SOME NEW Z-EIGENVALUE LOCALIZATION SETS FOR TENSORS AND THEIR APPLICATIONS

In this paper some new Z-eigenvalue localization sets for general tensors are established, which are proved to be tighter than those newly derived by Wang et al. [Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 187–198]. Also, some relationships between the Z-eigenvalue inclusion sets presented by Wang et al. and the new Z-eigenvalue localization sets for tensors are given. Besides, we discuss the effects of orthonormal transformations for the proposed sets. As applications of the proposed sets, some improved upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors are given. Numerical examples are also given to verify the advantages of our proposed results over some known ones.


Introduction
For a positive integer n, let N = {1, 2, . . ., n} and C (R) denote the set of all complex (real) numbers.A ∈ C [m,n] (R [m,n] ) denotes the order m dimension n tensor with n m complex (real) entries, where i j ∈ N for j = 1, 2, . . ., m.A real tensor A = (a i1...im ) is called symmetric [29,42,16,17,20,21] if where Π m is the permutation group of m indices.Furthermore, a real tensor of order m dimension n is called the unit tensor [18], denoted by I, if its entries are δ i1...im for i 1 , . . ., i m ∈ N , where 100 ZHENGGE HUANG, LIGONG WANG, ZHONG XU, AND JINGJING CUI Throughout this paper, for A = (a i1...im ) ∈ R [m,n] and i, j ∈ N , j = i, the following notations are utilized repeatedly in our proofs.For each vector x = (x 1 , . . ., x n ) T of dimension n, real or complex, we define the following vector of dimension n [24]: The definitions of H-eigenvalue and Z-eigenvalue of tensors were first introduced and studied by Qi and Lim [29,27]. ) T .Here x T denotes the transpose of x.Furthermore, we call (λ, x) an H-eigenpair, if both λ and x are real.This definition was introduced by Qi in [29] where he assumed that A = (a i1...im ) ∈ R [m,n] is symmetric and m is even.Independently, in [27], Lim gave such a definition but restricted x to be a real vector and λ to be a real number.It is noteworthy that the definition of eigenvalues of tensors proposed by Lim in [27] is different from Definition 1.1 in the case of odd order as the right hand side of (1.1) is taken absolute value.Definition 1.2 ([29, 27]).A pair (λ, x) ∈ C × (C n \ {0}) is called an E-eigenpair of A, if they satisfy the equation We call (λ, x) a Z-eigenpair, if both λ and x are real.Definition 1.3 ([29, 39, 11, 24]).We define σ(A) the Z-spectrum of A by the set of all Z-eigenvalues of A. Assume σ(A) = ∅.Then the Z-spectral radius of A is defined by ρ(A) = max{|λ| : λ ∈ σ(A)}.
However, as mentioned in [37], we can not judge that Z-eigenvalues generated by the aforementioned algorithms are the largest Z-eigenvalue in general.To overcome this difficulty, one should study and develop the Z-eigenvalue inclusion set which includes all Z-eigenvalues of general tensors.For eigenvalue inclusion sets for tensors, many researchers have investigated this topic and many related results have been developed recently, see [29,16,17,18,2,20,21,22,14,1,13,38] and references therein.Nevertheless, characterizations of inclusion set for Z-eigenvalue are still underdeveloped for general tensors.Based on this fact, Wang et al. [37] proposed some Z-eigenvalue inclusion sets which capture all Z-eigenvalues of general tensors.Very recently, several new Z-eigenvalue inclusion sets for tensors have been put forward, see [41,35] and the references therein.To capture all Z-eigenvalues of a tensor more precisely than some newly proposed results, in this paper we derive some new Z-eigenvalue localization sets for general tensors, which are proved to be sharper than the ones in [37].
