to the space of 4th order tensors, a riemannian approach is given to guarantee positive definiteness for a fourth order tensor [14]. In [11], the authors explain the definition of the smallest Z-eigenvalue and present a computational method for calculating it. Very recently, much literature has focused on the properties of Z-eigenvalues of tensors [15][16][17][18][19][20][21][22][23][24], but there are no Z-eigenvalues based sufficient conditions for the positive definiteness of an even-order real symmetric tensor.
In this paper, based on the Z-eigenvalue localization sets of structured fourth order tensors, a new sufficient condition for the positive definiteness of fourth order tensors is given.

New Z-Eigenvalue Localization Set for Structured Fourth Order Tensors
In this section, a Geršhgorin-type theorem for Z-eigenvalues of structured fourth order tensors is obtained. For any k ∈ N, let where we assume that We give our main results in this section as follows. Then Taking modulus in the above equation, and using the triangle inequality and . If |x s | > 0, from equality (2), we similarly get Multiplying inequalities (3) with (4), we have Thus, we complete the proof.

Upper Bound for the Z-Spectral Radius of Weakly Symmetric Nonnegative Tensors
In this section, we obtain a sharp upper bound for weakly symmetric nonnegative tensors. Firstly, let us recall the definition of the Z-spectral radius of tensor A. We need the following Perron-Frobenius Theorem for the Z-eigenvalue of nonnegative tensors [23].

Lemma 1.
Suppose that the m-order n-dimensional tensor A is weakly symmetric, nonnegative and irreducible. Then ρ(A) is a positive Z-eigenvalue with a positive Z-eigenvector.
Based on the above Lemma, we give the main result of this section. [4,n] be weakly symmetric, nonnegative and irreducible and Then Then, solving for ρ(A) we get Thus, we complete the proof.

Z-Eigenvalue Based Sufficient Condition for the Positive Definiteness of Fourth Order Tensors
In this section, we provide a new checkable sufficient condition for the positive definiteness of fourth order tensors, which is based on the inclusion set for Z-eigenvalues of structured fourth order tensors.
then A is positive definite.
Proof. Assume that λ ≤ 0 is a Z-eigenvalue of A. From Theorem 3, we have λ ∈ Υ(A), hence, there are i 0 , j 0 ∈ N such that This is a contradiction. Hence, λ > 0. Then, the symmetric tensor A is positive definite.
Assume B is a symmetric tensor whose (i 1 i 2 i 3 i 4 )-th entry is respectively defined as follows: If B is positive definite, then A is positive definite.
Proof. Let x ∈ R n be a nonzero vector. Since B is positive definite, from the definition of the positive definiteness of symmetric tensors, we have Then, we have Thus A is positive definite.
By Theorems 5 and 6, we have the following sufficient condition for the positive definiteness of symmetric fourth order tensors.
If for all i, j ∈ N, j = i, then A is positive definite.

Definition 4. Let A ∈ R
Then, the fourth order tensor A is called Z-eigenvalue based quasi-doubly strictly diagonally dominated(Z-QDSDD), if for all i, j ∈ N, j = i,

Numerical Examples
In this section, some examples are given to show the efficiency of our results. First, an example is given to show the efficiency of the result in Theorem 3. By Theorem 3, The Z-eigenvalue inclusion sets Υ(A) and the exact Z-eigenvalues are drawn in Figure 1. We can see that, Υ(A) can capture all Z-eigenvalues of A, and the Z-eigenvalue inclusion set Υ(A) is located on the right side of the coordinate axis, which is better than the Z-eigenvalue inclusion sets K(A) and L(A).
We now show the efficiency of the new upper bound in Theorem 4 by the following example. By Theorem 4,   By computation, we get that, a 1111 = 1 < R 1 (A) = 32.8, a 2222 = 2 < R 2 (A) = 63.6.
Hence, A is not a QSDD tensor. Then, we cannot use Theorem 2 (2) to determine the positiveness of A. However, it is easy to find C 1 = 1, C 2 = 2, In other words, A satisfies all the conditions of Theorem 7, i.e., A is a Z-QDSDD tensor. Hence, from Theorem 7, A is a positive definite tensor. In fact, From the definition of positive definite tensors, A is positive definite.

Conclusions
In this paper, focused the fourth order tensors, a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors is given. As an application, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a Z-eigenvalue based sufficient condition for the positive definiteness of structured fourth order tensors is also given. A positive definite diffusion tensor is a convex optimization problem with a convex quadratic objective function constrained by the nonnegativity requirement on the smallest Z-eigenvalue of the diffusivity function [11], but it is difficult to compute all the Z-eigenvalues or the smallest Z-eigenvalue of a fourth order tensor when n is large. Finally, we introduce the definition of Z-eigenvalue based doubly strictly diagonally dominated(Z-QDSDD) symmetric fourth order tensors and show that, if a tensor A is Z-QDSDD, then A is positive definite.