Bounds for eigenvalues of nonsingular H-tensor.pdf

Electronic Journal of Linear Algebra

Volume 29 Article 2

2015

Bounds for eigenvalues of nonsingular H-tensorXue-Zhong WangSchool of Mathematics and Statistics, Hexi University, [email protected]

Yimin WeiSchool of Mathematics and Computer Science, Guizhou Normal University, [email protected]

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Recommended CitationWang, Xue-Zhong and Wei, Yimin. (2015), "Bounds for eigenvalues of nonsingular H-tensor", Electronic Journal of Linear Algebra,Volume 29, pp. 3-16.DOI: http://dx.doi.org/10.13001/1081-3810.3116

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BOUNDS FOR EIGENVALUES OF NONSINGULAR H-TENSOR∗

XUE-ZHONG WANG† AND YIMIN WEI‡

Abstract. The bounds for the Z-spectral radius of nonsingular H-tensor, the upper and lower

bounds for the minimum H-eigenvalue of nonsingular (strong) M-tensor are studied in this paper.

Sharper bounds than known bounds are obtained. Numerical examples illustrate that our bounds

give tighter bounds.

Dedicated to Professor Ravindra B. Bapat on the occasion of his 60th birthday

Key words. M-tensor, H-tensor, Z-spectral radius, Minimum H-eigenvalue.

AMS subject classifications. 15A18, 15A69, 65F15, 65F10.

1. Introduction. Eigenvalue problems of higher order tensors have become an

important topic in applied mathematics branch, numerical multilinear algebra, and it

has a wide range of practical applications [2, 3, 4, 1, 8, 12, 13, 14, 15, 16, 19, 20, 21].

A tensor can be regarded as a higher-order generalization of a matrix. Let C(respectively, R) be the complex (respectively, real) field. An m-order n-dimensional

square tensor A with nm entries can be defined as follows,

A = (ai1i2...im), ai1i2...im ∈ C, 1 ≤ i1, i2, . . . , im ≤ n.

Let A be an m-order n-dimensional tensor, and x ∈ Cn. Then

(1.1) Axm =

n∑

i1,i2,...,im=1

ai1i2...imxi1xi2 . . . xim ,

and Axm−1 is a vector in Cn, with its ith component defined by

(Axm−1)i =n∑

i2,i3,...,im=1

aii2...imxi2xi3 · · ·xim .

∗Received by the editors on February 5, 2015. Accepted for publication on July 20, 2015. Handling

Editor: Manjunatha Prasad Karantha.†School of Mathematics and Statistics, Hexi University, Zhangye, 734000, P. R. China (xuezhong-

[email protected] ). This author is supported by the National Natural Science Foundation of China

under grant 11171371.‡School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001,

P.R. China (ymwei [email protected]). This author is supported by the National Natural Science

Foundation of China under grant 11271084.

3

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 29, pp. 3-16, September 2015

http://repository.uwyo.edu/ela

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4 X.Z. Wang, and Y. Wei

Let r be a positive integer. Then x[r] = [xr1, x

r2, . . . , x

rn]

⊤ is a vector in Cn, with its

ith component defined by xri .

The following two definitions were first introduced and studied by Qi and Lim,

respectively.

Definition 1.1. ([8, 11, 14]) Let A be an m-order n-dimensional real tensor.

A pair (λ, x) ∈ C×(Cn\{0}) is called an eigenvalue-eigenvector (or simply eigenpair)

of A, if it satisfies the equation

Axm−1 = λx[m−1].

We call (λ, x) an H-eigenpair, if both λ and x are real.

Definition 1.2. ([8, 11, 14]) Let A be an m-order n-dimensional real tensor.

A pair (λ, x) ∈ C× (Cn \ {0}) is called an E-eigenvalue and E-eigenvector (or simply

E-eigenpair) of A, if they satisfy the equation

{ Axm−1 = λx,

x⊤x = 1.

We call (λ, x) a Z-eigenpair, if both λ and x real. Here x⊤ denotes the transpose of

x.

In [8], He and Huang presented the definition of the Z-spectral radius of A as

follows.

Definition 1.3. ([1, 8]) Suppose that A is an m-order n-dimensional real

tensor. Let σ(A) denote the Z-spectrum of A by the set of all Z-eigenvalues of A.

Assume that σ(A) 6= ∅. Then the Z-spectral radius of A is denoted by

ρ(A) = sup{|λ| : λ ∈ σ(A)}.

