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:
[email protected]
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.
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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] , n ≥ . Then
σ (A ) ⊆ K (A ) =
Ki,j (A),
i,j∈N,j=i
where
j
Ki,j (A) = z ∈ C : |z – ai···i | – ri (A) |z – aj···j | ≤ |aij···j |rj (A) .
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 and S̄, where S̄ is the complement of S in N .
Lemma . ([]) Let A = (ai ···im ) ∈ C[m,n] , n ≥ , and S be a nonempty proper subset of N .
Then
Ki,j (A) ∪
Ki,j (A) ,
σ (A ) ⊆ K S (A ) =
i∈S,j∈S̄
i∈S̄,j∈S
where Ki,j (A) (i ∈ S, j ∈ S̄ or i ∈ S̄, 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 ..
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Page 4 of 19
Lemma . ([]) Let A = (ai ···im ) ∈ C[m,n] , n ≥ , and S be a nonempty proper subset of N .
Then
σ (A) ⊆ (A) =
j
i (A),
i,j∈N,j=i
where
j
j
i (A) = z ∈ C : (z – ai···i )(z – aj···j ) – aij···j aji···i ≤ |z – aj···j |ri (A) + |aij···j |rji (A) .
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 KS (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 .
2 Preliminaries
In this section, we start with some lemmas on tensors. They will be useful in the following
proofs.
Lemma . ([]) If A ∈ R[m,n] is irreducible nonnegative, then ρ(A) is a positive eigenvalue with an entrywise positive eigenvector x, i.e., x > , corresponding to it.
Lemma . ([]) Let A ∈ R[m,n] be a nonnegative tensor. Then ρ(A) ≥ maxi∈N {ai···i }.
Lemma . ([]) Suppose that  ≤ A < C . Then ρ(A) ≤ ρ(C ).
Lemma . ([]) Let A be a strong M-tensor and denoted by τ (A) the minimum value of
the real part of all eigenvalues of A. Then τ (A) is an eigenvalue of A with a nonnegative
eigenvector. Moreover, if A is irreducible, then τ (A) is a unique eigenvalue with a positive
eigenvector.
Lemma . ([]) Let A be an irreducible strong M-tensor. Then τ (A) ≤ mini∈N {ai···i }.
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Page 5 of 19
Lemma . ([]) A tensor A is semi-positive if and only if there exists x ≥  such that
Axm– > .
Lemma . ([]) A Z-tensor is a nonsingular M-tensor if and only if it is semi-positive.
Lemma . ([]) Let A, B ∈ Z, assume that A is an M-tensor and B ≥ A. Then B is an
M-tensor, and τ (A) ≤ τ (B ).
3 Main results
3.1 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 .-..
Theorem . Let A = (ai ···im ) ∈ C[m,n] with n ≥ . And let S be a nonempty proper subset
of N . Then
σ (A) ⊆ ϒ (A) :=
S
j
ϒi (A)
∪
i∈S,j∈S̄
j
ϒi (A)
,
()
i∈S̄,j∈S
where
j
j
ϒi (A) = z ∈ C : (z – ai···i )(z – aj···j ) – aij···j aji···i ≤ |z – aj···j |ri (A) + |aij···j |rji (A) .
Proof For any λ ∈ σ (A), let x = (x , . . . , xn )T ∈ Cn /{} be an eigenvector corresponding to
λ, i.e.,
Axm– = λx[m–] .
()
Let |xp | = maxi∈S {|xi |} and |xq | = maxi∈S̄ {|xi |}. Then, xp =  or xq = . Now, let us distinguish
two cases to prove.
(i) |xp | ≥ |xq |, so |xp | = maxi∈N {|xi |} and |xp | > . For any j ∈ S̄, it follows from () that
n
api ···im xi · · · xim = λxm–
p ,
m–
a
x
·
·
·
x
=
λx
.
im
j
i ,...,im = ji ···im i
in ,...,im =
Hence, we have
⎧
m–
m–
m–
⎪
⎨ δpi ···im = , api ···im xi · · · xim + ap···p xp + apj···j xj = λxp ,
δji ···im =
m–
m–
m–
⎪
⎩ δji ···im = , aji ···im xi · · · xim + aj···j xj + ajp···p xp = λxj ,
δpi ···im =
i.e.,
⎧
m–
m–
⎪
⎨ δpi ···im = , api ···im xi · · · xim = (λ – ap···p )xp – apj···j xj ,
δji ···im =
m–
m–
⎪
⎩ δji ···im = , aji ···im xi · · · xim = (λ – aj···j )xj – ajp···p xp .
δpi ···im =
