Sang & Zhao, Cogent Mathematics (2017), 4: 1320831https://doi.org/10.1080/23311835.2017.1320831
COMPUTATIONAL SCIENCE | RESEARCH ARTICLE
A new eigenvalue inclusion set for tensors with its applicationsCaili Sang1 and Jianxing Zhao1*
Abstract: In this paper, we give a new eigenvalue localization set for tensors and show that the new set is tighter than those presented by Qi (2005) and Li et al. (2014). As applications, we give a new sufficient condition of the positive (semi-)definiteness for an even-order real symmetric tensor and new lower and upper bounds of the minimum eigenvalue for -tensors.
1. IntroductionFor a positive integer n, n ≥ 2, N denotes the set {1, 2, … , n}. ℂ (respectively, ℝ) denotes the set of all complex (respectively, real) numbers. We call = (ai1⋯im
) a complex (real) tensor of order m dimension n, denoted by ℂ[m,n](ℝ[m,n]), if ai1…im
∈ ℂ(ℝ), where ij ∈ N for j = 1, 2, … , m.
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: [email protected], [email protected]
Reviewing editor:Song Wang, Curtin University, Australia
Additional information is available at the end of the article
ABOUT THE AUTHORJianxing 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.
PUBLIC INTEREST STATEMENTOne 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 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 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.
Received: 30 December 2016Accepted: 12 April 2017First Published: 20 April 2017
Sang & Zhao, Cogent Mathematics (2017), 4: 1320831https://doi.org/10.1080/23311835.2017.1320831
A real tensor = (ai1…im) is called symmetric (Qi, 2005) if
where Πm is the permutation group of m indices.
A tensor = (ai1 i2…im) 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 = (aii2…im)
be a nonnegative tensor, G = (gij) ∈ ℝn×n, gij =
∑j∈{i2,…, im}
aii2 … im. is called weakly reducible if G is a
reducible matrix. 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).
For a general tensor = (ai1 … im) ∈ ℂ
[m,n], 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.
Given a tensor = (ai1 … im) ∈ ℂ
[m,n], if there are � ∈ ℂ and x = (x1, x2, … , xn)T ∈ ℂ
n�{0} such that
then � is called an eigenvalue of and x an eigenvector of associated with �, where xm−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).
Let = (ai1 … im) ∈ ℝ
[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 > 𝜌(). 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 𝜏() > 0 is the unique eigenvalue with a positive eigenvector; for details, see Zhang et al. (2014) and Ding, Qi, and Wei (2013).
�i1…im=
{1, if i1 = ⋯ = im,
0, otherwise.
ai1…im= a
�(i1…im), ∀� ∈ Πm,
ai1 i2…im= 0, ∀ i1 ∈ �,∀ i2,⋯ , im ∉ �.
xm−1 = �x[m−1],
(xm−1)i =∑
i2,…, im∈N
aii2 …imxi2
… xim,
x[m−1] = (xm−11 , xm−1
2 , … , xm−1n )T .
�() = max{|�|:� ∈ �()},
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Given an even-order symmetric tensor = (ai1 ⋯ im) ∈ 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 in-cludes 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 and Li (2016), 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 supersym-metric tensors, which can be easily extended to general tensors (Li et al., 2014; Yang & Yang, 2010).
Theorem 1.1 (Qi, 2005, Theorem 6) Let = (ai1… im
) ∈ ℂ[m,n]. Then
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 local-ization sets for tensors.
Theorem 1.2 (Li et al., 2014, Theorem 2.1) Let = (ai1… im
) ∈ ℂ[m,n]. Then
where
One of many applications of eigenvalue inclusion sets is to bound the minimum H-eigenvalue of -tensors (He & Huang, 2014; Huang et al., 2016; Wang & Wei, 2015; Zhao & Sang, 2016). In He and Huang (2014) provided some inequalities on �() for an irreducible -tensor as follows.
Theorem 1.3 (He & Huang, 2014, Theorem 2.1) Let = (ai1… im
) ∈ ℝ[m,n] be an irreducible -tensor.
Then
where Ri() =∑
i2,…, im∈N
aii2… im
.
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 �() ≤ min
i∈N{ai… i}.
𝜎() ⊆ Γ() =⋃
i∈N
Γi(),
Γi() = {z ∈ ℂ:|z − ai… i| ≤ ri()}, ri() =∑
�ii2… i
m=0
|aii2… im
|.
𝜎() ⊆ () =⋃
i, j∈N, j≠i
i, j(),
i, j() = {z ∈ ℂ:(|z − ai… i| − rj
i())|z − aj… j| ≤ |aij… j|rj()},
rj
i() =
∑
�ii2… im
= 0,
�ji2… im
= 0
|aii2… im
| = ri() − |aij…j|.