Particularly, if A is an m-order n-dimensional nonnegative tensor, then

ρ(A) = max{|λ| : λ ∈ σ(A)}.

Recently, many contributions have been made on the bounds of the spectral radius

of nonnegative tensor in [1, 10, 13, 14]. Similarly, bounds for the Z-spectral radius

were given in [8] for the H-tensors. Also, in [7], He and Huang obtained the upper

and lower bounds for the minimum H-eigenvalue of nonsingular (strong) M-tensors.

In this paper, our purpose is to propose sharper bounds for the Z-spectral radius

of nonsingular H-tensors and for the minimum H-eigenvalue of nonsingular (strong)

M-tensors.



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Bounds for Eigenvalues of Nonsingular H-tensor 5

2. Preliminaries. We start this section with some fundamental notions and

properties on tensors. An m-order n-dimensional tensor A is called nonnegative ([2,

3, 9, 16, 20, 21]), if each entry is nonnegative. Similar to Z-matrices, we denote tensors

with all non-positive off-diagonal entries by Z-tensors. The m-order n-dimensional

identity tensor, denoted by I = (δi1i2...im), is the tensor with entries

δi1i2...im =

{

1, i1 = i2 = · · · = im,

0, otherwise.

The tensor D = (di1i2...im) is the diagonal tensor of A = (ai1i2...im), if

{

di1i2...im = ai1i2...im , i1 = i2 = · · · = im,

0, otherwise.

Definition 2.1. ([18]) Let A and B be two m-order n-dimensional tensors. If

there exists matrices P and Q of n-order with PIQ = I such that B = PAQ, then

we say that the two tensors are similar.

Let the tensor F be associated with an undirected d-partite graph G(F) =

(V,E(F)), the vertex set of which is the disjoint union V =⋃d

j=1 Vj , with Vj =

[mj ], j ∈ [d]. The edge (ik, il) ∈ Vk × Vl, k 6= l belongs to E(F) if and only if

fi1,i2,...,id > 0 for some d− 2 indices i1, . . . , id \ {ik, il}. The tensor F is called weakly

irreducible if the graph G(F) is connected. We call F irreducible if for each proper

nonempty subset ∅ 6= I $ V , the following condition holds: let J := V \I. Then there

exists k ∈ [d], ik ∈ I ∩ Vk and ij ∈ J ∩ Vj for each j ∈ [d] \ {k} such that fi1,...,id > 0.

This definition of irreducibility agrees with [2, 13].

Friedland et al. [6] showed that if F is irreducible then F is weakly irreducible

and presented the following results.

Lemma 2.2. ([6]) If the nonnegative tensor A is irreducible, then A is weakly

irreducible. For m = 2, A is irreducible if and only if A is weakly irreducible.

Lemma 2.2 illustrates that a nonnegative irreducible tensor must be weakly irre-

ducible. For a general tensor A = (ai1i2...im), ai1i2...im ∈ C, we can draw the following

conclusion.

Lemma 2.3. If a tensor A is irreducible, then A is weakly irreducible. For m = 2,

A is irreducible if and only if A is weakly irreducible.

Proof. Let A = D−E , where D is the diagonal tensor of A. If A is irreducible, it

is equivalent that E is irreducible. Note that |E| is a nonnegative tensor, by Lemma

2.2, |E| is weakly irreducible, and then A is weakly irreducible. Similar to the proof

of [6], we can get case m = 2.



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Lemma 2.4. ([14]) The product of the eigenvalues λi of tensor A is equal to

det(A), that is,

det(A) =

n(m−1)n−1

∏

i=1

λi.

We call tensor A is nonsingular, if det(A) 6= 0.

Definition 2.5. ([5, 23]) We call a tensor A an M-tensor, if there exist a

nonnegative tensor B and a positive real number η ≥ ρ(B) such that

A = ηI − B.

If η > ρ(B) then A is called a nonsingular (strong) M-tensor.

In [23], Zhang et al. obtained the following result for the H-eigenvalues of a

nonsingular (strong) M-tensor.

Lemma 2.6. ([23]) Let A be a nonsingular (strong) M-tensor and τ(A) denote

the minimal value of the real part of all eigenvalues of A. Then τ(A) > 0 is an H-

eigenvalue of A with a nonnegative eigenvector. If A is weakly irreducible Z-tensor,

then τ(A) > 0 is the unique eigenvalue with a positive eigenvector.

