A new S-type eigenvalue inclusion set for tensors and its applications

In this paper, a new S-type eigenvalue localization set for a tensor is derived by dividing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N=\{1,2,\ldots,n\}$\end{document}N={1,2,…,n} into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of strong M-tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).


Introduction
Eigenvalue problems of higher order tensors have become an important topic in the applied mathematics branch of numerical multilinear algebra, and they have a wide range of practical applications, such as best-rank one approximation in data analysis [], higher order Markov chains [], molecular conformation [], and so forth. In recent years, tensor eigenvalues have caused concern of lots of researchers [, , , -].
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; for details, see [, , ].
However, as mentioned in [, , ], it is not easy to compute the smallest Heigenvalue of tensors when the order and dimension are very large, we always try to give a set including all eigenvalues in the complex. Some sets including all eigenvalues of tensors have been presented by some researchers [-, -]. In particular, if one of these sets for an even-order real symmetric tensor is in the right-half 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 study the new eigenvalue inclusion set for tensors called the new S-type eigenvalue inclusion set, which is sharper than some existing ones.
For a positive integer n, N denotes the set N = {, , . . . , n}. The set of all real numbers is denoted by R, and C denotes the set of all complex numbers. Here, we call A = (a i  ···i m ) a complex (real) tensor of order m dimension n, denoted by C [m,n] (R [m,n] ), if a i  ···i m ∈ C(R), where i j ∈ N for j = , , . . . , m [].
Let A ∈ R [m,n] , and x ∈ C n . Then An m-order n-dimensional tensor A is called nonnegative [, , , , ], if each entry is nonnegative. We call a tensor A a Z-tensor, if all of its off-diagonal entries are nonpositive, which is equivalent to writing A = sI -B, where s >  and B is a nonnegative tensor (B ≥ ), denoted by Z the set of m-order and n-dimensional Z-tensors. A Z-tensor The tensor A is called reducible if there exists a nonempty proper index subset J ⊂ N such that a i  i  ···i m = , ∀i  ∈ J, ∀i  , . . . , i m / ∈ J. If A is not reducible, then we call A is irreducible []. The spectral radius ρ(A) [] of the tensor A is defined as Denote by τ (A) the minimum value of the real part of all eigenvalues of the nonsingular where m is the permutation group of m indices. Let Recently, much literature has focused on the bounds of the spectral radius of nonnegative tensor in [, , , , -, , ]. In addition, in [], He and Huang obtained the upper and lower bounds for the minimum H-eigenvalue of nonsingular M-tensors. Wang and Wei [] presented some new bounds for the minimum H-eigenvalue of nonsingular M-tensors, and they showed those are better than the ones in [] in some cases. As applications of the new S-type eigenvalue inclusion set, the other main results of this paper is to provide sharper bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of nonsingular M-tensors, which improve some existing ones.
Before presenting our results, we review the existing results that relate to the eigenvalue inclusion sets for tensors. In , Qi [] generalized the Geršgorin eigenvalue inclusion theorem from matrices to real supersymmetric tensors, which can be easily extended to general tensors [, ].
In addition, in order to reduce computations of determining the sets σ (A), Li et al. [] also presented the following S-type eigenvalue localization set by breaking N into disjoint subsets S andS, whereS is the complement of S in N . n] , n ≥ , and S be a nonempty proper subset of N . Then where K i,j (A) (i ∈ S, j ∈S or i ∈S, j ∈ S) is defined as in Lemma ..
Based on the results of [], in the sequel, Li et al.
[] exhibited a new tensor eigenvalue inclusion set, which is proved to be tighter than the sets in Lemma .. [m,n] , n ≥ , and S be a nonempty proper subset of N . Then In this paper, we continue this research on the eigenvalue inclusion sets for tensors; inspired by the ideas of [, ], we obtain a new S-type eigenvalue inclusion set for tensors. It is proved to be tighter than the tensor Geršgorin eigenvalue inclusion set (A) in Lemma ., the Brauer eigenvalue localization set K(A) in Lemma ., the S-type eigenvalue localization set K S (A) in Lemma ., and the set (A) in Lemma .. As applications, we establish some new bounds for spectral radius of nonnegative tensors and the minimum H-eigenvalue of strong M-tensors. Numerical examples are implemented to illustrate this fact.
The remainder of this paper is organized as follows. In Section , we recollect some useful lemmas on tensors which are utilized in the next sections. In Section ., a new Stype eigenvalue inclusion set for tensors is given, and proved to be tighter than the existing ones derived in Lemmas .-.. Based on the results of Section ., we propose a new upper bound for the spectral radius of nonnegative tensors in Section .; comparison results for this new bound and that derived in [] are also investigated in this section. Section . is devoted to the exhibition of new upper and lower bounds for the minimum H-eigenvalue of strong M-tensors, which are proved to be sharper than the ones obtained by He and Huang []. Finally, some concluding remarks are given to end this paper in Section .