In [29], Qi extended the well-known Geršgorin eigenvalue localization set of matrices to real symmetric tensors of higher order, and this result can be easily generalized to general tensors [42,20].However, Example 1 in [37] shows that the Geršgorin eigenvalue localization set can not be generalized to Z-eigenvalues of a general tensor and the authors in [37] established the Geršgorin Z-eigenvalue inclusion theorem for general tensors as follows.
102 ZHENGGE HUANG, LIGONG WANG, ZHONG XU, AND JINGJING CUI Lemma 1.6 ([37]).Let A = (a i1...im ) ∈ C [m,n] , n ≥ 2. Then all Z-eigenvalues of A are located in the union of the following sets: To obtain sharper Z-eigenvalue localization sets than K(A), they also developed other three Z-eigenvalue localization sets for general tensors in the following lemmas.
Lemma 1.7 ([37]).Let A = (a i1...im ) ∈ C [m,n] , n ≥ 2. Then all Z-eigenvalues of A are located in the union of the following sets: ,n] and n ≥ 2. Then all Z-eigenvalues of A are located in the union of the following sets: Lemma 1.9 ([37]).Let A = (a i1...im ) ∈ C [m,n] and n ≥ 2. Then all Z-eigenvalues of A are located in the union of the following sets: where [37] showed that the sets in Lemmas 1.7-1.9 are better than the one in Lemma 1.6.
Spectral theory of nonnegative tensors has developed rapidly and attracted a lot of researchers in late years.In the meanwhile, they have put forward various bounds to estimate the spectral radius of nonnegative tensors [42,40,39,36,26], and the upper bounds for the largest Z-eigenvalue of tensors are no exception.In [5], Chang et al. proposed upper bounds for the Z-spectral radius of nonnegative tensors.On the basis of the relationship between the Gelfand formula and the spectral radius, Song and Qi [36] developed the new upper bounds for the Z-spectral radius, which improve the ones in [5].After that, He and Huang [11] obtained the Ledermannlike upper bound for the largest Z-eigenvalue of the weakly symmetric positive tensors.To get the sharper upper bounds for the Z-spectral radius, Li et al. [24] gave some new upper bounds and proved that these bounds are tighter than the one in [11].Very recently, Wang et al. [37] developed some new upper bounds for the largest Z-eigenvalue of weakly symmetric nonnegative tensors by applying the proposed Z-eigenvalue inclusion sets in [37].Numerical results of [37] show that the proposed bounds are tighter than existing bounds of [11,24,5].As mentioned before, estimating the largest Z-eigenvalue is of significance.Hence, we continue this research on the Z-spectral radius of weakly symmetric nonnegative tensors and propose some new tighter bounds compared with those in [11,24,37,5] on the basis of the new set derived in this paper.
To improve the Z-eigenvalue inclusion sets in [37, Theorems 3.1 and 3.3-3.4],some new Z-eigenvalue localization sets for general tensors are employed in this paper.They are proved to be more precise than the Z-eigenvalue localization sets in Lemmas 1.6, 1.8 and 1.9.Meanwhile, some relationships between the Z-eigenvalue inclusion sets derived in [37] and the new Z-eigenvalue localization sets for tensors are given.Additionally, the effects of orthonormal transformations for the proposed sets are investigated.As applications, some new bounds for the Z-spectral radius of weakly symmetric nonnegative tensors are established, which outperform some known results.Several numerical results illustrate these facts.