()
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Page 6 of 19
Premultiplying by (λ – aj···j ) in the first equation of () results in
(λ – aj···j )
api ···im xi · · · xim
δpi ···im = ,
δji ···im =
= (λ – aj···j )(λ – ap···p )xm–
– apj···j (λ – aj···j )xm–
.
p
j
()
Combining () and the second equation of () one derives
(λ – aj···j )
api ···im xi · · · xim + apj···j
δpi ···im = ,
δji ···im =
aji ···im xi · · · xim
δji ···im = ,
δpi ···im =
= (λ – aj···j )(λ – ap···p )xm–
– apj···j ajp···p xm–
p
p
m–
= (λ – aj···j )(λ – ap···p ) – apj···j ajp···p xp .
Taking absolute values and using the triangle inequality yield
(λ – aj···j )(λ – ap···p ) – apj···j ajp···p |xp |m–
p
≤ |λ – aj···j |rpj (A)|xp |m– + |apj···j |rj (A)|xp |m– .
Note that |xp | > , thus
(λ – aj···j )(λ – ap···p ) – apj···j ajp···p ≤ |λ – aj···j |rj (A) + |apj···j |rp (A),
p
j
()
j
j
which implies that λ ∈ ϒp (A) ⊆ i∈S,j∈S̄ ϒi (A) ⊆ ϒ S (A).
(ii) |xp | ≤ |xq |, so |xq | = maxi∈N {|xi |} and |xq | > . For any i ∈ S, it follows from () that
n
aii ···im xi · · · xim = λxm–
,
i
m–
a
x
·
·
·
x
=
λx
im
q .
i ,...,im = qi ···im i
in ,...,im =
Using the same method as the proof in (i), we deduce that
(λ – ai···i )
aqi ···im xi · · · xim + aqi···i
δqi ···im = ,
δii ···im =
aii ···im xi · · · xim
δii ···im = ,
δqi ···im =
= (λ – aq···q )(λ – ai···i )xm–
– aiq···q aqi···i xm–
q
q
m–
= (λ – aq···q )(λ – ai···i ) – aiq···q aqi···i xq .
Taking the modulus in the above equation and using the triangle inequality we obtain
(λ – aq···q )(λ – ai···i ) – aiq···q aqi···i |xq |m–
q
≤ |λ – ai···i |rqi (A)|xq |m– + |aqi···i |ri (A)|xq |m– .
Note that |xq | > , thus
(λ – aq···q )(λ – ai···i ) – aiq···q aqi···i ≤ |λ – ai···i |ri (A) + |aqi···i |rq (A).
q
i
()
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
This means that λ ∈ ϒqi (A) ⊆
rem ..
Page 7 of 19
j
S
i∈S̄,j∈S ϒi (A) ⊆ ϒ (A).
This completes our proof of Theo
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
ϒ S (A) = ϒ (A) ∪ ϒ (A) = (A).
j
j
Besides, if n ≥ , |S|(n – |S|) < n(n – ), then ϒ S (A) ⊂ (A) if i (A) ∩ i (A) = ∅ for
any i , i , j , j ∈ N , i = i or j = j . 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 A = (ai ···im ) ∈ C[m,n] with n ≥ . Then
ϒ S (A) ⊆ KS (A) ⊆ K(A) ⊆ (A),
ϒ S (A) ⊆ (A).
()
Proof According to Remark ., it is obvious that ϒ S (A) ⊆ (A). By Theorem . in
[], we know that KS (A) ⊆ K(A) ⊆ (A). Hence, we only prove ϒ S (A) ⊆ KS (A). Let
z ∈ ϒ S (A), then
z∈
j
ϒi (A) or z ∈
i∈S,j∈S̄
j
ϒi (A).
i∈S̄,j∈S
j
Without loss of generality, we assume that z ∈ i∈S,j∈S̄ ϒi (A) (we can prove it similarly if
j
q
z ∈ i∈S̄,j∈S ϒi (A)). Then there exist p ∈ S and q ∈ S̄ such that z ∈ ϒp (A), that is,
(z – ap···p )(z – aq···q ) – apq···q aqp···p ≤ |z – aq···q |rq (A) + |apq···q |rp (A).
p
q
Inasmuch as
(z – ap···p )(z – aq···q ) – |apq···q aqp···p | ≤ (z – ap···p )(z – aq···q ) – apq···q aqp···p ,
z satisfies
(z – ap···p )(z – aq···q ) – |apq···q aqp···p | ≤ |z – aq···q |rq (A) + |apq···q |rp (A),
p
q
which yields
|z – aq···q | |z – ap···p | – rpq (A) ≤ |apq···q | rqp (A) + |aqp···p | = |apq···q |rq (A).
This means that
z ∈ Kp,q (A) ⊆
i∈S,j∈S̄
Ki,j (A) ⊆ KS (A),
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Page 8 of 19
which implies that
ϒ S (A) ⊆ KS (A).
This proof is completed.
3.2 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 A ∈ R[m,n] be an irreducible nonnegative tensor with n ≥ . And let S be
a nonempty proper subset of N . Then
ρ(A) ≤ ηs (A) = max ηS (A), ηS̄ (A) ,
where
η S (A ) =


