# M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

Authors: Gang Wang; Linxuan Sun; Lixia Liu
Journal: Complexity
 (2020)

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## Abstract

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.


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## Body

## 1. Introduction

Letℝ be the set of all real numbers, ℝn be the set of all dimension n real vectors, and n=1,2,…,n a fourth-order real tensor, denoted by A=aijkl∈ℝn1×n2×n3×n4, consists of n1×n2×n3×n4 components:(1)aijkl∈ℝ,i∈n1,j∈n2,k∈n3,l∈n4.Specifically,A=aijkl∈ℝm×n×m×n is called partially symmetric, if its components are invariant under the following permutation of subscripts:(2)aijkl=akjil=ailkj=aklij,i,k∈m,j,l∈n.In fact, the tensor of elastic moduli for elastic materials exactly is partially symmetric [1], and the components of such tensor are regarded as the coefficients of the following biquadratic homogeneous polynomial optimization problem:(3)minfAx,y=Axyxy=∑i,k∈m∑j,l∈naijklxiyjxkyls.t.xTx=1,yTy=1,x∈ℝm,y∈ℝn.This optimization problem induced by tensorA, finds applications in nonlinear elastic materials analysis [2], the ordinary ellipticity and strong ellipticity [1, 3], and stability study of nonlinear autonomous systems [4, 5]. As we know, the strong ellipticity condition is essential in theory of elasticity, which guarantees the existence of solutions of basic boundary-value problems of elastostatics and ensures an elastic material to satisfy some mechanical properties. Qi et al. [6] pointed out that the strong ellipticity condition holds if and only if the optimal value of problem (3) is positive. To establish the criteria in identifying the strong ellipticity in elastic mechanics, Qi et al. [6, 7] introduced the following definition.Definition 1.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric real tensor. For λ∈ℝ,x∈ℝm,y∈ℝn, if(4)A⋅yxy=λx,Axyx⋅=λy,xTx=1,yTy=1,where A⋅yxyi=∑k∈m,j,l∈naijklyjxkyl and Axyx⋅l=∑i,k∈m,j∈naijklxiyjxk, then the scalar λ is called an M-eigenvalue of the tensor A and x and y are called left and right M-eigenvectors of A, respectively, which are associated with the M-eigenvalue λ. Denote σMA as the set of all M-eigenvalues of A. Then, the M-spectral radius of A is denoted by(5)ρMA=maxλ:λ∈σMA.
Note thatfAx,y is positive definite if and only if M-eigenvalues of A are positive [7]. Hence, effective algorithms for finding M-eigenvalue and the corresponding eigenvector have been implemented [8–16]. Due to the complexity of the tensor eigenvalue problem [17, 18], it is difficult to compute all M-eigenvalues. Thus, some researchers turned to investigating the inclusion sets of M-eigenvalue [19–21]. For example, Che et al. [19] proposed a Gershgorin-type M-inclusion set as follows.Lemma 1 (Theorem 2.1 of [19]).
SupposeA=aijkl∈ℝm×n×m×n be a partially symmetric real tensor. Then,(6)σMA⊆ΓA≔∪i∈mΓiA,where(7)ΓiA=z∈ℂ:z≤RiA,RiA=∑k∈m;j,l∈naijkl.
Unfortunately, the mentioned inclusion sets always include zero and cannot identify the positive definiteness offAx,y.Example 1.
Consider the following partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(8)aijkl=a1111=20,a1122=a1221=1,a1212=8;a2222=10,a2112=a2211=1,a2121=7;aijkl=0,otherwise.
From Lemma1, it holds that(9)ΓA=∪i∈2ΓiA=λ∈ℂ:λ≤30.
By computation, we can obtain that the correspondingM-eigenvalues are 7,20. Hence, A is positive definite. However, we could not use ΓA to identify the positive definiteness of A. To overcome the drawback above, we present new M-eigenvalue inclusion intervals with n parameters, which can be used to identify the positive definiteness of fourth-order partially symmetric tensors.
This paper is organized as follows. In Section2, we establish two M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors. In Section 3, we propose some checkable sufficient conditions of the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors. Numerical examples are proposed to verify the efficiency of the obtained results.