Yang and Yang [20], Yuan and You [22] showed that if

(2.1) B = D−(m−1)AD(m−1),

where D is a diagonal nonsingular matrix, then A and B are similar. It is easy to see

that the similarity relation is an equivalent relation, and similar tensors have the same

characteristic polynomials, and thus they have the same spectrum (as a multi-set).

Now, we introduce the comparison tensor of any tensor A.

Definition 2.7. ([5]) Let A = (ai1...im) be an m-order and n-dimensional

tensor. We call a tensor M(A) = (mi1i2...im) the comparison tensor of A if

mi1i2...im =

{ |ai1i2...im |, (i1i2 . . . im) = (i1i1 . . . i1),

−|ai1i2...im |, (i1i2 . . . im) 6= (i1i1 . . . i1).

In the following, some basic definitions are given, which will be used in the sub-

sequent discussion. In [5], Ding et al. extended H-matrices to H-tensors as follows.

Definition 2.8. ([5]) We call a tensor A an H-tensor, if its comparison tensor

is an M-tensor; we call it as a nonsingular H-tensor, if its comparison tensor is a

nonsingular M-tensor.



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Very recently, Kannan et al. [17] established some properties of strong H-tensors

and general H-tensors.

Remark 2.9. It follows from definition 2.8 that an M-tensor is an H-tensor and

a nonsingular M-tensor is a nonsingular H-tensor.

Definition 2.10. ([5]) Let A be an m-order and n-dimensional tensor. A is

quasi-diagonally dominant, if there exists a positive vector x = (x1, x2, . . . , xn)⊤ such

that

(2.2) |aii...i|xm−1i ≥

∑

(i2i3...im) 6=(ii...i)

|aii2...im |xi2xi3 . . . xim , i = 1, 2, . . . , n.

If the strict inequality holds in (2.2) for all i, A is called quasi-strictly diagonally

dominant.

Lemma 2.11. ([5]) A tensor A is a nonsingular H-tensor if and only if it is

quasi-strictly diagonally dominant.

3. Bounds for the spectral radius of H-tensors. In this section, we present

some bounds for the Z-spectral radius of H-tensors. For convenience, let N =

{1, 2, . . . , n} . We denote by Ri(A) and R(A) the sum of the ith row and the maximal

row sum of A, respectively, i.e.,

Ri(A) =n∑

i2,i3,...,im=1

|aii2...im |, R(A) = maxi

Ri(A).

In [1], Chang, Pearson, and Zhang have given the following bounds for the Z-

eigenvalues of an m-order n-dimensional tensor A.

Lemma 3.1. ([1]) Let A be an m-order and n-dimensional tensor with σ(A) 6= ∅.Then

ρ(A) ≤ √nmax

i∈N

n∑

i2,i3,...,im=1

|aii2...im | = √nR(A).

For positively homogeneous operators, Song and Qi [19] established the relation-

ship between the Gelfand formula and the spectral radius, as well as the upper bound

of the spectral radius. Following the Corollary 4.5 in [19], He and Huang [8] presented

the following lemma.

Lemma 3.2. ([8, 19]) Let A be an m-order and n-dimensional tensor with

σ(A) 6= ∅. Then

ρ(A) ≤ maxi∈N

n∑

i2,i3,...,im=1

|aii2...im | = R(A).



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Based on the above lemma, we obtain some upper bounds for the Z-spectral

radius when A is a nonsingular H-tensor as follows.

Theorem 3.3. Let A be an m-order and n-dimensional nonsingular H-tensor

with σ(A) 6= ∅. Then

ρ(A) ≤ 2maxi∈N

|aii...i|.

Proof. Since A is a nonsingular H-tensor, there exists a positive diagonal matrix

X =diag(x1, x2, . . . , xn) such that AX(m−1) is strictly diagonally dominant. Then

X−(m−1)AX(m−1),

is also strictly diagonally dominant, i.e.,

|aii...i| >∑

(i2,i3,...,im)

6=(i,i,...,i)

|aii2...im |xi2xi3 . . . xim

xm−1i

, i ∈ N.

Because X−(m−1)AX(m−1) and A are similar, it follows that

ρ(A) = ρ(X−(m−1)AX(m−1)) ≤ R(X−(m−1)AX(m−1))

= maxi

n∑

i2,i3,...,im=1

|aii2...im |xi2xi3 ...xim

xm−1i

= maxi

(|aii...i|+∑

(i2,i3,...,im)

6=(i,i,...,i)

|aii2...im |xi2xi3 ...xim

xm−1i

)

< 2maxi

|aii...i|.

