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Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 59.
Published online 2017 March 9. doi:  10.1186/s13660-017-1331-1
PMCID: PMC5344962

An eigenvalue localization set for tensors and its applications

Abstract

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al. (Linear Algebra Appl. 481:36-53, 2015) and Huang et al. (J. Inequal. Appl. 2016:254, 2016). As an application of this set, new bounds for the minimum eigenvalue of -tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al., the advantage of our results is that, without considering the selection of nonempty proper subsets S of N = {1, 2, …, n}, we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of -tensors. Finally, numerical examples are given to verify the theoretical results.

Keywords: -tensors, nonnegative tensors, minimum eigenvalue, localization set

Introduction

For a positive integer n, n ≥ 2, N denotes the set {1, 2, …, n}. (respectively, ) denotes the set of all complex (respectively, real) numbers. We call 𝒜 = (ai1im) a complex (real) tensor of order m dimension n, denoted by [m,n](ℝ[m,n]), if

ai1im ∈ ℂ(ℝ), 

where ij ∈ N for j = 1, 2, …, m. 𝒜 is called reducible if there exists a nonempty proper index subset 𝕁 ⊂ N such that

ai1i2im = 0,  ∀i1 ∈ 𝕁, ∀i2, …, im ∉ 𝕁.

If 𝒜 is not reducible, then we call 𝒜 irreducible [3].

Given a tensor 𝒜 = (ai1im) ∈ ℂ[m,n], if there are λ ∈ ℂ and x = (x1,x2,…,xn)T ∈ ℂ∖{0} such that

𝒜xm−1λx[m−1]

then λ is called an eigenvalue of 𝒜 and x an eigenvector of 𝒜 associated with λ, where 𝒜xm−1 is an n dimension vector whose ith component is

(Axm1)i=i2,,imNaii2imxi2xim

and

x[m1]=(x1m1,x2m1,,xnm1)T.

If λ and x are all real, then λ is called an H-eigenvalue of 𝒜 and x an H-eigenvector of 𝒜 associated with λ; see [4, 5]. Moreover, the spectral radius ρ(𝒜) of 𝒜 is defined as

ρ(𝒜) = max {|λ|:λ ∈ σ(𝒜)}, 

where σ(𝒜) is the spectrum of 𝒜, that is, σ(𝒜) = {λ:λ is an eigenvalue of 𝒜}; see [3, 6].

A real tensor 𝒜 is called an -tensor if there exist a nonnegative tensor and a positive number α > ρ(ℬ) such that 𝒜 = αℐ − ℬ, where is called the unit tensor with its entries

δi1im={1ifi1==im,0otherwise.

Denote by τ(𝒜) the minimal value of the real part of all eigenvalues of an -tensor 𝒜. Then τ(𝒜) > 0 is an eigenvalue of 𝒜 with a nonnegative eigenvector. If 𝒜 is irreducible, then τ(𝒜) is the unique eigenvalue with a positive eigenvector [79].

Recently, many people have focused on locating eigenvalues of tensors and using obtained eigenvalue inclusion theorems to determine the positive definiteness of an even-order real symmetric tensor or to give the lower and upper bounds for the spectral radius of nonnegative tensors and the minimum eigenvalue of -tensors. For details, see [1, 2, 1014].

In 2015, Li et al. [1] proposed the following Brauer-type eigenvalue localization set for tensors.

Theorem 1

[1], Theorem 6

Let 𝒜 = (ai1im) ∈ ℂ[m,n]. Then

σ(A)Δ(A)=i,jN,jiΔij(A),

where

Δij(A)={zC:|(zaii)(zajj)aijjajii||zajj|rij(A)+|aijj|rji(A)},ri(A)=δii2im=0|aii2im|,rij(A)=δii2im=0,δji2im=0|aii2im|=ri(A)|aijj|.

To reduce computations, Huang et al. [2] presented an S-type eigenvalue localization set by breaking N into disjoint subsets S and S, where S is the complement of S in N.

Theorem 2

[2], Theorem 3.1

Let 𝒜 = (ai1im) ∈ ℂ[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then

σ(A)ΔS(A)=(iS,jS¯Δij(A))(iS¯,jSΔij(A)).

Based on Theorem 2, Huang et al. [2] obtained the following lower and upper bounds for the minimum eigenvalue of -tensors.

Theorem 3

[2], Theorem 3.6

Let 𝒜 = (ai1im) ∈ ℝ[m,n] be an -tensor, S be a nonempty proper subset of N, S be the complement of S in N. Then

min{miniSmaxjS¯Lij(A),miniS¯maxjSLij(A)}τ(A)max{maxiSminjS¯Lij(A),maxiS¯minjSLij(A)},

where

Lij(A)=12{aii+ajjrij(A)[(aiiajjrij(A))24aijjrj(A)]12}.

The main aim of this paper is to give a new eigenvalue inclusion set for tensors and prove that this set is tighter than those in Theorems 1 and 2 without considering the selection of S. And then we use this set to obtain new lower and upper bounds for the minimum eigenvalue of -tensors and prove that new bounds are sharper than those in Theorem 3.

