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Open Mathematics

formerly Central European Journal of Mathematics

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New bounds for the minimum eigenvalue of 𝓜-tensors

Jianxing Zhao
• Corresponding author
• College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
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• Other articles by this author:
/ Caili Sang
Published Online: 2017-03-16 | DOI: https://doi.org/10.1515/math-2017-0018

Abstract

A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.

MSC 2010: 15A18; 15A42; 15A69

1 Introduction

Let ℂ(ℝ) be the set of all complex (real) numbers, n be positive integer, n ≥ 2, and N = {1, 2, ⋯, n}. We call 𝒜 = (ai1i2 im) a complex (real) tensor of order m dimension n, if $ai1i2⋯im∈C(R),$ where ijN for j = 1, ⋯, m. Obviously, a vector is a tensor of order 1 and a matrix is a tensor of order 2. We call 𝒜 nonnegative if 𝒜 is real and each of its entries ai1 im ≥ 0. Let ℝ[m, n] denote the set of real tensors with order m dimension n.

A tensor 𝒜 = (ai1i2im) of order m dimension n is called reducible if there exists a nonempty proper index subset αN such that $ai1i2⋯im=0,∀i1∈α,∀i2,⋯,im∉α.$

If 𝒜 is not reducible, then we call 𝒜 irreducible [1].

For a complex tensor 𝒜 = (ai1i2 im) of order m dimension n, if there are a complex number λ and a nonzero complex vector x = (x1, x2, ⋯, xn)T that are solutions of the following homogeneous polynomial equations $Axm−1=λx[m−1],$ then λ is called an eigenvalue of 𝒜 and x an eigenvector of 𝒜 associated with λ, where 𝒜xm − 1 and x[m − 1] are vectors, whose i th component are $(Axm−1)i=∑i2,⋯,im∈Naii2⋯imXi2⋯Xim,$ and $(x[m−1])i=xim−1,$ respectively. This definition was introduced by Qi in [2], where he assumed that 𝒜 is an order m dimension n supersymmetric tensor and m is even. Independently, in [3], Lim gave such a definition but restricted x to be a real vector and λ to be a real number. In this case, we call λ an H-eigenvalue of 𝒜 and x an H-eigenvector of 𝒜 associated with λ.

Moreover, the spectral radius ρ(𝒜) of the tensor 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 [1, 4].

The class of 𝓜-tensors introduced in [5, 6] is related to nonnegative tensors, which is an generalization of M-matrices [7].

Definition 1.1

([5, 6)]. Let 𝒜 = (ai1i2 im) ∈ ℝ[m, n]. 𝒜 is called

1. a 𝒵-tensor if all of its off-diagonal entries are non-positive;

2. an 𝓜-tensor if 𝒜 is a 𝒵-tensor with the from 𝒜 = cℐ − ℬ such thatis a nonnegative tensor and c > ρ(ℬ), where ρ(ℬ) is the spectral radius of ℬ, andis called the unit tensor with its entries $δi1⋯im=1,i1=⋯=im,0,otherwise.$

Theorem 1.2

[5, 8] Let 𝒜 be an 𝓜-tensor and denote by τ(𝒜) the minimal value of the real part of all eigenvalues of 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.

Eigenvalue problems of tensors have become an important topic of study in numerical multilinear algebra, and received much attention in the literature; see [519]. In [8], He and Huang provided some lower and upper bounds on τ(𝒜) for an irreducible 𝓜-tensor 𝒜.

Theorem 1.3

([8, Theorem 2.1]). Let 𝒜 = (ai1i2 im) ∈ ℝ[m, n] be an irreducible 𝓜-tensor. Then $τ(A)≤mini∈Naii⋯i,andmini∈NRi(A)≤τ(A)≤maxi∈NRi(A),$ where $Ri(A)=∑i2,⋯,im∈Naii2⋯im.$

In order to obtain more sharper bounds of the minimum eigenvalue for an irreducible 𝓜-tensor, Zhao and Sang [9] gave a lower bound which estimates the minimum eigenvalue more precisely than that in Theorem 1.3.