�() ≤ mini∈N
aii… i , and mini∈N
Ri() ≤ �() ≤ maxi∈N
Ri(),
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In this paper, we continue this research on the eigenvalue inclusion sets for tensors and its appli-cations. 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).
2. A new eigenvalue inclusion set for tensorsIn 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.
Theorem 2.1 Let = (ai1… im
) ∈ ℂ[m,n]. Then
where
Proof For any � ∈ �(), let x = (x1, … , xn)
T ∈ ℂn�{0} be an eigenvector corresponding to �, i.e.
Let
(where the last term above is defined to be zero if n = 2). Then, |xp| > 0. From (1), we have
Taking modulus in the above equation and using the triangle inequality give
equivalently,
If |xq| = 0, by |xp| > 0, we have |𝜆 − ap… p| − r̃p() ≤ 0. Then for any j ≠ p,
which implies that 𝜆 ∈ Ωp, j() ⊆ Ω(). Otherwise, |xq| > 0. Similarly, from (1), we can obtain
𝜎() ⊆ Ω() =⋃
i, j∈N,
i≠j
Ωi, j(),
Ωi, j() ={z ∈ ℂ:
(|z − ai… i| − r̃i()
)|z − aj… j| ≤ r̂i()rj()
},
r̂i() =∑
k≠i
|aik… k|, r̃i() = ri() − r̂i().
(1)xm−1 = �x[m−1].
|xp| ≥ |xq| ≥ max{|xk|:k ∈ N, k ≠ p, q}
(� − ap… p)xm−1
p =∑
�pi2… i
m=0,
(i2i3… i
m)≠(jj… j)
api2… im
xi2… xim
+∑
j≠p
apj… jxm−1
j .
|𝜆 − ap… p||xp|m−1
≤∑
𝛿pi2… i
m=0,
(i2i3… i
m)≠(jj… j)
|api2…im
||xi2|… |xim | +
∑
j≠p
|apj⋯j||xj|m−1
≤∑
𝛿pi2… i
m=0,
(i2i3… i
m)≠(jj… j)
|api2… im
||xp|m−1 +
∑
j≠p
|apj… j||xq|m−1
= r̃p()|xp|m−1 + r̂p()|xq|
m−1,
(2)(|𝜆 − ap… p| − r̃p())|xp|
m−1≤ r̂p()|xq|
m−1.
(|𝜆 − ap… p| − r̃p())|𝜆 − aj… j| ≤ 0 ≤ r̂p()rj(),
(3)|� − aq… q||xq|m−1
≤ rq()|xp|m−1.
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Multiplying (2) with (3) and noting that |xp|m−1|xq|
m−1> 0, we have
and 𝜆 ∈ Ωp, q() ⊆ Ω(). Hence, 𝜎() ⊆ Ω(). ✷
Next, a comparison theorem is given for Theorems 1.1, 1.2 and 2.1.
Theorem 2.2 Let = (ai1… im
) ∈ ℂ[m,n]. Then
Proof According to Theorem 2.3 in Li et al. (2014), () ⊆ Γ(). Hence it suffices to show that Ω() ⊆ (). Let z ∈ Ω(), then there exist p, q ∈ N, p ≠ q such that z ∈ Ωp, q(), i.e.
The following proof will be divided into two cases according to a certain rule.
(I) Suppose that r̂p()rq() = 0. Then r̂p() = 0 or rq() = 0.
(i) If r̂p() = 0, then |apq… q| = 0,̃rp() = rqp () = rp(), and
which implies that z ∈ p, q() ⊆ (), consequently, Ω() ⊆ ().
(ii) If rq() = 0, by rqp () ≥ r̃p(), we have
We can also have z ∈ p, q() ⊆ () and Ω() ⊆ ().
(II) Suppose that r̂p()rq() > 0. Dividing (5) by r̂p()rq(), we have
which implies
or
Let a = |z − ap… p|, b = r̃p(), c = r̂p() − |apq… q|, d = |apq… q|.
(i) When (7) holds and |apq… q| > 0, then from Lemma 2.2 in Li and Li (2016) and (6), we have
Sang & Zhao, Cogent Mathematics (2017), 4: 1320831https://doi.org/10.1080/23311835.2017.1320831
and other aijkl = 0. The eigenvalue inclusion sets Γ(), (), Ω() and the exact eigenvalues are drawn in Figure 1, where Γ(), () and Ω() are represented by green boundary, blue boundary, and red boundary, respectively. The exact eigenvalues are plotted by black “+". It is easy to see 𝜎() ⊆ Ω() ⊂ () ⊂ Γ(), i.e. Ω() can capture all eigenvalues of more precisely than () and Γ().