Main results

Now, we give a new eigenvalue inclusion set for tensors and establish the comparison between this set with those in Theorems 1 and 2.

Theorem 4

Let 𝒜 = (ai1im) ∈ ℂ[m,n]. Then

σ(A)Δ(A)=iNjN,jiΔij(A).

Proof

For any λ ∈ σ(𝒜), let x = (x1,…,xn)T ∈ ℂn∖{0} be an associated eigenvector, i.e.,

𝒜xm−1λx[m−1].
1

Let |xp| = max {|xi|:i ∈ N}. Then |xp| > 0. For any j ∈ Nj ≠ p, then from (1) we have

λxpm1=δpi2im=0,δji2im=0api2imxi2xim+appxpm1+apjjxjm1

and

λxjm1=δji2im=0,δpi2im=0aji2imxi2xim+ajjxjm1+ajppxpm1,

equivalently,

(λapp)xpm1apjjxjm1=δpi2im=0,δji2im=0api2imxi2xim
2

and

(λajj)xjm1ajppxpm1=δji2im=0,δpi2im=0aji2imxi2xim.
3

Solving xpm1 from (2) and (3), we get

((λapp)(λajj)apjjajpp)xpm1=(λajj)δpi2im=0,δji2im=0api2imxi2xim+apjjδji2im=0,δpi2im=0aji2imxi2xim.

Taking absolute values and using the triangle inequality yields

|(λapp)(λajj)apjjajpp||xp|m1|λajj|rpj(A)|xp|m1+|apjj|rjp(A)|xp|m1.

Furthermore, by |xp| > 0, we have

|(λapp)(λajj)apjjajpp||λajj|rpj(A)+|apjj|rjp(A),

which implies that λΔpj(A). From the arbitrariness of j, we have λjN,jpΔpj(A). Furthermore, we have λiNjN,jiΔij(A). The conclusion follows.

Next, a comparison theorem is given for Theorems 1, 2 and 4.

Theorem 5

Let 𝒜 = (ai1im) ∈ ℂ[m,n], S be a nonempty proper subset of N. Then

Δ(𝒜) ⊆ ΔS(𝒜) ⊆ Δ(𝒜).

Proof

By Theorem 3.2 in [2], ΔS(𝒜) ⊆ Δ(𝒜). Here, only Δ(𝒜) ⊆ ΔS(𝒜) is proved. Let z ∈ Δ(𝒜), then there exists some i0 ∈ N such that zΔi0j(A),jN,ji0. Let S be the complement of S in N. If i0 ∈ S, then taking jS¯, obviously, zi0S,jS¯Δi0j(A)ΔS(A). If i0S¯, then taking j ∈ S, obviously, zi0S¯,jSΔi0j(A)ΔS(A). The conclusion follows.

Remark 1

Theorem 5 shows that the set Δ(𝒜) in Theorem 4 is tighter than those in Theorems 1 and 2, that is, Δ(𝒜) can capture all eigenvalues of 𝒜 more precisely than Δ(𝒜) and ΔS(𝒜).

In the following, we give new lower and upper bounds for the minimum eigenvalue of -tensors.

Theorem 6

Let 𝒜 = (ai1im) ∈ ℝ[m,n] be an irreducible -tensor. Then

miniNmaxjiLij(A)τ(A)maxiNminjiLij(A).

Proof

Let x = (x1,x2,…,xn)T be an associated positive eigenvector of 𝒜 corresponding to τ(𝒜), i.e.,

𝒜xm−1τ(𝒜)x[m−1].
4

(I) Let xq = min {xi:i ∈ N}. For any j ∈ Nj ≠ q, we have by (4) that

τ(A)xqm1=δqi2im=0,δji2im=0aqi2imxi2xim+aqqxqm1+aqjjxjm1

and

τ(A)xjm1=δji2im=0,δqi2im=0aji2imxi2xim+ajjxjm1+ajqqxqm1,

equivalently,

(τ(A)aqq)xqm1aqjjxjm1=δqi2im=0,δji2im=0aqi2imxi2xim
5

and

(τ(A)ajj)xjm1ajqqxqm1=δji2im=0,δqi2im=0aji2imxi2xim.
6

Solving xqm1 by (5) and (6), we get

((τ(A)aqq)(τ(A)ajj)aqjjajqq)xqm1=(τ(A)ajj)δqi2im=0,δji2im=0aqi2imxi2xim+aqjjδji2im=0,δqi2im=0aji2imxi2xim.

From Theorem 2.1 in [9], we have τ(𝒜) ≤ miniNaii and

((aqqτ(A))(ajjτ(A))aqjjajqq)xqm1=(ajjτ(A))δqi2im=0,δji2im=0|aqi2im|xi2xim+|aqjj|δji2im=0,δqi2im=0|aji2im|xi2xim.