Preliminaries
In this section, we start with some lemmas on tensors. They will be useful in the following proofs.
is a positive eigenvalue with an entrywise positive eigenvector x, i.e., x > , corresponding to it.

A new S-type eigenvalue inclusion set for tensors
In this section, we propose a new S-type eigenvalue set for tensors and establish the comparisons between this new set with those in Lemmas .-..
Proof For any λ ∈ σ (A), let x = (x  , . . . , x n ) T ∈ C n /{} be an eigenvector corresponding to λ, i.e., Then, x p =  or x q = . Now, let us distinguish two cases to prove.
Hence, we have i.e., Premultiplying by (λa j···j ) in the first equation of () results in Combining () and the second equation of () one derives Taking absolute values and using the triangle inequality yield Note that |x p | > , thus Using the same method as the proof in (i), we deduce that Taking the modulus in the above equation and using the triangle inequality we obtain Note that |x q | > , thus . This completes our proof of Theorem ..
Remark . Note that |S| < n, where |S| is the cardinality of S. If n = , then |S| =  and n(n -) = |S|(n -|S|) = , which implies that Furthermore, how to choose S to make ϒ S (A) as sharp as possible is very interesting and important. However, this work is difficult especially the dimension of the tensor A is large. At present, it is very difficult for us to research this problem, we will continue to study this problem in the future.
Next, we establish a comparison theorem for the new S-type eigenvalue inclusion set derived in this paper and those in Lemmas .-..

Theorem . Let
Without loss of generality, we assume that z ∈ i∈S,j∈S ϒ j i (A) (we can prove it similarly if z ∈ i∈S,j∈S ϒ j i (A)). Then there exist p ∈ S and q ∈S such that z ∈ ϒ q p (A), that is, (za p···p )(za q···q )a pq···q a qp···p ≤ |za q···q |r q p (A) + |a pq···q |r p q (A).
This means that which implies that This proof is completed.

A new upper bound for the spectral radius of nonnegative tensors
Based on the results of Section ., we discuss the spectral radius of nonnegative tensors, and we give their upper bounds, which are better than those of Theorem . in [].

Theorem . Let
Hence, we have Premultiplying by (ρ(A)a j···j ) in the first equation of () results in It follows from () and the second equation of () that Note that x p ≥ x j for any j ∈S and by Lemma ., we deduce that i.e., Solving the quadratic inequality () yields It is not difficult to verify that () can be true for any j ∈S. Thus So we obtain In a similar manner to the proof of (i) i.e., which yields It is easy to see that () can be true for any j ∈ S. Thus which implies that This completes our proof in this theorem.
Next, we extend the results of Theorem . to general nonnegative tensors; without the condition of irreducibility, compare with Theorem .. Proof Let A k = A +  k ε, where k = , , . . . , and ε denotes the tensor with every entry being . Then A k is a sequence of positive tensors satisfying

Theorem . Let
By Lemma ., {ρ(A k )} is a monotone decreasing sequence with lower bound ρ(A). So ρ(A k ) has a limit. Let

And we denote
As m and n are finite numbers, then by the properties of the sequence, it is easy to see that Furthermore, since A k is an irreducible nonnegative tensor, it follows from Theorem . that Letting k → +∞ results in from which one may get the desired bound ().

New upper and lower bounds for the minimum H-eigenvalue of nonsingular M-tensors
In this section, by making use of the results in Section ., we investigate the bounds for the minimum H-eigenvalue of strong M-tensors and derive sharper bounds for that. This bounds are proved to be tighter than those in Theorem . of []. Let x p = max i∈S {x i } and x q = max i∈S {x i }. We distinguish two cases to prove.

Theorem . Let
(ii) x q ≥ x p > , so x q = max i∈N {x i }. For any i ∈ S, it follows from () that So we obtain Using the same technique as the proof of (i), we have which is equivalent to which results in It is not difficult to verify that () can be true for any j ∈ S. Thus In the same manner as applied in the above proof, we can deduce the following results: Therefore, the conclusions follow from the above discussions. In the sequel, we extend the results of Theorem . to a more general case, which needs a weaker condition compared with Theorem .. Proof Since A is a nonsingular M-tensor, A ∈ Z, by Lemma . and Lemma ., there exists x = (x  , . . . , x n ) T ≥  such that Ax m- > , that is, for any i  ∈ N , n i  ,...,i m = a i  ···i m x i  · · · x i m > .

Theorem . Let
Let So x max >  by x ≥ . By replacing the zero entries of A with - k , where k is a positive integer, we see that the Z-tensor A k is irreducible. Here, we use a i  ···i m (- k ) to denote the entries of A k . We choose k > [