By the above theorem, the following corollary can be obtained easily.

Corollary 3.4. If A is an m-order and n-dimensional nonsingular H-tensor

with σ(A) 6= ∅, then

ρ(A) ≤ min

{

R(A), 2maxi∈N

|aii...i|}

.

Corollary 3.5. If A is an m-order and n-dimensional nonsingular M-tensor

with σ(A) 6= ∅, then

ρ(A) ≤ min

{

R(A), 2maxi∈N

aii...i

}

.



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Remark 3.6. In fact, the bound of Theorem 3.3 is not better than the bound in

Lemma 3.2 for diagonally dominant H-tensors. However, by Lemma 2.11 we know

that H-tensors are not necessary diagonally dominant. Thus, the bound given in

Theorem 3.3 is sharper than the one given in Lemma 3.2 for non-diagonally dominant

H-tensors. The following example illustrates the same.

Example 3.7. Let A = (aijk) be an 3-order 2-dimension tensor with the form,

a111 = 1.1, a112 = −1, a121 = −1, a122 = 1,

a211 = −1, a221 = 1, a212 = 1, a222 = 1.1.

It is easy to check that A is quasi-strictly diagonally dominant and then A is an

nonsingular H-tensor. By Lemma 3.1, we have,

ρ(A) ≤ √nR(A) = 5.7974.

By Lemma 3.2, we obtain the upper bound,

ρ(A) ≤ R(A) = 4.1.

Now from Theorem 3.3, we have the following bound:

ρ(A) ≤ 2maxi

|aiii| = 2.2.

Obviously, the bound given in Theorem 3.3 is sharper than those given in Lemma

3.2 and Lemma 3.1.

4. Bounds for the minimum eigenvalue of M-tensors. In this section, we

consider the minimum H-eigenvalue of M-tensors. We adopt the following notation

throughout this section. We define a nonnegative matrix M(A), where

(M(A))ij =

{

ri(A), i = j,

aij...j , i 6= j.rji (A) =

∑

δii2...im=0

δji2 ...im=0

ri(A) − |aij...j |,

and

△ij(A) = [aii...i − ajj...j + rji (A)]2 − 4aij...jrj(A),

with

ri(M(A)) =∑

j 6=i

M(A)ij , r̃i(A) = ri(A)− ri(M(A)),



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and

△̃ij(A) = [aii...i − ajj...j + r̃i(A)]2 + 4ri(M(A))rj(A).

Lemma 4.1. Let A be a weakly irreducible M-tensor and ti =∑

k 6=i,j

|aik...k|, i ∈ N .

(1) If 0 ≤ ti ≤ 2[aii...i−ajj...j+rji (A)−2rj(A)], i, j ∈ N , then △ij(A) ≥ △̃ij(A).

(2) If ti ≥ 2[aii...i − ajj...j + rji (A)− 2rj(A)], i, j ∈ N , then △ij(A) ≤ △̃ij(A).

Proof. For convenience, denote a = aii...i − ajj...j + rji (A), notice that r̃i(A) =

rji (A)− ti. Thus

△ij(A)− △̃ij(A) = a2 − 4aij...jrj(A)− (a− ti)2 − 4[(t− aij...j)rj(A)]

= −t2i + 2[a− 2rj(A)]ti.

The equation −t2i +2[a− 2rj(A)]ti = 0 has two roots ti1 = 0 and ti2 = 2[a− 2rj(A)].

Therefore, if 0 ≤ ti ≤ 2[a−2rj(A)]. Thus △ij(A) ≥ △̃ij(A), and if ti ≥ 2[a−2rj(A)],

then △ij(A) ≤ △̃ij(A).

In [7], He and Huang gave the following bounds for the minimum H-eigenvalue

of irreducible M-tensors.

Lemma 4.2. ([7]) Let A be an irreducible M-tensor. Then τ(A) ≤ mini∈N

{aii...i}.

Lemma 4.3. ([7]) Let A = (ai1i2...im) be an irreducible M-tensor. Then

(4.1)

mini,j∈Nj 6=i

12

{

aii...i + ajj...j − rji (A)−△

12

ij(A)}

≤ τ(A) ≤

maxi,j∈N

j 6=i

12

{


12

ij(A)}

.

For the weakly irreducible M-tensor, we have a result similar to that of Lemma

4.2 in the following.

Lemma 4.4. Let A be a weakly irreducible M-tensor. Then τ(A) ≤ mini∈N

{aii...i}.