j
max min ai···i + aj···j + ri (A) + i,j (A) ,
 i∈S j∈S̄
()
with

j
i,j (A) = ai···i – aj···j + ri (A) + aij···j rj (A).
Proof Since A is an irreducible nonnegative tensor, by Lemma ., there exists x =
(x , . . . , xn )T >  such that
Axm– = ρ(A)x[m–] .
()
Let xp = maxi∈S {xi } and xq = maxi∈S̄ {xi }. Below we distinguish two cases to prove.
(i) xp ≥ xq > , so xp = maxi∈N {xi }. For any j ∈ S̄, it follows from () that
n
api ···im xi · · · xim = ρ(A)xm–
p ,
m–
a
x
·
·
·
x
=
ρ(
A
)x
.
ji
···i
i
i
m
 m 
j
i ,...,im =
in ,...,im =
Hence, we have
⎧
m–
m–
⎪
⎨ δpi ···im = , api ···im xi · · · xim = (ρ(A) – ap···p )xp – apj···j xj ,
δ
ji ···im =
m–
m–
⎪
⎩ δji ···im = , aji ···im xi · · · xim = (ρ(A) – aj···j )xj – ajp···p xp .
()
δpi ···im =
Premultiplying by (ρ(A) – aj···j ) in the first equation of () results in
ρ(A) – aj···j
api ···im xi · · · xim
δpi ···im = ,
δji ···im =
= ρ(A) – aj···j ρ(A) – ap···p xm–
– apj···j ρ(A) – aj···j xm–
.
p
j
()
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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It follows from () and the second equation of () that
ρ(A) – aj···j
api ···im xi · · · xim + apj···j
δpi ···im = ,
δji ···im =
aji ···im xi · · · xim
δji ···im = ,
δpi ···im =
= ρ(A) – aj···j ρ(A) – ap···p xm–
– apj···j ajp···p xm–
p
p
m–
= ρ(A) – aj···j ρ(A) – ap···p – apj···j ajp···p xp .
Note that xp ≥ xj for any j ∈ S̄ and by Lemma ., we deduce that
p
ρ(A) – aj···j ρ(A) – ap···p – apj···j ajp···p ≤ ρ(A) – aj···j rpj (A) + apj···j rj (A),
i.e.,
ρ(A) – ap···p + aj···j + rpj (A) ρ(A) + aj···j ap···p + rpj (A) – apj···j rj (A) ≤ .
()
Solving the quadratic inequality () yields
ρ(A) ≤



(A ) .
ap···p + aj···j + rpj (A) + p,j

()
It is not difficult to verify that () can be true for any j ∈ S̄. Thus
ρ(A) ≤



(A ) ,
min ap···p + aj···j + rpj (A) + p,j
 j∈S̄
which implies that
ρ(A) ≤


j
max min ai···i + aj···j + ri (A) + i,j (A) .
 i∈S j∈S̄
()
(ii) xq ≥ xp > , so xq = maxi∈N {xi }. For any i ∈ S, it follows from () that
n
aii ···im xi · · · xim = ρ(A)xm–
,
i
m–
i ,...,im = aqi ···im xi · · · xim = ρ(A)xq .
in ,...,im =
So we obtain
⎧
m–
m–
⎪
⎨ δii ···im = , aii ···im xi · · · xim = (ρ(A) – ai···i )xi – aiq···q xq ,
δqi ···im =
m–
m–
⎪
⎩ δqi ···im = , aqi ···im xi · · · xim = (ρ(A) – aq···q )xq – aqi···i xi .
()
δii ···im =
In a similar manner to the proof of (i)
q
ρ(A) – aq···q ρ(A) – ai···i – aiq···q aqi···i ≤ ρ(A) – ai···i rqi (A) + aqi···i ri (A),
i.e.,
ρ(A) – ai···i + aq···q + rqi (A) ρ(A) + ai···i aq···q + rqi (A) – aqi···i ri (A) ≤ ,
()
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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which yields
ρ(A) ≤