## 2.M-Eigenvalue Inclusion Intervals for Fourth-Order Partially Symmetric Tensors

In this section, inspired byH-eigenvalue inclusion theorems [22–26] and Z-eigenvalue inclusion intervals [27–31], we establish two M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors. We begin our work by introducing M-identity tensor.Definition 2.
We callℐM∈ℝm×n×m×n an M-identity tensor if its entries are(10)ℐMijkl=1,ifi=k,j=l,0,otherwise,where i,k∈m,j,l∈n.
Obviously,ℐM∈ℝm×n×m×n is a partially symmetric tensor and has the following property:(11)ℐM⋅yxy=x,ℐMxyx⋅=y,with xTx=1,yTy=1 for all x∈ℝm,y∈ℝn.Theorem 1.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For any α=α1,⋯,αmT∈ℝm, then(12)σMA⊆GA,α=∪i∈mGiA,α,where(13)GiA,α=z∈ℝ:z−αi≤RiA,αi,RiA,αi=∑k∈mj,l∈naijkl−αiℐMijkl.Further, σMA⊆∩α∈ℝmGA,α.Proof.
Letλ,x,y be an M-eigenpair of A and ℐM be an M-identity tensor. From the definition of M-identity tensor and (11), it holds(14)Ayxy=λx=λℐMyxy.
Settingxt=maxi∈mxi, by xTx=1, one has 0<xt≤1. From the tth equality of (14), we obtain(15)∑k∈mj,l∈nλℐMtjklyjxkyl=∑k∈mj,l∈natjklyjxkyl.
Hence, for any real numberαt, it follows that(16)λ−αtxt=∑k∈mj,l∈nλ−αtℐMtjklyjxkyl=∑k∈mj,l∈natjkl−αtℐMtjklyjxkyl.
Taking modulus in the above equation, one has(17)λ−αtxt=∑k∈mj,l∈natjkl−αtℐMtjklyjxkyl≤∑k∈mj,l∈natjkl−αtℐMtjklyjxkyl≤∑k∈mj,l∈natjkl−αtℐMtjklxt.
Therefore,(18)λ−αt≤∑k∈mj,l∈natjkl−αtℐMtjkl,which implies that λ∈GtA,α⊆GA,α. From the arbitrariness of α, the conclusion follows.Theorem 2.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For any α=α1,…,αmT∈ℝm, then(19)σMA⊆KA,α=∪i∈m∩v≠i,v∈mKi,vA,α,where(20)RivA,αi=∑j,l∈naijvl−αiℐMijvlKi,vA,α=λ∈ℝ:λ−αi−RiA,αi−RivA,αiλ−αv≤RivA,αiRvA,αv.Further, σMA⊆∩α∈ℝmKA,α.Proof.
Letλ,x,y be an M-eigenpair of A and ℐM be an M-identity tensor. Set xt=maxi∈mxi. Since xTx=1, it holds that 0<xt≤1. From the tth equation of A⋅yxy=λx in (4), for any p∈m,p≠t and any real number αt, we have(21)λ−αtxt=∑k∈mj,l∈natjklyjxkyl−∑k∈mj,l∈nαtℐMtjklyjxkyl=∑k≠p,k∈mj,l∈natjkl−αtℐMtjklyjxkyl+∑j,l∈natjpl−αtℐMtjplyjxpyl.
Taking modulus in the above equation and using the triangle inequality give(22)λ−αtxt≤∑k≠p,k∈mj,l∈natjkl−αtℐMtjklyjxtyl+∑j,l∈natjpl−αtℐMtjplyjxpyl≤RtA,αt−RtpA,αtxt+RtpA,αtxp.
Thus,(23)λ−αt−RtA,αt−RtpA,αt≤RtpA,αtxpxt.
Ifxp=0, then(24)λ−αt≤RtA,αt−RtpA,αt,which shows λ∈Kt,pA,α. Otherwise, for xp>0, we obtain(25)λ−αpxp≤∑k∈mj,l∈napjkl−αpℐMpjklyjxkyl≤RpA,αpxt.
That is,(26)λ−αp≤RpA,αpxtxp.
Multiplying (23) with (26) yields(27)λ−αt−RtA,αt−RtpA,αtλ−αp≤RtpA,αtRpA,αp,which implies that λ∈Kt,pA,α. From the arbitrariness of p, it follows that λ∈∩v≠t,v∈mKt,vA,α. Further, λ∈∪i∈m∩v≠i,v∈mKi,vA,α. It follows from the arbitrariness of α that σMA⊆∩α∈ℝmKA,α.Remark 1.
(i)
It is clear that Theorems1 and 2 reduce to Theorems 2.1 and 2.2 of [19] if one takes α=0, respectively. Consequently, the upper bounds of ρMA in Theorems 1 and 2 are smaller than those in Theorems 2.1 and 2.2 of [19].(ii)