Two new Z-eigenvalue localization sets for tensors
In this section, we investigate Z-eigenvalue localization sets and present some new Z-eigenvalue localization sets for tensors, and the comparison results of the proposed sets with those in Lemmas 1.6, 1.8 and 1.9 are given.By making use of the technique applied in [22, Theorem 2.1], we establish the following theorem.
Theorem 2.1.Let A = (a i1...im ) ∈ C [m,n] and n ≥ 2. Then all Z-eigenvalues of A are located in the following localization sets: where Proof.Let λ be a Z-eigenvalue of A and the corresponding eigenvector be Then it is evident that |x t | > 0 and 2) Taking absolute values in the above equation and using the triangle inequality yield Recalling that |x s | > 0, then from inequality (2.4) we have . This completes our proof of Theorem 2.1.
The following theorem gives some comparisons between B(A) and the results in Lemmas 1.6, 1.8 and 1.9.To prove the relations, we need the following lemma.
Proof.We prove this theorem in view of the methods of [22,Theorem 2.3].By Corollaries 3.2 and 3.3 in [37], it can be seen that N (A) ⊆ K(A) and M(A) ⊆ K(A) hold true.Thus, we only need to show the relations B(A) ⊆ N (A) and Without loss of generality, we first assume that z ∈ i∈N B 1 i (A).Under this condition, there exists one index i 0 ∈ N such that It follows from (2.7) that there exist p, q ∈ N and p = q such that and (2.9) ) and (2.9) results in By multiplying (2.8) with (2.10), we derive This means that z ∈ N q,p (A) ⊆ N (A).Afterwards, if P p p (A)(R q (A) − P p q (A)) > 0, dividing (2.9) by P p p (A)(R q (A) − P p q (A)) yields Let a = |z| ≥ 0, b + c = P p q (A) ≥ 0 with b, c ≥ 0 and d = R q (A) − P p q (A) > 0. If which is equivalent to This implies that z ∈ N p,q (A) ⊆ N (A).Furthermore, if |z|−P p q (A) Rq(A)−P p q (A) ≤ 1, then (2.10) holds.Multiplying (2.8) with (2.10) leads to In what follows, we show the relation B(A) ⊆ M(A).Similar to the above proof, we first suppose that z ∈ i∈N B 1 i (A), then there exists one index i 0 ∈ N such that |z| ≤ R i0 (A) − P i0 i0 (A).For j 0 ∈ N and j 0 = i 0 , we have |a i0j0...j0 | ≤ P i0 i0 (A), from which it follows that For j 0 ∈ N and j 0 = i 0 , it holds that )-(2.9) hold true.If P p p (A)(R q (A) − P p q (A)) = 0, then P p p (A) = 0 or R q (A) = P p q (A).When P p p (A) = 0, then |a pq...q | = 0 and therefore If R q (A) = P p q (A), then it follows from |z| > R p (A)−P p p (A) and inequality (2.9) that |z| ≤ P p q (A).If |z| = P p q (A), then it is obvious that z ∈ M p,q (A) ⊆ M(A). Otherwise, 0, and d = |a pq...q |.Thus, from inequality (2.11), we have that by virtue of Lemma 2.2.From inequality (2.12) we derive (|z| − [R p (A) − |a pq...q |])(|z| − P p q (A)) ≤ |a pq...q |(R q (A) − P p q (A)), from which it follows that z ∈ M p,q (A) ⊆ M(A).If |z| ≥ P p q (A) and |a pq...q | = 0, then |z| ≤ R p (A) = R p (A) − |a pq...q |, which leads to ).This implies that z ∈ M p,q (A) ⊆ M(A).Therefore, B(A) ⊆ M(A).The conclusions follow immediately from what we have proved.
Thus, the Z-eigenvalue localization set of Theorem 2.1 is more precise than those in Lemmas 1.6, 1.8 and 1.9.Now we give an example to verify the fact.