aq···q + ai···i + rqi (A) + q,i
(A ) .

()
It is easy to see that () can be true for any j ∈ S. Thus
ρ(A) ≤



(A ) ,
min aq···q + aj···j + rqj (A) + q,j
 j∈S
which implies that
ρ(A) ≤


j
max min ai···i + aj···j + ri (A) + i,j (A) .
 i∈S̄ j∈S
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 ..
Theorem . Let A ∈ R[m,n] be a nonnegative tensor with n ≥ . And let S be a nonempty
proper subset of N . Then
ρ(A) ≤ ηs = max ηS (A), ηS̄ (A) ,
()
where
η S (A ) =


j
max min ai···i + aj···j + ri (A) + i,j (A) ,
 i∈S j∈S̄
with

j
i,j (A) = ai···i – aj···j + ri (A) + aij···j rj (A).
Proof Let Ak = A + k ε, where k = , , . . . , and ε denotes the tensor with every entry being . Then Ak is a sequence of positive tensors satisfying
 ≤ A < · · · < Ak+ < Ak < · · · < A .
By Lemma ., {ρ(Ak )} is a monotone decreasing sequence with lower bound ρ(A). So
ρ(Ak ) has a limit. Let
lim ρ(Ak ) = λ ≥ ρ(A).
k→+∞
()
By Lemma ., we see that ρ(Ak ) is the eigenvalue of Ak with a positive eigenvector yk ,
i.e., Ak ym–
= ρ(Ak )y[m–]
. In a manner similar to Theorem . in [], we have
k
k
lim ρ(Ak ) = ρ(A).
k→+∞
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
j
Page 11 of 19

And we denote i,j (A) =  {ai···i + aj···j + ri (A) + i,j (A)} (i ∈ S, j ∈ S̄ or i ∈ S̄, j ∈ S). Then
i,j (Ak ) =



nm– – 
j
ai···i + aj···j + + ri (A) +
+ i,j (Ak ) ,

k
k
where
nm– –  

nm– – 
j
rj (A) +
.
+  aij···j +
i,j (Ak ) = ai···i – aj···j + ri (A) +
k
k
k
As m and n are finite numbers, then by the properties of the sequence, it is easy to see that
lim i,j (Ak ) = i,j (A).
k→+∞
Furthermore, since Ak is an irreducible nonnegative tensor, it follows from Theorem .
that
ρ(Ak ) ≤ max ηS (Ak ), ηS̄ (Ak ) .
Letting k → +∞ results in
ρ(A) ≤ max ηS (A), ηS̄ (A) ,
from which one may get the desired bound ().
Remark . Now, we compare the upper bound in Theorem . with that in Theorem .
in []. It is not difficult to see that
η S (A ) =
≤


j
max min ai···i + aj···j + ri (A) + i,j (A)
 i∈S j∈S̄


j
max ai···i + aj···j + ri (A) + i,j (A)
 i∈S,j∈S̄
and


j
max min ai···i + aj···j + ri (A) + i,j (A)
 i∈S̄ j∈S


j
≤ max ai···i + aj···j + ri (A) + i,j (A) .
 i∈S̄,j∈S
ηS̄ (A) =
This shows that the upper bound in Theorem . improves the corresponding one in Theorem . of [].
We have showed that our bound is sharper than the existing one in []. Now we take an
example to show the efficiency of the new upper bound established in this paper.
Example . Let A = (aijk ) ∈ R[,] be nonnegative with entries defined as follows: a =
a = a = a = a = a = a = , a = a = a = , a = , a = , and the
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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other aijk = . It is easy to compute
r (A) = ,
r (A) = ,
r (A) = ;
r (A) = ,
r (A) = ,
r (A) = ;
r (A) = ,
r (A) = ,
r (A) = .
We choose S = {, }. Evidently, S̄ = {}. By Theorem . of [], we have
ρ(A) ≤ ..
By Theorem ., we obtain
ρ(A) ≤ .,
which means that the upper bound in Theorem . is much better than that in Theorem . of [].
3.3 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 [].
Theorem . Let A ∈ R[m,n] be an irreducible nonsingular M-tensor with n ≥ . And let
S be a nonempty proper subset of N . Then
min φ S (A), φ S̄ (A) ≤ τ (A) ≤ max χ S (A), χ S̄ (A) ,
where
χ S (A ) =