By using the equationAxyx=λy, we can establish some conclusions similar to Theorems 1 and 2.Corollary 1.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For any α=α1,⋯,αmT∈ℝm, then(28)σMA⊆KA,α⊆GA,α.Proof.
For anyλ∈KA,α, without loss of generality, there exists t∈m such that λ∈Kt,kA,α, for all t≠k. Thus,(29)λ−αt−RtA,αt−RtkA,αtλ−αk≤RtkA,αtRkA,αk.
We now break up the argument into two cases.Case 1.
IfRtkA,αtRkA,αk=0, then(30)λ−αt−RtA,αt−RtkA,αt≤0orλ=αk.
Hence,(31)λ−αt≤RtA,αt−RtkA,αt≤RtA,αtorλ=αk.Therefore, λ∈GtA,α∪GkA,α⊆GA,α.Case 2.
IfRtkA,αtRkA,αk>0, then(32)λ−αt−RtA,αt−RtkA,αtRtkA,αtλ−αkRkA,αk≤1,which implies that(33)λ−αt−RtA,αt−RtkA,αtRtkA,αt≤1,or(34)λ−αkRkA,αk≤1.Thus, λ∈GtA,α∪GkA,α⊆GA,α.
In summary,σMA⊆KA,α⊆GA,α and the desired result follows.
The following example exhibits the superiority of the results given in Theorems1 and 2.Example 2.
Consider the tensorA=aijkl∈ℝ2×2×2×2 in Example 1.
Setα=14,8.5T. For this tensor, the bounds via different estimations given in the literature are shown in Table 1.
It is easy to see that the results given in Theorems1 and 2 are sharper than some existing results.
We observe that the suitable parameterα has a great influence on the numerical effects (Table 2).Table 1
References
Inclusion interval
Theorem 2.1 of [19]
Γ
A
=
−
30,30
Theorem 2.2 of [19]
ℒ
A
=
−
29.2971
,
29.2971
Theorem 2.4 of [19]
ℳ
A
=
−
28.3523
,
28.3523
Theorem 2.6 of [19]
N
A
=
−
29.2971
,
29.2971
Theorem1 of [20]
Γ
A
=
−
29,29
Theorem2 of [20]
Θ
A
=
−
28.4081
,
28.4081
Theorem1
G
A
,
14,8.5
=
0,28
Theorem2
K
A
,
14,8.5
=
0.7154
,
26.5539Table 2
α
20,10
T
14,8.5
T
7,6
T
Theorem1
5,34
0,28
−
9,23
Theorem2
5.1185
,
32.4455
0.7154
,
26.5539
−
6.1521
,
20
For the medium-sized tensors, we show the validity of the estimations given by our theorems.Example 3.
All testing partially symmetric tensorsA=aijkl∈ℝm×n×m×n are generated with m=n as aijij=4i+j and other elements are generated randomly in −0.5,0.5.
The choice of parameterα is derived as follows: αi=∑j∈naijij/n. For the tensors with different dimensions, the values presented in the table are the average values of 10 examples (Table 3).Table 3
References
n
=
20
n
=
30
n
=
40
Inclusion interval
Inclusion interval
Inclusion interval
Theorem 2.1 of [1]
−
3392.3
,
3392.3
−
9420.1
,
9420.1
−
19808.2
,
19808.2
Theorem 2.2 of [1]
−
3313.1
,
3313.1
−
9200.3
,
9200.3
−
19345.4
,
19345.4
Theorem 2.4 of [1]
−
3205.7
,
3205.7
−
8901.9
,
8901.9
−
18718.8
,
18718.8
Theorem 2.6 of [1]
−
3391.9
,
3391.9
−
9420.1
,
9420.1
−
19806.2
,
19806.2
Theorem1 of [14]
−
3279.2
,
3279.2
−
9106.1
,
9106.1
−
19147.5
,
19147.5
Theorem2 of [14]
−
3211.4
,
3211.4
−
8917.7
,
8917.7
−
18752.5
,
18752.5
Theorem1
−
1704.2
,
1814.4
−
5649.1
,
5793.2
−
13226.6
,
13419.2
Theorem2
−
1192.9
,
1717.2
−
3954.4
,
5482.8
−
9258.1
,
12700.9