Hence,

((aqqτ(A))(ajjτ(A))|aqjj||ajqq|)xqm1(ajjτ(A))δqi2im=0,δji2im=0|aqi2im|xqm1+|aqjj|δji2im=0,δqi2im=0|aji2im|xqm1.

From xq > 0, we have

(aqqτ(A))(ajjτ(A))|aqjj||ajqq|(ajjτ(A))δqi2im=0,δji2im=0|aqi2im|+|aqjj|δji2im=0,δqi2im=0|aji2im|=(ajjτ(A))rqj(A)+|aqjj|rjq(A),

equivalently,

(aqqτ(A))(ajjτ(A))(ajjτ(A))rqj(A)|aqjj|rj(A)0,

that is,

τ(A)2(aqq+ajjrqj(A))τ(A)+aqqajjajjrqj(A)+aqjjrj(A)0.

Solving for τ(𝒜) gives

τ(A)12{aqq+ajjrqj(A)[(aqqajjrqj(A))24aqjjrj(A)]12}=Lqj(A).

For the arbitrariness of j, we have τ(𝒜) ≤ minjqLqj(𝒜). Furthermore, we have

τ(A)maxiNminjiLij(A).

(II) Let xp = max {xi:i ∈ N}. Similar to (I), we have

τ(A)miniNmaxjiLij(A).

The conclusion follows from (I) and (II).

Similar to the proof of Theorem 3.6 in [2], we can extend the results of Theorem 6 to a more general case.

Theorem 7

Let 𝒜 = (ai1im) ∈ ℝ[m,n] be an -tensor. Then

miniNmaxjiLij(A)τ(A)maxiNminjiLij(A).

By Theorems 3, 6 and 7 in [13], the following comparison theorem is obtained easily.

Theorem 8

Let 𝒜 = (ai1im) ∈ ℝ[m,n] be an -tensor, S be a nonempty proper subset of N, S be the complement of S in N. Then

miniNRi(A)minjiLij(A)min{miniSmaxjS¯Lij(A),miniS¯maxjSLij(A)}miniNmaxjiLij(A)maxiNminjiLij(A)max{maxiSminjS¯Lij(A),maxiS¯minjSLij(A)},

where Ri(𝒜) = ∑i2,…,imNaii2im.

Remark 2

Theorem 8 shows that the bounds in Theorem 7 are shaper than those in Theorem 3, Theorem 2.1 of [9] and Theorem 4 of [13] without considering the selection of S, which is also the advantage of our results.

Numerical examples

In this section, two numerical examples are given to verify the theoretical results.

Example 1

Let 𝒜 = (aijk) ∈ ℝ[3,4] be an irreducible -tensor with elements defined as follows:

A(:,:,1)=(62342422131333322),A(:,:,2)=(04331282212242231),A(:,:,3)=(21211112246344422),A(:,:,4)=(42211231233222461).

By Theorem 2.1 in [9], we have

2=miniNRi(A)τ(A)min{maxiNRi(A),miniNaii}=28.

By Theorem 4 in [13], we have

τ(A)minjiLij(A)=2.3521.

By Theorem 3, we have

if S={1},S¯={2,3,4},3.6685τ(A)24.2948;if S={2},S¯={1,3,4},3.6685τ(A)19.7199;if S={3},S¯={1,2,4},2.3569τ(A)27.7850;if S={4},S¯={1,2,3},2.3521τ(A)27.8536;if S={1,2},S¯={3,4},2.3569τ(A)27.7850;if S={1,3},S¯={2,4},3.6685τ(A)23.0477;if S={1,4},S¯={2,3},3.6685τ(A)23.9488.

By Theorem 7, we have

3.6685 ≤ τ(𝒜) ≤ 19.7199.

In fact, τ(𝒜) = 14.4049. Hence, this example verifies Theorem 8 and Remark 2, that is, the bounds in Theorem 7 are sharper than those in Theorem 3, Theorem 2.1 of [9] and Theorem 4 of [13] without considering the selection of S.

Example 2

Let 𝒜 = (aijkl) ∈ ℝ[4,2] be an -tensor with elements defined as follows:

a1111 = 6,  a1222 = −1,  a2111 = −2,  a2222 = 5, 

other aijkl = 0. By Theorem 7, we have

4 ≤ τ(𝒜) ≤ 4.

In fact, τ(𝒜) = 4.

Conclusions

In this paper, we give a new eigenvalue inclusion set for tensors and prove that this set is tighter than those in [1, 2]. As an application, we obtain new lower and upper bounds for the minimum eigenvalue of -tensors and prove that the new bounds are sharper than those in [2, 9, 13]. Compared with the results in [2], the advantage of our results is that, without considering the selection of S, we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of -tensors.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11361074, 11501141), the Foundation of Guizhou Science and Technology Department (Grant No. [2015]2073) and the Natural Science Programs of Education Department of Guizhou Province (Grant No. [2016]066).

Footnotes

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.

Contributor Information

Jianxing Zhao, moc.361@402018xjz.

Caili Sang, moc.621@lcgnas.

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