j
max min ai···i + aj···j – ri (A) – i,j (A)
 i∈S j∈S̄
()
φ S (A ) =


j
min max ai···i + aj···j – ri (A) – i,j (A) ,
 i∈S j∈S̄
()
and
with

j
i,j (A) = ai···i – aj···j – ri (A) – aij···j rj (A).
Proof Since A is an irreducible nonsingular M-tensor, by Lemma ., there exists x =
(x , . . . , xn )T >  such that
Axm– = τ (A)x[m–] .
Let xp = maxi∈S {xi } and xq = maxi∈S̄ {xi }. We distinguish two cases to prove.
()
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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(i) xp ≥ xq > , so xp = maxi∈N {xi }. For any j ∈ S̄, it follows from () that
n
api ···im xi · · · xim = τ (A)xm–
p ,
m–
a
x
·
·
·
x
=
τ
(
A
)x
.
im
j
i ,...,im = ji ···im i
in ,...,im =
Hence, we have
⎧ m–
m–
⎪
⎨ – δpi ···im = , api ···im xi · · · xim = (ap···p – τ (A))xp + apj···j xj ,
δji ···im =
m–
m–
⎪
⎩ – δji ···im = , aji ···im xi · · · xim = (aj···j – τ (A))xj + ajp···p xp .
()
δpi ···im =
Premultiplying by (aj···j – τ (A)) in the first equation of () results in
– aj···j – τ (A)
api ···im xi · · · xim
δpi ···im = ,
δji ···im =
= aj···j – τ (A) ap···p – τ (A) xm–
+ apj···j aj···j – τ (A) xm–
.
p
j
()
It follows from () and the second equation of () that
– aj···j – τ (A)
api ···im xi · · · xim + apj···j
δpi ···im = ,
δji ···im =
aji ···im xi · · · xim
δji ···im = ,
δpi ···im =
= aj···j – τ (A) ap···p – τ (A) xm–
– apj···j ajp···p xm–
p
p
m–
= aj···j – τ (A) ap···p – τ (A) – apj···j ajp···p xp .
Combining xp ≥ xj for any j ∈ S̄ with Lemma . results in
p
aj···j – τ (A) ap···p – τ (A) – apj···j ajp···p ≤ aj···j – τ (A) rpj (A) – apj···j rj (A),
i.e.,
τ (A) – ap···p + aj···j – rpj (A) τ (A) + aj···j ap···p – rpj (A) + apj···j rj (A) ≤ .
()
Solving the quadratic inequality () yields
τ (A ) ≥



ap···p + aj···j – rpj (A) – p,j
(A ) .

()
It is not difficult to verify that () can be true for any j ∈ S̄. Thus
τ (A ) ≥



max ap···p + aj···j – rpj (A) – p,j
(A ) ,
 j∈S̄
and therefore
τ (A ) ≥


j
min max ai···i + aj···j – ri (A) – i,j (A) .
 i∈S j∈S̄
()
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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(ii) xq ≥ xp > , so xq = maxi∈N {xi }. For any i ∈ S, it follows from () that
n
aii ···im xi · · · xim = τ (A)xm–
,
i
m–
i ,...,im = aqi ···im xi · · · xim = τ (A)xq .
in ,...,im =
So we obtain
⎧ m–
m–
⎪
⎨ – δii ···im = , aii ···im xi · · · xim = (ai···i – τ (A))xi + aiq···q xq ,
δqi ···im =
m–
m–
⎪
⎩ – δqi ···im = , aqi ···im xi · · · xim = (aq···q – τ (A))xq + aqi···i xi .
()
δii ···im =
Using the same technique as the proof of (i), we have
q
aq···q – τ (A) ai···i – τ (A) – aiq···q aqi···i ≤ ai···i – τ (A) rqi (A) – aqi···i ri (A),
which is equivalent to
τ (A) – aq···q + ai···i – rqi (A) τ (A) + ai···i aq···q – rqi (A) + aqi···i ri (A) ≤ ,
()
which results in
τ (A ) ≥



aq···q + ai···i – rqi (A) – q,i
(A ) .

