A new eigenvalue inclusion set for tensors with its applications

An m-order n-dimensional tensor  is called nonnegative, if each entry is nonnegative. A tensor of order m dimension n is called the unit tensor, denoted by , if its entries are i1...im for i1, ... , im ∈ N, where *Corresponding author: Jianxing Zhao, College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, P.R. China E-mails: zjx810204@163.com, zhaojianxing@gzmu.edu.cn

ABOUT THE AUTHOR Jianxing Zhao has obtained PhD in applied mathematics from Yunnan University. Currently, he is an associate professor in College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, China. His main research interests include criteria for H-tensors and its applications, H(Z)-eigenvalue inclusion set for general tensors with its applications, and estimates of the minimum eigenvalue for -tensors.

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One of many practical applications of eigenvalues of tensors is that one can identify the positive (semi-)definiteness for an even-order real symmetric tensor by using the smallest H-eigenvalue of a tensor; consequently, one can identify the positive (semi-)definiteness of the multivariate homogeneous polynomial determined by this tensor. However, it is not easy to compute the smallest H-eigenvalue of tensors when the order and dimension are very large, we always try to give a set including all eigenvalues in the complex. In particular, if one of these sets for an even-order real symmetric tensor is in the righthalf complex plane, then we can conclude that the smallest H-eigenvalue is positive, consequently, the corresponding tensor is positive definite. Therefore, the main aim of this paper is to give a new eigenvalue inclusion set for tensors, and using the set to obtain a weaker sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor.
A real tensor  = (a i 1 …i m ) is called symmetric (Qi, 2005) if where Π m is the permutation group of m indices.
A tensor  = (a i 1 i 2 …i m ) is called reducible if there exists a nonempty proper index subset ⊂ N such that If  is not reducible, then we call  is irreducible (Chang, Zhang, & Pearson, 2008). Let  = (a ii 2 …i m ) be a nonnegative tensor, If  is not weakly reducible, then it is called weakly irreducible; for details, see Friedland, Gaubert, and Han (2013) and Zhang, Qi, and Zhou (2014). Wang and Wei (2015) proved that if  is irreducible, then  is weakly irreducible, and for m = 2,  is irreducible if and only if  is weakly irreducible.
then is called an eigenvalue of  and x an eigenvector of  associated with , where x m−1 is an n dimension vector whose ith component is and If and x are all real, then is called an H-eigenvalue of  and x an H-eigenvector of  associated with . This definition was introduced by Qi (2005) where he assumed that  ∈ ℝ [m, n] is symmetric and m is even. Independently, Lim (2015) gave such a definition but restricted x to be a real vector and to be a real number. Moreover, the spectral radius () of the tensor  is defined as where () is the spectrum of , i.e. () = { : is an eigenvalue of } (see Chang et al., 2008;Yang & Yang, 2010). m, n] .  is called a -tensor, if all of its off-diagonal entries are non-positive, which is equivalent to write  = s − , where s > 0 and  is a nonnegative tensor. A -tensor  = s −  is an -tensor if s > (). Here, we denote by () the minimal value of the real part of all eigenvalues of an -tensor , and note that if  is a weakly irreducible -tensor, then () > 0 is the unique eigenvalue with a positive eigenvector; for details, see Zhang et al. (2014) and Ding, Qi, and Wei (2013).
Given an even-order symmetric tensor  = (a i 1 ⋯ i m ) ∈ ℝ [m, n] , the positive (semi-)definiteness of  is determined by the sign of its smallest H-eigenvalue, that is, if the smallest H-eigenvalue is positive (nonnegative), then  is positive (semi-)definite. However, when m and n are very large, it is not easy to compute the smallest H-eigenvalue of . Then we can try to give a set in the complex which includes all eigenvalues of . If this set is in the right-half complex plane, then we can conclude that the smallest H-eigenvalue is positive, consequently,  is positive definite; for details, see Qi (2005), Li, Li, and Kong (2014), , Li, Jiao, and Li (2016), Li, Chen, and Li (2015) and Huang, Wang, Xu, and Cui (2016).
Therefore, one of the main aims of this paper is to give a new eigenvalue inclusion set for tensors, and use this set to determine positive (semi-)definiteness of tensors.
In Qi (2005) generalized Gerŝgorin eigenvalue inclusion theorem from matrices to real supersymmetric tensors, which can be easily extended to general tensors (Li et al., 2014;Yang & Yang, 2010). Theorem 1.1 (Qi, 2005, Theorem 6 where To get tighter eigenvalue inclusion sets than Γ(), Li et al. (2014) extended the Brauer's eigenvalue localization set of matrices (Varga, 2004) and proposed the following Brauer-type eigenvalue localization sets for tensors.
For the weakly irreducible -tensor, Wang and Wei (2015) obtained the following results on ().
Theorem 1.4 (Wang and Wei, 2015, Lemma 4.4) Let  be a weakly irreducible -tensor. Then In this paper, we continue this research on the eigenvalue inclusion sets for tensors and its applications. We obtain a new eigenvalue inclusion set for tensors and prove that the new set is tighter than Theorems 1.1 and 1.2. As applications, we establish a sufficient condition for the positive (semi-)definiteness of tensors and give new lower and upper bounds of the minimum H-eigenvalue for -tensors, which are the correction of Theorem 4.5 in Wang and Wei (2015).