## 3. Applications

In this section, based on the inclusion intervalsGA,α and KA,α in Theorems 1 and 2, we propose some sufficient conditions for the positive definiteness and make bound estimations on the M-spectral radius of nonnegative fourth-order partially symmetric tensors.


### 3.1. Positive Definiteness of Fourth-Order Partially Symmetric Tensors

Theorem 3.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For i∈m, if there exists positive real vector α=α1,…,αmT such that(35)αi>RiA,αi,then A is positive definite and fAx,y defined in (3) is positive definite.Proof.
Suppose on the contrary thatλ≤0. From Theorem 1, there exists i0∈m such that λ∈Gi0A,α, i.e.,(36)λ−αi0≤Ri0A,αi0.
On the other hand, byαi0>0 and λ≤0, we have(37)αi0≤λ−αi0≤Ri0A,αi0,which contradicts (35). Hence, λ>0. Since A is partially symmetric and all M-eigenvalues are positive, A is positive definite and fAx,y defined in (3) is positive definite.Theorem 4.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For i∈m, if there exist a positive real vector α=α1,…,αmT and k≠v such that(38)αi−RiA,αi−RivA,αiαv>RivA,αiRvA,αv,then A is positive definite and fAx,y defined in (3) is positive definite.Proof.
Suppose on the contrary thatλ≤0. From Theorem 2, there exist i0∈m such that λ∈Ki0,pA,α, i.e.,(39)λ−αi0−Ri0A,αi0−Ri0pA,αi0λ−αp≤Ri0pA,αi0RpA,αp,∀p≠i0.
Further, it follows fromαi>0 and λ≤0 that(40)αi0−Ri0A,αi0−Ri0pA,αi0αp≤λ−αi0−Ri0A,αi0−Ri0pA,αi0λ−αp≤Ri0pA,αi0RpA,αp,which contradicts (38). Hence, λ>0. Since A is partially symmetric and all M-eigenvalues are positive, A is positive definite and fAx,y defined in (3) is positive definite.
The following example show Theorems3 and 4 can judge the positive definiteness of fourth-order partially symmetric tensors.Example 4.
Consider the partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(41)aijkl=a1111=10,a1122=a1221=−0.5,a1212=4;a2222=3,a2112=a2211=−0.5,a2121=5;aijkl=0,otherwise.
From the calculation method provided in Theorem 7 of [7], we obtain that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are λ¯,x¯,y¯=3,0,1,0,1. Hence, A is positive definite.
Setα=8,4T. According to Theorem 3, we have(42)α1=8>R1A,α1=7,α2=4>R2A,α2=3.Hence, A satisfies all conditions of Theorem 3, which implies that A is positive definite.
According to Theorem4, it holds(43)α1−R1A,α1−R12A,α1α2=8>R12A,α1R2A,α2=4,α2−R2A,α2−R21A,α2α1=16>R21A,α2R1A,α1=7.Hence, A satisfies all conditions of Theorem 4, which implies that A is positive definite.
The following example reveals that Theorem4 can judge the positive definiteness of partially symmetric tensors more accurately than Theorem 3.Example 5.
Consider the partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(44)aijkl=a1111=10,a1212=8,a1122=a1221=0.5;a1222=−1.5,a1112=a1211=−0.1,a1121=1.5;a2222=3,a2121=5,a2112=a2211=0.5;a2212=−1.5,a2221=a2122=−0.1,a2111=1.5.
By Theorem 7 of [7], we observe that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are λ¯,x¯,y¯=2.5774,0.2724,0.9622,−0.0452,0.9990. Hence, A is positive definite. For any positive real number α2, we have(45)R2A,α2=3−α2+5−α2+4.2>α2,which implies that the condition of Theorem 3 is not satisfied. Thus, Theorem 3 is not suitable to check the positive definiteness of A. However, taking α=10,5T, from Theorem 4, we have(46)α1−R1A,α1−R12A,α1α2=39>R12A,α1R2A,α2=24.8,α2−R2A,α2−R21A,α2α1=28>R21A,α2R1A,α1=24.8,which can show the positive definiteness of A.