3. Determining the positive definiteness for an even-order real symmetric tensorAs shown in Qi (2005), Li et al. (2014), Li and Li 2016, Li et al. (2016), Li et al. (2015) and 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 sec-tion provide some sufficient conditions for the positive definiteness and positive semi-definiteness of tensors, respectively.
Theorem 3.1 Let = (ai1⋯ im
) ∈ ℝ[m,n] be an even-order symmetric tensor with ak… k > 0 for all k ∈ N.
If for any i, j ∈ N, i ≠ j,
then is positive definite.
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 ak… k > 0, k ∈ N, we have
This is a contradiction. Hence, 𝜆 > 0, and is positive definite. The conclusion follows. ✷
Similar to the proof of Theorem 3.1, we can easily obtain the following conclusion:
Let = (ai1 … im) ∈ ℝ
[m,n] be an even-order symmetric tensor with ak… k ≥ 0 for all k ∈ N. If for any i, j ∈ N, i ≠ j,
(ai… i − r̃i()
)aj… j > r̂i()rj().
(|𝜆 − ai… i| − r̃i()
)|𝜆 − aj… j| ≤ r̂i()rj().
(|𝜆 − ai… i| − r̃i()
)|𝜆 − aj… j| ≥
(ai… i − r̃i()
)aj… j > r̂i()rj().
Figure 1. Comparisons of Γ(), () and Ω().
−5 0 5 10 15 20 25 30
−15
−10
−5
0
5
10
15
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then is positive semi-definite.
4. New bounds for the minimum eigenvalue of -tensorsIn 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).
THEOREM 4.1 Let ∈ ℝ[m,n] be a weakly irreducible -tensor. Then
where
Proof Because �() is an eigenvalue of , from Theorem 2.1, there are i, j ∈ N, j ≠ i, such that
From Theorem 2.1, we can get
equivalently,
Solving for �() gives
Next, we prove that the second inequality in (9) holds. Suppose that x = (x1, … , xn)
(10)𝜏()2 − (ai… i + aj… j − r̃i())𝜏() + aj… j(ai… i − r̃i()) − r̂i()rj() ≤ 0.
𝜏() ≥1
2
{ai… i + aj… j − r̃i() −
[(ai… i + aj… j − r̃i())2 − 4(aj… j(ai… i − r̃i()) − r̂i()rj())
] 1
2
}
=1
2
{ai… i + aj… j − r̃i() −
[(ai… i − aj… j − r̃i())2 + 4r̂i()rj()
] 1
2
}
≥ mini, j∈N
j≠i
1
2
{ai… i + aj… j − r̃i() −
[(ai… i − aj… j − r̃i())2 + 4r̂i()rj()
] 1
2
}.
(11)xm−1 = �()x[m−1],
xl ≤ xu ≤ min{xk:k ∈ N, k ≠ l, u}.
(12)
(au…u − �())xm−1
u = −∑
�ui2… i
m=0
aui2… im
xi2… xim
≥ ru()xm−1
l ,
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i.e.
Multiplying (12) with (13) and noting that |xl|m−1|xu|
m−1> 0, we have
Then solving for �(A) gives
This proof is completed. ✷
Remark 4.1 Note that Lij() ≠Wij(). 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), respec-tively); 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 = (aijkl) 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.
Theorem 4.2 Let ∈ ℝ[m,n] be an -tensor. Then
Proof Because is an -tensor, by Theorems 2 and 3 in Ding et al. (2013), there is x = (x1, … , xn)
T≥ 0
such that xm−1> 0. Let
Then xmax
> 0. Let k = −1
k, 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 >[nm−1xm−1
max
𝛿
]+ 1, then for any i ∈ N,
(al… l − 𝜏())xm−1
l = −∑
𝛿li2… i
m=0
(i2i3…im
)≠(jj… j)
ali2… im
xi2… xim
−∑
j≠l
alj… jxm−1
j
≥ r̃l()xm−1
l + r̂l()xm−1
u ,
(13)(al… l − 𝜏() − r̃l())xm−1
l ≥ r̂l()xm−1
u .
(14)(au…u − 𝜏())(al… l − 𝜏() − r̃l()) ≥ r̂l()ru().
Sang & Zhao, Cogent Mathematics (2017), 4: 1320831https://doi.org/10.1080/23311835.2017.1320831
which implies that kxm−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 subse-quence. 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) results in � = �(), which means that
Using Theorem 4.1 for k, we have
where
Letting k→ +∞ in (17), we have that (15) holds. ✷
Similar to the proof of Theorem 4.2, we can extend the results of Theorem 1.3 to a more general case.
Theorem 4.3 Let be an -tensor. Then
Next, we compare the bounds in Theorem 4.2 with those in Theorem 4.3.
Theorem 4.4 Let = (ai1… im
) ∈ ℝ[m,n] be an -tensor. Then
Proof Similar to the proof of Theorem 5 in Zhao and Sang (2016), we can obtain mini∈N
Ri() ≤ mini, j∈N
j≠i
Lij() easily. Next, we only prove that the last inequality in (18) holds.