A new eigenvalue inclusion set for tensors
In this section, we propose a new eigenvalue inclusion set for tensors and establish the comparisons between this new set with those in Theorems 1.1 and 1.2.

Let
(where the last term above is defined to be zero if n = 2). Then, |x p | > 0. From (1), we have Taking modulus in the above equation and using the triangle inequality give equivalently, (1), we can obtain (1) Multiplying (2) with (3) and noting that |x p | m−1 |x q | m−1 > 0, we have Next, a comparison theorem is given for Theorems 1.1, 1.2 and 2.1. n] . Then Proof According to Theorem 2.3 in Li et al. (2014), The following proof will be divided into two cases according to a certain rule.
(ii) When (8)  In the following, a numerical example is given to verify Theorem 2.2.
Example 2.1 Let  ∈ ℝ [4, 2] with entries be defined as follows:   Huang et al. (2016), an eigenvalue localization set can provide a sufficient condition for the positive definiteness and positive semi-definiteness of tensors. As applications of the results in Section 2, we in this section provide some sufficient conditions for the positive definiteness and positive semi-definiteness of tensors, respectively. m, n] be an even-order symmetric tensor with a k … k > 0 for all k ∈ N. If for any i, j ∈ N, i ≠ j, then  is positive definite.

Determining the positive definiteness for an even-order real symmetric tensor
Proof Let be an H-eigenvalue of . Suppose that ≤ 0. By Theorem 2.1, we have ∈ Ω(), that is, there are some i, j ∈ N, i ≠ j such that From a k … k > 0, k ∈ N, we have This is a contradiction. Hence, > 0, and  is positive definite. The conclusion follows. ✷ Similar to the proof of Theorem 3.1, we can easily obtain the following conclusion: m, n] be an even-order symmetric tensor with a k … k ≥ 0 for all k ∈ N. If for any i, j ∈ N, i ≠ j,

New bounds for the minimum eigenvalue of -tensors
In this section, new lower and upper bounds for the minimum H-eigenvalue of -tensors are given, which are the correction and generalization of Theorem 4.5 in Wang and Wei (2015). Next, we prove that the second inequality in (9) holds. Suppose that x = (x 1 , … , x n ) T > 0 is an eigenvalue of  corresponding to (), i.e. and From (11), we have and a i … i −r i () a j … j ≥r i ()r j ().

Then solving for (A) gives
This proof is completed. ✷ Remark 4.1 Note that L ij () ≠ W ij (). Hence, the bounds (9) in Theorem 4.1 are slightly different from the bounds in Theorem 4.5 of Wang and Wei (2015). In fact, the bounds (9) are the correction of the bounds in Theorem 4.5 of Wang and Wei (2015). Because the left (right) inequality of (4.2) in Theorem 4.5 of Wang and Wei (2015) obtained by solving for (A) from inequality (4.2) ( (14), respectively); for details, see the proof of Theorem 4.5 in Wang and Wei (2015). However, solving for () by inequalities (10) and (14) gives the bounds (9). In the following, a counterexample is given to show that the result in Theorem 4.5 in Wang and Wei (2015) is false. Consider the tensor  = (a ijkl ) of order 4 dimension 2 with entries defined as follows: By Theorem 4.5 in Wang and Wei (2015), we have 11 ≤ () ≤ 11. By Theorem 2.1, we have 8.9585 ≤ () ≤ 12.6893. In fact, () = 10.8851.
Next, we can extend the results of Theorem 4.1 to a more general case. m, n] be an -tensor. Then Proof Because  is an -tensor, by Theorems 2 and 3 in Ding et al. (2013), there is where k = 1, 2, …, and  denote the tensor with every entry being 1.
Then  k is an irreducible -tensor, and { k } is a monotonically increasing sequence.
Taking k > n m−1 x m−1 max + 1, then for any i ∈ N, which implies that  k x m−1 > 0. Then, by Theorems 2 and 3 in Ding et al. (2013), we can conclude that  k is an irreducible -tensor. By Theorem 4.1 in He and Huang (2014), { ( k )} is a monotonically increasing sequence with upper bound (), so ( k ) has a limit, and let By Theorem 2.6 in Wang and Wei (2015), we see that ( k ) is the eigenvalue of  k with a positive eigenvector y (k) , i.e.  k (y (k) ) m−1 = ( k )(y (k) ) [m−1] . As homogeneous multivariable polynomials, we can restrict y (k) on the unit ball ‖y (k) ‖ = 1. Then {y (k) } is a bounded sequence, so it has a convergent subsequence. Without loss of generality, suppose that it is the sequence itself. Let y (k) → y as k → +∞, we get y ≥ 0 and ‖y‖ = 1. Letting k → +∞, we have y = y [m−1] from  k (y (k) ) m−1 = ( k )(y (k) ) [m−1] . So is an eigenvalue of , furthermore, ≥ (). Together with (16) () ≤ min i∈N a ii … i , and min i∈N R i () ≤ () ≤ max i∈N R i (). then [a i … i − a j … j −r i ()] 2 + 4r i ()r j () r j () ≥ a j … j − a i … i +r i () +r i ().