Proof. The proof is similar to that of Theorem 2.1 in [7], and omit it.

Based on the above lemma, we derive the bounds for the minimum H-eigenvalue

of weakly irreducible M-tensors as follows.



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Theorem 4.5. Let A = (ai1i2...im) be a weakly irreducible M-tensor. Then

(4.2)

mini,j∈Nj 6=i

12

{

aii...i + ajj...j − r̃i(A)− △̃12

ij(A)}

≤ τ(A) ≤

maxi,j∈Nj 6=i

12

{

aii...i + ajj...j − r̃i(A)− △̃12

ij(A)}

.

Proof. Let x > 0 be an eigenvector of A corresponding to τ(A). i.e.,

(4.3) Axm−1 = τ(A)x[m−1].

Suppose that

xt ≥ xs ≥ maxi∈N

{xi : i 6= t, i 6= s}.

From (4.3), we have

[τ(A) − att...t]xm−1t =

∑

δii2...im=0

(i2i3...im) 6=(jj...j)

ati2...imxi2xi3 . . . xim +∑

j 6=t

atj...jxm−1j .

Taking modulus in the above equation and using the triangle inequality gives,

|τ(A) − att...t|xm−1t ≤ ∑

δii2...im=0

(i2i3...im) 6=(jj...j)

|ati2...im |xi2xi3 . . . xim +∑

j 6=t

|atj...j |xm−1j

≤ ∑

δii2...im=0

(i2i3...im) 6=(jj...j)

|ati2...im |xm−1t +

∑

j 6=t

|atj...j |xm−1s

= r̃t(A)xm−1t + rt(M)xm−1

s .

Note that τ(A) ≤ att...t, and

[att...t − τ(A)]xm−1t ≤ r̃t(A)xm−1

t + rt(M)xm−1s .

Equivalently

(4.4) [att...t − τ(A) − r̃t(A)]xm−1t ≤ rt(M)xm−1

s .

From (4.3), we also obtain

(4.5) [ass...s − τ(A)]xm−1s ≤ rs(A)xm−1

t .

Multiplying inequalities (4.4) with (4.5), we have

[att...t − τ(A) − r̃t(A)][ass...s − τ(A)]xm−1t xm−1

s ≤ rt(M)rs(A)xm−1s xm−1

t .



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Note that xm−1t xm−1

s < 0, and

(4.6) [att...t − τ(A) − r̃t(A)][ass...s − τ(A)] ≤ rt(M)rs(A).

This is

τ(A)2 − [att...t + ass...s − r̃t(A)]τ(A) − rt(M)rs(A) + [att...t − r̃t(A)]ass...s ≤ 0.

Note that

[att...t + ass...s − r̃t(A)]2 − 4[ass...s − r̃t(A)]att...t = [att...t − ass...s + r̃t(A)]2.

This gives the following bound for τ(A),

τ(A) ≥ 12{att...t + ass...s − r̃t(A)−△

12ts(A)}

≥ mini,j∈Nj 6=i

12{aii...i + ajj...j − r̃i(A)− △̃

12ij(A)}.

On the other hand, let

xl ≤ xu ≤ mini∈N

{xi : i 6= t, i 6= s}.

From (4.3), we have

(4.7) (auu...u − τ(A))xm−1u = −

∑

δui2...im=0

aui2...imxi2xi3 . . . xim ≥ ru(A)xm−1l .

and

(all...l − τ(A))xm−1l = − ∑

δli2...im=0

(i2i3...im) 6=(jj...j)

ali2...imxi2xi3 . . . xim − ∑

j 6=l

alj...jxm−1j

≥ r̃l(A)xm−1l + rl(M)xm−1

u .

Then

(4.8) [all...l − τ(A) − r̃l(A)]xm−1l ≥ rl(M)xm−1

u .

Multiplying inequalities (4.7) with (4.8), we have

(4.9) [auu...u − τ(A)][all...l − τ(A) − r̃l(A)] ≥ rl(M)ru(A).

Inequality (4.9) is equivalent to

τ(A)2 − [all...l + auu...u − r̃l(A)]τ(A) − rl(M)ru(A) + [all...l − r̃l(A)]auu...u ≥ 0.

This gives the following bound for τ(A),

τ(A) ≤ 12

{

all...l + auu...u − r̃l(A)−△12

lu(A)}

≤ maxi,j∈N

j 6=i

12

{

aii...i + ajj...j − r̃j(A)− △̃12ij(A)

}

.