### 3.2. Bound Estimations on theM-Spectral Radius

Based on Theorems1 and 2, we present sharp bound estimations on M-spectral radius of fourth-order partially symmetric nonnegative tensors, which improves the corresponding results in [19, 20]. For M-eigenvalues and associated left and right M-eigenvectors of fourth-order partially symmetric tensors, Qi and Luo [7] provided several related results.Lemma 2 (Theorem 1 of [7]).
M-eigenvalues always exist. If x and y are left and right M-eigenvectors of A, associated with an M-eigenvalue λ, then λ=Axyxy.Lemma 3.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric nonnegative tensor. The M-spectral radius of A is exactly its largest M-eigenvalue. Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the M-spectral radius.Proof.
Assume thatλ∗ is the largest M-eigenvalue of A. It is clear that λ∗≤ρMA. In the following, we show λ∗≥ρMA. It follows from Lemma 2 that there exist left and right M-eigenvectors x∗,y∗ of λ∗ such that(47)λ∗=maxfAx,y=Axyxy:xTx=1andyTy=1.
Obviously,λ∗≥0. Next, we show λ∗,x∗,y∗ is a M-eigenpair of A. Since A is nonnegative, x∗Tx∗=x∗Tx∗=1 and y∗Ty∗=y∗Ty∗=1, we obtain(48)λ∗=fAx∗,y∗=∑i,j,k,l∈Naijklxi∗yj∗xk∗yl∗≤∑i,j,k,l∈Naijklxi∗yj∗xk∗yl∗=fAx∗,y∗≤λ∗,which implies λ∗=fAx∗,y∗. Consequently, λ∗,x∗,y∗ is a nonnegative M-eigenpair of A. Meanwhile, let x¯,y¯ be a corresponding M-eigenvector of λ¯ with λ¯=ρMA. Since A is nonnegative and λ∗ is largest value of fAx,y, we have(49)ρMA=fAx¯,y¯=∑i,k∈m,j,l∈naijklx¯iy¯jx¯ky¯l≤∑i,k∈m,j,l∈naijklx¯iy¯jx¯ky¯l≤λ∗,which shows(50)ρMA≤λ∗.Thus, ρMA=λ∗.Lemma 4.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor. If A is nonnegative, then(51)ρMA≥maxmaxi∈m,j∈naijij,∑i∈mRiAmn.Proof.
Without loss of generality, we assume thatρMA=λ∗ is the largest M-eigenvalue of A by Lemma 3. It follows from Lemma 2 that(52)ρMA=maxx,yfAx,y=Axyxy:xTx=1andyTy=1.
Letai∗j∗i∗j∗=maxi∈m,j∈naijij. Setting a feasible solution of (52)(53)x∗,y∗=xi∗=1,yj∗=1,ifi=i∗,j=j∗;xi=0,yj=0,otherwise,we have(54)ρMA=maxx,yfAx,y≥fAx∗,y∗=ai∗j∗i∗j∗=maxi∈m,j∈naijij,which implies ρMA≥maxi∈m,j∈naijij.
Meanwhile, taking a feasible solutionx¯,y¯=1/m,…,1/m,1/n,…,1/n, from (52), we obtain(55)ρMA≥fAx¯,y¯=∑i,k∈m∑j,l∈naijklmn=∑i∈mRiAmn.
From (52) and (55), the conclusion follows.Theorem 5.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric nonnegative tensor and ℐM be an M-identity tensor. For real vector α=α1,…,αmT∈ℝm with αi≤,maxi∈m,j∈naijij, then(56)ρMA≤maxi∈mαi+RiA,αi.Proof.
Without loss of generality, we assume thatρA=λ∗ is the largest M-eigenvalue of A by Lemma 3. It follows from Theorem 1 that there exists t∈N such that(57)ρMA−αt≤RtA,αt.
SinceA is nonnegative and αi≤maxi∈m,j∈naijij, from Lemma 4 and (57), we deduce(58)αt≤maxi∈m,j∈naijij≤ρMA,ρMA≤αt+RtA,αt.
Furthermore,(59)ρMA≤maxi∈mαi+RiA,αi.Theorem 6.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric nonnegative tensor and ℐM be an M-identity tensor. For real vector α=α1,…,αmT∈ℝm with αi≤maxi∈m,j∈naijij, then(60)ρMA≤maxi∈mminv≠i,v∈m12αi+αv+RiA,αi−RivA,αi+Δi,v1/2A,where Δi,vA=αi−αv+RiA,αi−RivA,αi2+4RivA,αiRvA,αv.Proof.
Without loss of generality, we assume thatρMA=λ∗ is the largest M-eigenvalue of A by Lemma 3. It follows from Theorem 2 that there exists t∈N such that(61)λ−αi−RiA,αi−RivA,αiλ−αv≤RivA,αiRvA,αv,∀v≠t,
Noting thatA is nonnegative and αi≤maxi∈m,j∈naijij, from Lemma 4 and (61), we have(62)αi≤maxi∈m,j∈naijij≤ρMA,(63)ρMA−αt−RiA,αi−RivA,αiρMA−αv≤RivA,αiRvA,αv,∀v≠t.
Solving (63) for ρMA gives(64)ρMA≤12αt+αv+RtA,αt−RtvA,αt+Δt,v1/2A,∀v≠t,where Δt,vA=αt−αv+RtA,αt−RtvA,αt2+4RtvA,αtRvA,αv. Since v∈m is chosen arbitrarily, it holds(65)ρMA≤minv≠t,v∈m12αt+αv+RtA,αt−RtvA,αt+Δt,v1/2A.
Furthermore,(66)ρMA≤maxi∈mminv≠i,v∈m12αi+αv+RiA,αi−RivA,αi+Δi,v1/2A.
In the following, we use Example 1 of [20] to show the superiority of our results.Example 6.
Consider the partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(67)aijkl=a1111=2,a1211=a1112=3,a1121=6,a1212=2;a1222=10,a2111=6,a2212=10,a2222=5.aijkl=0,otherwise.
In fact,σMA=−7.6841,13.8616,−4.2541,6.6751. From Lemma 4, we compute 11.75≤ρMA. Set α=2,5T. For this tensor, the bounds via different estimations given in the literature are shown in Table 4.
It is easy to see that the result given in Theorem6 is sharper than some existing results.Table 4
References
Interval
Theorem 3.1 of [19]
ρ
M
A
≤
24
Theorem 3.3 of [19]
ρ
M
A
≤
24
Theorem 3.5 of [19]
ρ
M
A
≤
24
Theorem1 of [20]
ρ
M
A
≤
26
Theorem2 of [20]
ρ
M
A
≤
24
Lemma4 and Theorem 5
11.75
≤
ρ
M
A
≤
24
Lemma4 and Theorem 6
11.75
≤
ρ
M
A
≤
23.6941


