A new S-type eigenvalue inclusion set for tensors and its applications
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
DOI 10.1186/s13660-016-1200-3
RESEARCH
Open Access
A new S-type eigenvalue inclusion set for
tensors and its applications
Zheng-Ge Huang* , Li-Gong Wang, Zhong Xu and Jing-Jing Cui
*
Correspondence:
Department of Applied
Mathematics, Northwestern
Polytechnical University, Xi’an,
Shaanxi 710072, P.R. China
Abstract
In this paper, a new S-type eigenvalue localization set for a tensor is derived by
dividing 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).
MSC: 15A18; 15A69
Keywords: tensor eigenvalue; nonsingular M-tensor; minimum H-eigenvalue;
nonnegative tensor; spectral radius; positive definite
1 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.
© 2016 Huang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Page 2 of 19
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 = (ai ···im ) a
complex (real) tensor of order m dimension n, denoted by C[m,n] (R[m,n] ), if ai ···im ∈ C(R),
where ij ∈ N for j = , , . . . , m [].
Let A ∈ R[m,n] , and x ∈ Cn . Then
Ax
m–
:=
n
aii ···im xi · · · xim
i ,...,im =
,
≤i≤n
a pair (λ, x) ∈ C × (Cn /{}) is called an eigenpair of A [] if
Axm– = λx[m–] ,
m–
m– T
where x[m–] = (xm–
, x , . . . , xn ) []. Furthermore, we call (λ, x) an H-eigenpair, if
both λ and x are real [].
A real tensor of order m dimension n is called the unit tensor [], denoted by I , if its
entries are δi ···im for i , . . . , im ∈ N , where
δi ···im =
, if i = · · · = im ,
, otherwise.
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
A = sI – B is an M-tensor if s ≥ ρ(B ), and it is a nonsingular (strong) M-tensor if s > ρ(B )
[, ].
The tensor A is called reducible if there exists a nonempty proper index subset J ⊂
N such that ai i ···im = , ∀i ∈ J, ∀i , . . . , im ∈/ J. If A is not reducible, then we call A is
irreducible []. The spectral radius ρ(A) [] of the tensor A is defined as
ρ(A) = max |λ| : λ is an eigenvalue of A .
Denote by τ (A) the minimum value of the real part of all eigenvalues of the nonsingular
M-tensor A []. A real tensor A = (ai ···im ) is called symmetric [–, , , ] if
ai ···im = aπ (i ···im ) ,
∀π ∈ m ,
where m is the permutation group of m indices.
Let A = (ai ···im ) ∈ R[m,n] . For i, j ∈ N , j = i, denote
R i (A ) =
n
aii ···im ,
Rmax (A) = max Ri (A),
i∈N
i ,...,im =
ri (A) =
δii ···im =
|aii ···im |,
j
ri (A) =
δii ···im =,
δji ···im =
Rmin (A) = min Ri (A),
i∈N
|aii ···im | = ri (A) – |aij···j |.
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Page 3 of 19
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 [, ].
Lemma . ([]) Let A = (ai ···im ) ∈ C[m,n] , n ≥ . Then
σ (A ) ⊆ ( A ) =
i (A),
i∈N
where σ (A) is the set of all the eigenvalues of A and
i (A ) =
z ∈ C : |z – ai···i | ≤ ri (A) .
To get sharper eigenvalue inclusion sets than (A), Li et al. [] extended the Brauer
eigenvalue localization set of matrices [, ] and proposed the following Brauer-type
eigenvalue localization sets for tensors.
Lemma . ([]) Let A = (ai ···im ) ∈ C[m,n] (...truncated)