Example 2.4 ([37]
).Consider the tensor A = (a ijk ) ∈ R [3,3] defined by By some calculation, the comparative results are given in Table 1.As observed in this table, B(A) is the sharpest among the aforementioned localization sets.
In the following theorem, motivated by [1, Theorem 3.3], we develop another Z-eigenvalue localization set for tensors.
Table 1.Some Z-eigenvalue localization sets for A.
Then all Z-eigenvalues of A are located in the following localization sets: Proof.Let λ be a Z-eigenvalue of A with the corresponding eigenvector x, i.e., Then for all i ∈ N , it holds that (2.13) Now, we consider two cases as follows.
Then from (2.13), we obtain the inequalities The proof is completed.
In the following theorem we show that F(A) ⊆ K(A).
Proof.For any z ∈ F(A), if z / ∈ K(A), then |z| > R i (A) for all i ∈ N .In this case, for all a i1i2...im = 0, i j ∈ N and j = 1, 2, . . ., m, it follows that , which contradicts the fact that z ∈ F(A).
Next, enlightened by [19], we investigate the relationships of F(A), K(A), and N (A).

Theorem 2.7. Let
Now, we utilize the following example to confirm the result in Theorem 2.5.
Obviously, a 111 , a 222 = 0. We present the comparative results in Table 2.As seen in this table, the relationship Let λ be any eigenvalue of the tensor A. By Theorem 2.5 and R i (A) = 0 for all i ∈ N , there exists a h1...hm = 0 such that .
On squaring the above inequality, we have It follows from this inequality that at least one of the factors in its left-hand side is at most 1.Without loss of generality, we assume that there exists p with 1 where if p = m, then h p+1 = h 1 .This inequality implies that As a result, we derive F(A) ⊆ N (A) ⊆ K(A) by virtue of Theorem 2.3.The result follows.
It is noteworthy that the Z-eigenvalues of the tensors are invariant under orthonormal transformations [32] and have applications in physics and mechanics.Thus it is meaningful to analyze whether Z-eigenvalue localization sets for tensors obtained in Theorems 2.1 and 2.5 are invariant under orthonormal transformations, or what are the effects of orthonormal transformations.In physics and mechanics, it is most important that the results are independent from coordinate system choice.Now we introduce the definition of the orthogonal transformation for tensors.

Definition 3.1 ([32]
).For any given n × n real orthogonal matrix P = (p ij ) and tensor A = (a i1...im ) ∈ C [m,n] , the orthogonal transformation P m (A) is defined as For any given two tensors A = (a i1...im ), D = (d j1...jm ) ∈ C [m,n] , we say that A is orthogonally similar to D if there exists some orthogonal matrix P such that D = P m (A).
The following lemma shows that the Z-eigenvalues of the tensors are invariant under orthonormal transformations.

Lemma 3.2 ([32]
).Let A = (a i1...im ), D = (d j1...jm ) ∈ C [m,n] be orthogonally similar with D = P m (A), where P = (p ij ) is an n × n orthogonal matrix.Then A and D have the same Z-eigenvalues.In particular, if λ is a Z-eigenvalue of A with a Z-eigenvector x, then λ is also a Z-eigenvalue of D with a Z-eigenvector y = P x.
Next, we discuss whether Z-eigenvalue localization sets for tensors obtained in Theorems 2.1 and 2.5 are invariant under orthonormal transformations, or what are the effects of orthonormal transformations.
For a given orthogonal matrix P = (p ij ) ∈ R n×n , D = P m (A).Based on Lemma 3.2 and applying the same technique used in Theorem 2.1, we can derive where Similarly, we can establish the following Z-eigenvalue localization set by virtue of Theorem 2.5: The following example indicates that under orthonormal transformations, the Z-eigenvalue localization sets obtained in this paper may become sharper.
Example 3.3.Consider the tensor A = (a i1i2i3 ) ∈ R [3,3]  We report in Table 4 the Z-eigenvalue localization sets for A by Theorem 2.1, (3.1), Theorem 2.5, and (3.2).By comparing the results in this table, we conclude that under orthonormal transformations with the orthogonal matrix (3.3), the sets derived in Theorems 2.1 and 2.5 become better, and it implies that orthonormal transformations may make the Z-eigenvalue localization sets sharper.In summary, the Z-eigenvalue localization sets for tensors obtained in this paper are variant under orthonormal transformations.Under proper orthonormal transformations, they may be better.An interesting problem is: How to choose the orthogonal matrix P such that the Z-eigenvalue localization sets are best under orthonormal transformations.It is a difficult task for us at present, and we will investigate this problem in our further work.