## 3.1. Positive Definiteness of Fourth-Order Partially Symmetric Tensors

Theorem 3.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For i∈m, if there exists positive real vector α=α1,…,αmT such that(35)αi>RiA,αi,then A is positive definite and fAx,y defined in (3) is positive definite.Proof.
Suppose on the contrary thatλ≤0. From Theorem 1, there exists i0∈m such that λ∈Gi0A,α, i.e.,(36)λ−αi0≤Ri0A,αi0.
On the other hand, byαi0>0 and λ≤0, we have(37)αi0≤λ−αi0≤Ri0A,αi0,which contradicts (35). Hence, λ>0. Since A is partially symmetric and all M-eigenvalues are positive, A is positive definite and fAx,y defined in (3) is positive definite.Theorem 4.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor and ℐM be an M-identity tensor. For i∈m, if there exist a positive real vector α=α1,…,αmT and k≠v such that(38)αi−RiA,αi−RivA,αiαv>RivA,αiRvA,αv,then A is positive definite and fAx,y defined in (3) is positive definite.Proof.
Suppose on the contrary thatλ≤0. From Theorem 2, there exist i0∈m such that λ∈Ki0,pA,α, i.e.,(39)λ−αi0−Ri0A,αi0−Ri0pA,αi0λ−αp≤Ri0pA,αi0RpA,αp,∀p≠i0.
Further, it follows fromαi>0 and λ≤0 that(40)αi0−Ri0A,αi0−Ri0pA,αi0αp≤λ−αi0−Ri0A,αi0−Ri0pA,αi0λ−αp≤Ri0pA,αi0RpA,αp,which contradicts (38). Hence, λ>0. Since A is partially symmetric and all M-eigenvalues are positive, A is positive definite and fAx,y defined in (3) is positive definite.
The following example show Theorems3 and 4 can judge the positive definiteness of fourth-order partially symmetric tensors.Example 4.
Consider the partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(41)aijkl=a1111=10,a1122=a1221=−0.5,a1212=4;a2222=3,a2112=a2211=−0.5,a2121=5;aijkl=0,otherwise.
From the calculation method provided in Theorem 7 of [7], we obtain that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are λ¯,x¯,y¯=3,0,1,0,1. Hence, A is positive definite.
Setα=8,4T. According to Theorem 3, we have(42)α1=8>R1A,α1=7,α2=4>R2A,α2=3.Hence, A satisfies all conditions of Theorem 3, which implies that A is positive definite.
According to Theorem4, it holds(43)α1−R1A,α1−R12A,α1α2=8>R12A,α1R2A,α2=4,α2−R2A,α2−R21A,α2α1=16>R21A,α2R1A,α1=7.Hence, A satisfies all conditions of Theorem 4, which implies that A is positive definite.
The following example reveals that Theorem4 can judge the positive definiteness of partially symmetric tensors more accurately than Theorem 3.Example 5.
Consider the partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(44)aijkl=a1111=10,a1212=8,a1122=a1221=0.5;a1222=−1.5,a1112=a1211=−0.1,a1121=1.5;a2222=3,a2121=5,a2112=a2211=0.5;a2212=−1.5,a2221=a2122=−0.1,a2111=1.5.
By Theorem 7 of [7], we observe that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are λ¯,x¯,y¯=2.5774,0.2724,0.9622,−0.0452,0.9990. Hence, A is positive definite. For any positive real number α2, we have(45)R2A,α2=3−α2+5−α2+4.2>α2,which implies that the condition of Theorem 3 is not satisfied. Thus, Theorem 3 is not suitable to check the positive definiteness of A. However, taking α=10,5T, from Theorem 4, we have(46)α1−R1A,α1−R12A,α1α2=39>R12A,α1R2A,α2=24.8,α2−R2A,α2−R21A,α2α1=28>R21A,α2R1A,α1=24.8,which can show the positive definiteness of A.