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This completes the proof.

In what follows, we will show the bounds in Theorem 4.5 are tighter and sharper

than those of Lemma 4.3.

Theorem 4.6. Under the conditions of Lemma 4.1. If

0 ≤ ti ≤ 2[

aii...i − ajj...j + rji (A) − 2rj(A)

]

, i, j ∈ N,

then

mini,j∈Nj 6=i

12

{

aii...i + ajj...j − rji (A) −△

12

ij(A)}

≤ mini,j∈N

j 6=i

12

{

aii...i + ajj...j − r̃i(A) − △̃12ij(A)

}

.

Proof. From the Lemma 4.1, if 0 ≤ ti ≤ 2[aii...i − ajj...j + rji (A) − 2rj(A), then

△ij(A) ≥ △̃ij(A). Note that r̃i(A) = rji (A)− ti, and then


12

ij(A) ≤ aii...i + ajj...j − r̃i(A)− △̃12

ij(A), i, j ∈ N,

which implies that

mini,j∈N

j 6=i

12

{


12ij(A)

}

≤ mini,j∈Nj 6=i

12

{

aii...i + ajj...j − r̃i(A) − △̃12

ij(A)}

.

Remark 4.7. From Theorem 4.6, we can see that the lower bound of τ(A) in

Theorem 4.5 is sharper than those of Lemma 4.3, if

0 ≤ ti ≤ 2[

aii...i − ajj...j + rji (A) − 2rj(A)

]

, i, j ∈ N.

Theorem 4.8. Under the conditions of Lemma 4.1. If

ti ≥ 2[

aii...i − ajj...j + rji (A)− 2rj(A)

]

+ 1, i, j ∈ N,

then

maxi,j∈Nj 6=i

12

{

aii...i + ajj...j − r̃i(A)− △̃12

ij(A)}

≤ maxi,j∈N

j 6=i

12

{


12ij(A)

}

.



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Proof. From the proof of Lemma 4.1, we know that

(4.10) △ij(A) − △̃ij(A) + ti = −t2i + 2[(a− 2rj(A)) + 1]ti.

Because equation (4.10) has two roots ti1 = 0 and ti2 = 2[a− 2rj(A)] + 1. Therefore,

if ti2 ≥ 2(a− 2rj(A)) + 1, then

△ij(A) ≤ △̃ij(A)− ti.

Note that r̃i(A) = rji (A)− ti, we have

r̃i(A) + △̃12ij(A) ≥ r

ji (A) +△

12ij(A).

Hence

maxi,j∈Nj 6=i

12

{

aii...i + ajj...j − r̃i(A)− △̃12

ij(A)}

≤ maxi,j∈N

j 6=i

12

{


12

ij(A)}

.

Remark 4.9. From Theorem 4.8, we can see that the upper bound of τ(A) in

Theorem 4.5 is sharper than those in Lemma 4.3, if ti ≥ 2[aii...i − ajj...j + rji (A) −

2rj(A)] + 1, i, j ∈ N.

Remark 4.10. Since ti ≥ 0, if 0 ≤ ti ≤ 2[aii...i − ajj...j + rji (A) − 2rj(A)] for

some i, and ti ≥ 2[aii...i−ajj...j+rji (A)−2rj(A)]+1 for some other i, we can see that

the upper and lower bounds of τ(A) in Theorem 4.5 are tighter than those of Lemma

4.3. The following example shows this.

Example 4.11. Let A = (aijk) be an 4-order 3-dimension tensor with the form,

a111 = a222 = 5, a333 = a444 = 4, aijj = −1, i 6= j,

a121 = −0.5, a212 = −1, aijk = 0, otherwise.

By Lemma 4.3, we have the bound

0.2614 ≤ τ(A) ≤ 1.5635.

We have our new bounds from Theorem 4.5.

0.7251 ≤ τ(A) ≤ 1.2769.



ELA


5. Conclusion. In this paper, the Z-spectral radius for nonsingular H-tensor

and the minimum H-eigenvalue of nonsingular (strong) M-tensor are studied. Fur-

thermore, we prove that the results of this paper are sharper than those of [1, 8] and

[7].

Acknowledgement. The authors would like to thank the Editor, Prof. Manju-

natha Prasad Karantha and the referee for their valuable and detailed comments on

our manuscript. We also thank Prof. L. Qi for reminding us of the recent paper [17]

during our revision.

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ELA


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