()
It is not difficult to verify that () can be true for any j ∈ S. Thus
τ (A ) ≥



(A ) ,
max aq···q + aj···j – rqj (A) – q,j
 j∈S
which implies that
τ (A ) ≥


j
min max ai···i + aj···j – ri (A) – i,j (A) .
 i∈S̄ j∈S
Let xk = mini∈S {xi } and xl = mini∈S̄ {xi }. With a strategy quite similar to the one utilized in
the above proof, we can prove that
τ (A) ≤ max χ S (A), χ S̄ (A) ,
which implies this theorem.
Remark . We next prove the bounds in Theorem . are sharper than those of Theorem . in []; it is easy to see that
φ S (A ) =
≥


j
min max ai···i + aj···j – ri (A) – i,j (A)
 i∈S j∈S̄


j
min ai···i + aj···j – ri (A) – i,j (A) = ψ S (A)
 i∈S,j∈S̄
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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and


j
min max ai···i + aj···j – ri (A) – i,j (A)
 i∈S̄ j∈S


j
≥ min ai···i + aj···j – ri (A) – i,j (A) = ψ S̄ (A),
 i∈S̄,j∈S
φ S̄ (A) =
which implies that
min φ S (A), φ S̄ (A) ≥ min ψ S (A), ψ S̄ (A)
≥


j
min ai···i + aj···j – ri (A) – i,j (A) .
 i,j∈N,i=j
In the same manner as applied in the above proof, we can deduce the following results:
max χ S (A), χ S̄ (A) ≤ max θ S (A), θ S̄ (A)
≤


j
max ai···i + aj···j – ri (A) – i,j (A) ,
 i,j∈N,i=j
j

where θ S (A) =  maxi∈S,j∈S̄ {ai···i + aj···j – ri (A) – i,j (A)}. Therefore, the conclusions follow
from the above discussions.
Example . Consider the following irreducible nonsingular M-tensor:
A = A(, :, :), A(, :, :), A(, :, :) ∈ R[,] ,
where
⎛
⎞



⎜
⎟
A(, :, :) = ⎝ –. –⎠ ,
 – –
⎛
⎞
– –. –
⎜
⎟
A(, :, :) = ⎝ 

 ⎠,


–.
⎛
⎞
– – 
⎜
⎟
A(, :, :) = ⎝  – –⎠ .
 – 
We compare the results derived in Theorem . with those in Theorem . of [] and
Theorem . of [] in the correct forms. Let S = {, }, then S̄ = {}. By Theorem . of
[], we have
. ≤ τ (A) ≤ ..
By Theorem . of [], we get
. ≤ τ (A) ≤ ..
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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By Theorem ., we obtain
. ≤ τ (A) ≤ ..
This shows that the upper and lower bounds in Theorem . are sharper than those in
Theorem . of [] and Theorem . of [].
In the sequel, we extend the results of Theorem . to a more general case, which needs
a weaker condition compared with Theorem ..
Theorem . Let A ∈ R[m,n] be a nonsingular M-tensor with n ≥ . And let S be a
nonempty proper subset of N . Then
min φ S (A), φ S̄ (A) ≤ τ (A) ≤ max χ S (A), χ S̄ (A) ,
where
χ S (A ) =


j
max min ai···i + aj···j – ri (A) – i,j (A) ,
 i∈S j∈S̄
()
φ S (A ) =


j
min max ai···i + aj···j – ri (A) – i,j (A) ,
 i∈S j∈S̄
()
and
with

j
i,j (A) = ai···i – aj···j – ri (A) – aij···j rj (A).
Proof Since A is a nonsingular M-tensor, A ∈ Z, by Lemma . and Lemma ., there
exists x = (x , . . . , xn )T ≥  such that Axm– > , that is, for any i ∈ N ,
n
ai ···im xi · · · xim > .
i ,...,im =
Let
n
g = min
i ∈N
ai ···im xi · · · xim , xmax = max xi .
i∈N
i ,...,im =
So xmax >  by x ≥ . By replacing the zero entries of A with – k , where k is a positive
integer, we see that the Z-tensor Ak is irreducible. Here, we use ai ···im (– k ) to denote the
entries of Ak . We choose k > [
(nm– –)xm–
max
] + ,
g
then, for any i ∈ N , we have