## 3.2. Bound Estimations on theM-Spectral Radius

Based on Theorems1 and 2, we present sharp bound estimations on M-spectral radius of fourth-order partially symmetric nonnegative tensors, which improves the corresponding results in [19, 20]. For M-eigenvalues and associated left and right M-eigenvectors of fourth-order partially symmetric tensors, Qi and Luo [7] provided several related results.Lemma 2 (Theorem 1 of [7]).
M-eigenvalues always exist. If x and y are left and right M-eigenvectors of A, associated with an M-eigenvalue λ, then λ=Axyxy.Lemma 3.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric nonnegative tensor. The M-spectral radius of A is exactly its largest M-eigenvalue. Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the M-spectral radius.Proof.
Assume thatλ∗ is the largest M-eigenvalue of A. It is clear that λ∗≤ρMA. In the following, we show λ∗≥ρMA. It follows from Lemma 2 that there exist left and right M-eigenvectors x∗,y∗ of λ∗ such that(47)λ∗=maxfAx,y=Axyxy:xTx=1andyTy=1.
Obviously,λ∗≥0. Next, we show λ∗,x∗,y∗ is a M-eigenpair of A. Since A is nonnegative, x∗Tx∗=x∗Tx∗=1 and y∗Ty∗=y∗Ty∗=1, we obtain(48)λ∗=fAx∗,y∗=∑i,j,k,l∈Naijklxi∗yj∗xk∗yl∗≤∑i,j,k,l∈Naijklxi∗yj∗xk∗yl∗=fAx∗,y∗≤λ∗,which implies λ∗=fAx∗,y∗. Consequently, λ∗,x∗,y∗ is a nonnegative M-eigenpair of A. Meanwhile, let x¯,y¯ be a corresponding M-eigenvector of λ¯ with λ¯=ρMA. Since A is nonnegative and λ∗ is largest value of fAx,y, we have(49)ρMA=fAx¯,y¯=∑i,k∈m,j,l∈naijklx¯iy¯jx¯ky¯l≤∑i,k∈m,j,l∈naijklx¯iy¯jx¯ky¯l≤λ∗,which shows(50)ρMA≤λ∗.Thus, ρMA=λ∗.Lemma 4.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric tensor. If A is nonnegative, then(51)ρMA≥maxmaxi∈m,j∈naijij,∑i∈mRiAmn.Proof.
Without loss of generality, we assume thatρMA=λ∗ is the largest M-eigenvalue of A by Lemma 3. It follows from Lemma 2 that(52)ρMA=maxx,yfAx,y=Axyxy:xTx=1andyTy=1.
Letai∗j∗i∗j∗=maxi∈m,j∈naijij. Setting a feasible solution of (52)(53)x∗,y∗=xi∗=1,yj∗=1,ifi=i∗,j=j∗;xi=0,yj=0,otherwise,we have(54)ρMA=maxx,yfAx,y≥fAx∗,y∗=ai∗j∗i∗j∗=maxi∈m,j∈naijij,which implies ρMA≥maxi∈m,j∈naijij.
Meanwhile, taking a feasible solutionx¯,y¯=1/m,…,1/m,1/n,…,1/n, from (52), we obtain(55)ρMA≥fAx¯,y¯=∑i,k∈m∑j,l∈naijklmn=∑i∈mRiAmn.
From (52) and (55), the conclusion follows.Theorem 5.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric nonnegative tensor and ℐM be an M-identity tensor. For real vector α=α1,…,αmT∈ℝm with αi≤,maxi∈m,j∈naijij, then(56)ρMA≤maxi∈mαi+RiA,αi.Proof.