Theorem 1.4

([9, Theorem 4]). Let 𝒜 = (ai1i2 im) ∈ ℝ[m, n] be an irreducible 𝓜-tensor. Then $τ(A)≥mini,j∈Nj≠jLij(A),$where $Lij(A)=12{ai⋯i+aj⋯j−rij(A)−[ai⋯i−aj⋯j−rij(A))2−4aij⋯jrj(A)]12},ri(A)=∑i2,⋯,im∈N,δii2⋯im=0|aii2⋯im|,rij(A)=∑δii2⋯im=0,δji2⋯im=0|aii2⋯im|=ri(A)−|aij⋯j|.$

In this paper, we continue this research, and give a lower bound and an upper bound for τ(𝒜) of an 𝓜-tensor. It is proved that these bounds are better than the corresponding bounds in [8] and [9]. Finally, two numerical examples are given to verify the obtained results.

2 Main results

In this section, we give a new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensors and establish the comparison between the new bounds with those in Theorem 1.3 and Theorem 1.4. For simplification, we denote $Δi={(i2,i3,⋯im):ij=iforsomej∈{2,⋯,m},wherei,i2,⋯,jm∈N},Δ¯i={(i2,i3,⋯im):ij≠iforanyj∈{2,⋯,m},wherei,i2,⋯,jm∈N}.$

Given a tensor 𝒜 = (ai1i2 im) ∈ ℝ[m, n], let $riΔi(A)=∑(i2,⋯,im)∈Δi,δii2⋯im=0|aii2⋯im|,riΔ¯i(A)=∑(i2,⋯,im)∈Δ¯i|aii2⋯im|.$

Obviously, $ri(A)=riΔi(A)+riΔ¯i(A),rij(A)=riΔi(A)+riΔ¯i(A)−|aij⋯j|.$

Theorem 2.1

Let 𝒜 = (ai1 im) ∈ ℝ[m, n] be an irreducible 𝓜-tensor. Then $mini,j∈N,j≠iΩij(A)≤τ(A)≤maxi,j∈N,j≠iΩij(A),$(1) where $Ωij(A)=12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔ¯i(A)rj(A)]12}.$

Proof

1. Because τ(𝒜) is an eigenvalue of 𝒜, from Theorem 2.1 in [10], there are i, jN, ji, such that $(|τ(A)−ai⋯i|−riΔi(A))|τ(A)−aj⋯j|≤riΔ¯i(A)rj(A).$

From Theorem 1.3, we can get $(ai⋯i−τ(A)−riΔi(A))(aj⋯j−τ(A))≤riΔ¯i(A)rj(A),$ equivalently, $τ(A)2−(ai⋯i+aj⋯j−riΔi(A))τ(A)+aj⋯j(ai⋯i−riΔi(A))−riΔ¯i(A)rj(A)≤0.$

Solving for τ(𝒜) gives $τ(A)≥12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i+aj⋯j−riΔi(A))2−4(aj⋯j(ai⋯i−riΔi(A))−riΔ¯i(A)rj(A))]12}=12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔ¯i(A)rj(A)]12}≥mini,j∈Nj≠i12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔ¯i(A)rj(A)]12}.$

2. Next, we prove that the second inequality in (1) holds. Let x = (x1, x2, ⋯, xn)T be an associated positive eigenvector of 𝒜 corresponding to τ(𝒜), i.e., $Axm−1=τ(A)x[m−1].$(2)

Without loss of generality, suppose that $xtn≥xtn−1≥⋯≥xt2≥xt1.$

From (2), we have $∑i2,⋯,im∈Nat1i2⋯imxi2⋯xim=τ(A)xt1m−1,$ equivalently, $(at1⋯t1−τ(A))xt1m−1=∑(i2,⋯,im)∈Δt1δt1i2,⋯,im=0,|at1i2⋯im|xi2⋯xim+∑(i2,⋯,im)∈Δ¯t1|at1i2⋯im|xi2⋯xim.$

Hence, $(at1⋯t1−τ(A))xt1m−1≥∑(i2,⋯,im)∈Δt1,δt1i2,⋯,im=0|at1i2⋯im|xt1m−1+∑(i2,⋯,im)∈Δ¯t1|at1i2⋯im|xt2m−1=rt1Δt1(A)xt1m−1+rt1Δ¯t1(A)xt2m−1,$ i. e., $(at1⋯t1−rt1Δt1(A)−τ(A))xt1m−1≥rt1Δ¯t1(A)xt2m−1≥0.$(3)