(I) For any i, j ∈ N, j ≠ i, if Ri() ≤ Rj(), i.e. ai… i − r̃i() − r̂i() ≤ aj… j − rj(), then
Hence,
∑
i2,…, im∈N
(aii
2… im
−1
k
)xi
2… xim
≥ mini∈N
∑
i2,…, im∈N
aii2… im
xi2… xim
−nm−1
kxm−1
max
= 𝛿 −nm−1xm−1
max
k> 0,
(16)limk→+∞
�(k) = � ≤ �().
limk→+∞
�(k) = �().
(17)mini, j∈N
j≠i
Lij(k) ≤ �(k) ≤ maxi, j∈N
j≠i
Lij(k),
Lij(k) =1
2
{ai… i + aj… j −
2
k− r̃i(k) −
[(ai… i − aj… j − r̃i(k))
2 + 4r̂i(k)rj(k)] 1
2
},
ri(k) = ri() +nm−1
k, r̂i(k) = r̂i() +
n − 1
k, r̃i(k) = r̃i() +
nm−1 − (n − 1)
k.
�() ≤ mini∈N
aii… i , and mini∈N
Ri() ≤ �() ≤ maxi∈N
Ri().
(18)mini∈N
Ri() ≤ mini, j∈N
j≠i
Lij() ≤ maxi, j∈N
j≠i
Lij() ≤ maxi∈N
Ri().
r̂i() ≥ ai… i − aj… j − r̃i() + rj().
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When
we have
And when
i.e. ai… i − r̃i() ≤ ajj… j − 2rj(), we have
Therefore,
which implies
(II) For any i, j ∈ N, j ≠ i, if Rj() ≤ Ri(), i.e.
then
[ai… i − aj… j − r̃i()]2 + 4r̂i()rj()
≥ [ai… i − aj… j − r̃i()]2 + 4[ai… i − aj… j − r̃i() + rj()]rj()
= [ai… i − aj… j − r̃i()]2 + 4[ai… i − aj… j − r̃i()]rj() + 4[rj()]2
= [ai… i − aj… j − r̃i() + 2rj()]2.
ai… i − aj… j − r̃i() + 2rj() > 0,
ai… i + aj… j − r̃i() − [(ai… i − aj… j − r̃i())2 + 4r̂i()rj()]1
2
≤ ai… i + aj… j − r̃i() − [ai… i − aj… j − r̃i() + 2rj()]
= 2aj… j − 2rj()
= 2Rj().
ai… i − aj… j − r̃i() + 2rj() ≤ 0,
ai… i + aj… j − r̃i() − [(ai… i − aj… j − r̃i())2 + 4r̂i()rj()]1
2
≤ ai… i + aj… j − r̃i() − [(ai… i − aj… j − r̃i())2]1
2
= ai… i + aj… j − r̃i() − |ai… i − aj… j − r̃i()|
= ai… i + aj… j − r̃i() + [ai… i − aj… j − r̃i()]
= 2ai… i − 2r̃i()
≤ 2aj… j − 4rj()
≤ 2aj… j − 2rj()
= 2Rj().
Lij() =1
2
{ai… i + aj… j − r̃i() −
[(ai… i − aj… j − r̃i())2 + 4r̂i()rj()
] 1
2
}≤ Rj(),
maxi, j∈N
j≠i
Lij() ≤ maxj∈N
Rj().
aj… j − rj() ≤ ai… i − r̃i() − r̂i(),
rj() ≥ aj… j − ai… i + r̃i() + r̂i().
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Similarly, we can obtain
which implies
The conclusion follows from I and II. ✷
5. ConclusionIn this paper, a new eigenvalue localization set for tensors is given. It is proved that the new set is tighter than those in Qi (2005) and Li et al. (2014). As applications of the obtained results, a new suf-ficient condition of the positive (semi-)definiteness for an even-order real symmetric tensor, and new lower and upper bounds of the minimum eigenvalue for -tensors, which are the correction of the bounds in Wang and Wei (2015), are obtained. Finally, we extend Lemma 4.4 in Wang and Wei (2015) and Theorem 2.1 in He and Huang (2014) to a more general case.
Authors’ contributionsAll authors contributed equally to this work. All authors read and approved the final manuscript.
FundingThis work is supported by National Natural Science Foundations of China [grant number 11501141], Foundation of Guizhou Science and Technology Department [grant number [2015]2073] and Natural Science Programs of Education Department of Guizhou Province [grant number [2016]066].
Citation informationCite this article as: A new eigenvalue inclusion set for tensors with its applications, Caili Sang & Jianxing Zhao, Cogent Mathematics (2017), 4: 1320831.
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