Some new upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors
On the basis of the results in Section 2, we establish two new upper bounds for the Z-spectral radius of weakly symmetric tensors in this section, and compare those with some known bounds derived in [36,11,24,37].To this end, we start with a lemma which will be useful in the following proofs.
Since A is a weakly symmetric nonnegative tensor, we see that ρ(A) is the largest Z-eigenvalue of A by virtue of Lemma 4.1; then it follows from Theorem 2.1 that For the case that ρ(A) ∈ i,j∈N, i =j B 2 i,j (A) K i (A) , there exist p, q ∈ N and p = q such that ρ(A) ≤ R p (A) (4.2) and (ρ(A) − [R p (A) − P p p (A)])(ρ(A) − P p q (A)) ≤ P p p (A)(R q (A) − P p q (A)).(4.3) Solving the quadratic inequality (4.3) gives where Υ p,q (A) = (R p (A) − P p p (A) − P p q (A)) 2 + 4P p p (A)(R q (A) − P p q (A)).Combining (4.2) and (4.4) gives The conclusion follows from inequalities (4.1) and (4.5).We compare the upper bound exhibited in Theorem 4.5 with that in Corollary 4.5 of [36] in Table 6.From the observations in this table, it can be clearly seen that the upper bound in Theorem 4.5 is tighter than that in Corollary 4.5 of [36], which is in accordance with the conclusions in Theorem 2.6 and Remark 4.6.Table 6.Upper bounds for the Z-spectral radius of A. Remark 4.8.We have recently developed a new S-type eigenvalue localization set for tensors in [14].The differences between both papers are as follows: • The kinds of the eigenvalue localization sets of these two papers are different.Both papers introduce some eigenvalue localization sets for tensors.While the main aim of the present paper is to establish some new Z-eigenvalue (see Definition 1.2) localization sets for tensors, in [14] we derive a new S-type eigenvalue (see Definition 1.1) localization set for tensors.• There is a difference between the techniques used in these two papers.
Theorem 3.1 of [14] is derived by breaking N into disjoint subsets S and its complement, and the eigenvalue localization set proposed in [14] relies on the choice of S. But the Z-eigenvalue localization sets developed in the present paper do not contain the set S. • The applications of eigenvalue localization sets in the two papers are different.One of the applications of the eigenvalue localization set in [14] is obtaining new sufficient conditions for positive (semi-)definiteness of tensors, which involve the H -eigenvalues of tensors, but this application can not be established by the Z-eigenvalue localization set in the present paper.

Concluding remarks
To localize all Z-eigenvalues of a tensor much sharper compared with some known ones, some new Z-eigenvalue localization sets for tensors are derived in this paper, which are proved to be more precise than the ones in [37], and we also study the effects of orthonormal transformations for the proposed sets.Based on the new sets, we establish some new upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors.These bounds are tighter than those proposed by Wang et al. [37] in theory.Some numerical results worked out in this paper verify that the proposed results are superior to some existing ones.
Besides, we should mention that there exist some problems to be studied in the future: • Study the new Z-eigenvalue localization set which is sharper than the one in Lemma 1.7 in theory.• Research the relationship between the set B(A) and the set F(A).
• Choose the optimal orthogonal matrix P such that B(A) and F(A) are best under orthonormal transformations.

. 2 ) 1 )
Now we illustrate that the Z-eigenvalue localization sets for tensors obtained in Theorems 2.1 and 2.5 are variant under orthonormal transformations.For Example 2.4, we choose the orthogonal matrix generated by the function orth of Matlab , we obtain B(D) = {z ∈ C : |z| ≤ 7.1733}, while B(A) = {z ∈ C : |z| ≤ 4.7913} in terms of Theorem 2.1.This means that B(A) (no orthonormal transformations) is better than B(D).For Example 2.10, we also consider the orthogonal matrix (3.3); then we have the Z-eigenvalue localization set F(D) = {z ∈ C : |z| ≤ 6.1935} by Theorem 2.5 under this orthonormal transformation, which means that F(A) also becomes worse under this orthonormal transformation.

Theorem 4 . 2 .
Let A ∈ R[m,n] be a weakly symmetric nonnegative tensor.Then

Remark 4 . 3 .
Due to the fact that the upper bounds for ρ(A) in Theorems 4.6 and 4.7 in[37] are deduced from the Z-eigenvalue localization sets M(A) and N (A), respectively, and that in Theorem 4.2 is derived from the Z-eigenvalue localization set B(A), andB(A) ⊆ M(A) ⊆ K(A), B(A) ⊆ N (A) ⊆ K(A), we can prove that Θ max (A) ≤ v ≤ max i∈N R i (A) and Θ max (A) ≤ w ≤ max i∈N R i (A), where v and w are defined as in Theorems 4.6 and 4.7 in[37], respectively.