Similarly, we have, from (2), $∑i2,⋯,im∈Nat2i2⋯imxi2⋯xim=τ(A)xt2m−1.$

Furthermore, $(at2⋯t2−τ(A))xt2m−1=∑i2,⋯,im∈Nδt2i2⋯im=0|at2i2⋯im|xi2⋯xim≥∑i2,⋯,im∈Nδt2i2⋯im=0|at2i2⋯im|xt1m−1=rt2(A)xt1m−1≥0.$(4)

Multiplying inequality (3) and inequality (4) gives $(at1⋯t1−rt1Δt1(A)−τ(A))(at2⋯t2−τ(A))xt1m−1xt2m−1≥rt1Δ¯t1(A)rt2(A)xt1m−1xt2m−1.$

Note that xt2xt1 > 0, hence $(at1⋯t1−tt1Δt1(A)−τ(A))(at2⋯t2−τ(A))≥rt1Δ¯t1(A)rt2(A),$ that is, $τ(A)2−(at1⋯t1+at2⋯t2−rt1Δt1(A))τ(A)+at2⋯t2(at1⋯t1−rt1Δt1(A))−rt1Δ¯t1(A)rt2(A)≥0.$(5)

From (3), we have $\begin{array}{}\tau \left(\mathcal{A}\right)\le {a}_{{t}_{1}\cdots {t}_{1}}-{r}_{{t}_{1}}^{{\mathrm{\Delta }}_{{t}_{1}}}\left(\mathcal{A}\right).\end{array}$ From (4), we have τ(𝒜) ≤ at2t2. Then $τ(A)≤12{at1⋯t1+at2⋯t2−rt1Δt1(A)}.$

Thus, solving for τ(𝒜) from (5) gives $τ(A)≤12{at1⋯t1+at2⋯t2−rt1Δt1(A)−[(at1⋯t1+at2⋯t2−rt1Δt1(A))2−4(at2⋯t2(at1⋯t1−rt1Δt1(A))−rt1Δ¯t1(A)rt2(A))]12}=12{at1⋯t1+at2⋯t2−rt1Δt1(A)−[(at1⋯t1−at2⋯t2−rt1Δt1(A))2+4rt1Δ¯t1(A)rt2(A))]12}≤maxi,j∈Nj≠i12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A)2+4riΔ¯i(A)rj(A)]12}.$

The conclusion follows from I and II. □

Similarly to the proof of Theorem 3.6 in [11], we can extend the results of Theorem 2.1 to general 𝓜-tensors.

Theorem 2.2

Let 𝒜 = (ai1im) ∈ ℝ[m, n] be an 𝓜-tensor. Then $mini,j∈Nj≠iΩij(A)≤τ(A)≤maxi,j∈Nj≠iΩij(A).$

Next, we compare the bounds in Theorem 2.1 with those in Theorem 1.3 and Theorem 1.4.

Theorem 2.3

Let 𝒜 = (ai1 im) ∈ ℝ[m, n] be an irreducible 𝓜-tensor. Then $mini∈NRi(A)≤mini,j∈Nj≠iLij(A)≤mini,j∈Nj≠iΩij(A)≤maxi,j∈Nj≠iΩij(A)≤maxi∈NRi(A).$(6)

Proof

1. From Theorem 5 in [9], we have $\begin{array}{}\underset{i\in N}{min}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{R}_{i}\left(\mathcal{A}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{\underset{j\ne i}{i,j\in N}}{min}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{ij}\left(\mathcal{A}\right).\end{array}$ Obviously, the first inequality in (6) holds.

2. From Theorem 2.4 in [10], the proof of Theorem 4 in [9] and Theorem 2.1, it is easy to see that $\begin{array}{}\underset{\underset{j\ne i}{i,j\in N}}{min}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{ij}\left(\mathcal{A}\right)\phantom{\rule{thinmathspace}{0ex}}\le \phantom{\rule{thinmathspace}{0ex}}\underset{\underset{j\ne i}{i,j\in N}}{min}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\Omega }}_{ij}\left(\mathcal{A}\right),\end{array}$ that is, the second inequality in (6) holds.