Without loss of generality, we assume thatρA=λ∗ is the largest M-eigenvalue of A by Lemma 3. It follows from Theorem 1 that there exists t∈N such that(57)ρMA−αt≤RtA,αt.
SinceA is nonnegative and αi≤maxi∈m,j∈naijij, from Lemma 4 and (57), we deduce(58)αt≤maxi∈m,j∈naijij≤ρMA,ρMA≤αt+RtA,αt.
Furthermore,(59)ρMA≤maxi∈mαi+RiA,αi.Theorem 6.
LetA=aijkl∈ℝm×n×m×n be a partially symmetric nonnegative tensor and ℐM be an M-identity tensor. For real vector α=α1,…,αmT∈ℝm with αi≤maxi∈m,j∈naijij, then(60)ρMA≤maxi∈mminv≠i,v∈m12αi+αv+RiA,αi−RivA,αi+Δi,v1/2A,where Δi,vA=αi−αv+RiA,αi−RivA,αi2+4RivA,αiRvA,αv.Proof.
Without loss of generality, we assume thatρMA=λ∗ is the largest M-eigenvalue of A by Lemma 3. It follows from Theorem 2 that there exists t∈N such that(61)λ−αi−RiA,αi−RivA,αiλ−αv≤RivA,αiRvA,αv,∀v≠t,
Noting thatA is nonnegative and αi≤maxi∈m,j∈naijij, from Lemma 4 and (61), we have(62)αi≤maxi∈m,j∈naijij≤ρMA,(63)ρMA−αt−RiA,αi−RivA,αiρMA−αv≤RivA,αiRvA,αv,∀v≠t.
Solving (63) for ρMA gives(64)ρMA≤12αt+αv+RtA,αt−RtvA,αt+Δt,v1/2A,∀v≠t,where Δt,vA=αt−αv+RtA,αt−RtvA,αt2+4RtvA,αtRvA,αv. Since v∈m is chosen arbitrarily, it holds(65)ρMA≤minv≠t,v∈m12αt+αv+RtA,αt−RtvA,αt+Δt,v1/2A.
Furthermore,(66)ρMA≤maxi∈mminv≠i,v∈m12αi+αv+RiA,αi−RivA,αi+Δi,v1/2A.
In the following, we use Example 1 of [20] to show the superiority of our results.Example 6.
Consider the partially symmetric tensorA=aijkl∈ℝ2×2×2×2 defined by(67)aijkl=a1111=2,a1211=a1112=3,a1121=6,a1212=2;a1222=10,a2111=6,a2212=10,a2222=5.aijkl=0,otherwise.
In fact,σMA=−7.6841,13.8616,−4.2541,6.6751. From Lemma 4, we compute 11.75≤ρMA. Set α=2,5T. For this tensor, the bounds via different estimations given in the literature are shown in Table 4.
It is easy to see that the result given in Theorem6 is sharper than some existing results.Table 4
References
Interval
Theorem 3.1 of [19]
ρ
M
A
≤
24
Theorem 3.3 of [19]
ρ
M
A
≤
24
Theorem 3.5 of [19]
ρ
M
A
≤
24
Theorem1 of [20]
ρ
M
A
≤
26
Theorem2 of [20]
ρ
M
A
≤
24
Lemma4 and Theorem 5
11.75
≤
ρ
M
A
≤
24
Lemma4 and Theorem 6
11.75
≤
ρ
M
A
≤
23.6941


## 4. Conclusions

In this paper, we introducedM-identity tensor to establish sharp M-eigenvalue inclusion intervals. Further, we proposed some sufficient conditions for the positive definiteness of four-order partially symmetric tensors. The given experiments show the validity of the obtained results. It is worth noting that suitable parameter α has a great influence on the numerical effects and positive definiteness. Therefore, how to select the suitable parameter α is our further research.



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*Source: 2474278-2020-01-08.xml*