ai ···im – xi · · · xim
k
=
n
i ,...,im
≥ min
i ∈N
=g–
n
ai ···im xi · · · xim –
i ,...,im =
(nm– – )xm–
max
> ,
k
(nm– – )xm–
max
k
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
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which implies that Ak xm– >  and, by Lemma . and Lemma ., we infer that Ak is an
(nm– –)xm–
max
] + . It follows from the above discusirreducible nonsingular M-tensor if k > [
g
sions that Ak (k > [
satisfying
(nm– –)xm–
max
]
g
+ ) is a sequence of irreducible nonsingular M-tensors
A > · · · > Ak+ > Ak .
By Lemma ., {τ (Ak )} is a monotone increasing sequence with upper bound τ (A) so that
τ (Ak ) has a limit. Let
lim τ (Ak ) = λ ≤ τ (A).
()
k→+∞
By Lemma ., we see that τ (Ak ) is the eigenvalue of Ak with a positive eigenvector yk ,
= τ (Ak )y[m–]
. As homogeneous multivariable polynomials, we can restrict
i.e., Ak ym–
k
k
yk to a ball; that is, yk = . Then {yk } is a bounded sequence, so it has a convergent
subsequence. Without loss of generality, we can suppose it is the sequence itself. Let yk → y
= τ (Ak )y[m–]
and letting k → +∞, we
as k → +∞, we get y ≥  and y = . By Ak ym–
k
k
[m–]
have Ay = λy
. Thus λ is an eigenvalue of A, thus λ ≥ τ (A). Together with () this
results in λ = τ (A), which means that
lim τ (Ak ) = τ (A).
k→+∞
j
Besides, for i ∈ S, j ∈ S̄ (for i ∈ S̄, j ∈ S, we can define Mi and Mj similarly), we define the
following sets:
j
Mi = {aii ···im |aii ···im = , δii ···im =  and δji ···im = , i , . . . , im ∈ N},
Mj = {aji ···im |aji ···im =  and δji ···im = , i , . . . , im ∈ N}.
j
j
Let the numbers of entries in Mi and Mj be ni and nj , respectively, and we denote i,j (A) =
j

+ aj···j – ri (A) – i,j (A)}. Then

{a
 i···i
j


ni
j

i,j (Ak ) =
ai···i + aj···j – ri (A) – – i,j (Ak ) ,

k
where
j
εi,j
nj
n 
j
rj (A) +
,
i,j (Ak ) = ai···i – aj···j – ri (A) – i –  aij···j –
k
k
k
with
εi,j =
, if aij···j = ,
, if aij···j = .
By the properties of the sequence, it is not difficult to verify that
lim χ S (Ak ) = χ S (A),
k→+∞
lim φ S (Ak ) = φ S (A).
k→+∞
Huang et al. Journal of Inequalities and Applications (2016) 2016:254
Furthermore, since Ak is an irreducible nonsingular M-tensor for k > [
Theorem ., we have
Page 18 of 19
(nm– –)xm–
max
] + ,
g
by
min φ S (Ak ), φ S̄ (Ak ) ≤ τ (Ak ) ≤ max χ S (Ak ), χ S̄ (Ak ) .
Letting k → +∞ results in
min φ S (A), φ S̄ (A) ≤ τ (A) ≤ max χ S (A), χ S̄ (A) .
This completes our proof of Theorem ..
4 Concluding remarks
In this paper, a new S-type eigenvalue inclusion set for tensors is presented, which is
proved to be sharper than the ones in [, ]. As applications, we give new bounds for
the spectral radius of nonnegative tensors and the minimum H-eigenvalue of strong Mtensors, these bounds improve some existing ones obtained by Li et al. [] and He and
Huang []. In addition, we extend these new bounds to more general cases.
However, the new S-type eigenvalue inclusion set and the derived bounds depend on the
set S. How to choose S to make ϒ S (A) and the bounds exhibited in this paper as tight as
possible is very important and interesting, while if the dimension of the tensor A is large,
this work is very difficult. Therefore, future work will include numerical or theoretical
studies for finding the best choice for S.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the National Natural Science Foundations of China (No. 11171273) and sponsored by
Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX201628).
Received: 27 April 2016 Accepted: 5 October 2016
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