3. Next, we prove that the last inequality in (6) holds.

1. For any i, jN, ji, if Rj(𝒜) ≤ Rj(𝒜), i.e., $ai⋯i−riΔi(A)−riΔ¯i(A)≤aj⋯j−r(A),$ then $riΔ¯i(A)≥ai⋯i−aj⋯j−riΔi(A)+rj(A).$

Hence, $[ai⋯i−aj⋯j−riΔi(A)]2+4riΔ¯i(A)rj(A)≥[ai⋯i−aj⋯j−riΔi(A)]2+4[ai⋯i−aj⋯j−riΔi(A)+rj(A)]rj(A)=[ai⋯i−aj⋯j−riΔi(A)]2+4[ai⋯i−aj⋯j−riΔi(A)]rj(A)+4[rj(A)]2=[ai⋯i−aj⋯j−riΔi(A)+2rj(A)]2.$

When $ai⋯i−aj⋯j−riΔi(A)+2rj(A)>0,$ we have $ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔ¯i(A)rj(A)]12≤ai⋯i+aj⋯j−riΔi(A)−[ai⋯i−aj⋯j−riΔi(A)+2rj(A)]=2aj⋯j−2rj(A)=2Rj(A).$

When $ai⋯i−aj⋯j−riΔi(A)+2rj(A)≤0,$ that is, $ai⋯i−riΔi(A)≤ajj⋯j−2rj(A),$ we have $ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔ¯i(A)rj(A)]12≤ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2]12=ai⋯i+aj⋯j−riΔi(A)−|ai⋯i−aj⋯j−riΔi(A)|=ai⋯i+aj⋯j−riΔi(A)+[ai⋯i−aj⋯j−riΔi(A)]=2ai⋯i−2riΔi(A)≤2aj⋯j−4rj(A)≤2aj⋯j−2rj(A)=2Rj(A).$

Therefore, $Ωij(A)=12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔi¯(A)rj(A)]12}≤Rj(A),$ which implies $maxi,j∈Nj≠iΩij(A)≤maxj∈NRj(A).$

2. For any i, jN, ji, if Rj(𝒜) ≤ Rj(𝒜), i.e., $aj⋯j−rj(A)≤ai⋯i−riΔi(A)−riΔ¯i(A),$ then $rj(A)≥aj⋯j−ai⋯i+riΔi(A)+riΔ¯i(A).$

Similarly, we can obtain $Ωij(A)=12{ai⋯i+aj⋯j−riΔi(A)−[(ai⋯i−aj⋯j−riΔi(A))2+4riΔi¯(A)rj(A)]12}≤Ri(A),$ which implies $maxi,j∈Nj≠iΩij(A)≤maxi∈NRi(A).$

The conclusion follows from I, II and III. □

Remark 2.4

Theorem 2.3 shows that the bounds in Theorem 2.1 are better than those in Theorem 1.3 and Theorem 1.4.

3 Numerical examples

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

Example 3.1

Let 𝒜 = (aijk) ∈ ℝ[3,4] be an irreducible 𝓜-tensor with elements defined as follows: $A(1,:,:)= 63−2−1−3−3−3−4−3−2−3−1−2−3−4−3−2,A(2,:,:)= −2−3−4−4−346−4−1−3−1−4−2−1−3−3−3,A(3,:,:)= −3−3−3−3−2−2−3−2−3−253−2−2−2−1−2,A(4,:,:)= −2−3−2−1−2−2−3−3−4−2−2−3−3−3−254.$

By Theorem 1.3, we have $5≤τ(A)≤24.$

By Theorem 1.4, we have $τ(A)≥5.6402.$

Let $S=\left\{1,2\right\},\overline{S}=\left\{3,4\right\}.$ By Theorem 6 in [9], we have $τ(A)≥6.0200.$

By Theorem 2.1, we have $9.4206≤τ(A)≤21.3521.$

In fact, τ(𝒜) = 15.3013. Hence, this example shows that the bounds in Theorem 2.1 are better than those in Theorem 1.3, Theorem 1.4 and Theorem 6 in [9].

Example 3.2

Let 𝒜 = (aijkl) ∈ ℝ[4,2] be an irreducible 𝓜-tensor with elements defined as follows: $a1111=5,a1222=−1,a2111=−2,a2222=4,$ other aijkl = 0. By Theorem 1.3, we have $2≤τ(A)≤4.$

By Theorem 1.4, we have $τ(A)≥3.$

Let $S=\left\{1,2\right\},\overline{S}=\left\{3,4\right\}.$

By Theorem 6 in [9], we have $τ(A)≥3.$

By Theorem 2.1, we have $3≤τ(A)≤3.$

In fact, τ(𝒜) = 3. Hence, the bounds in Theorem 2.1 are tight and sharper than those in Theorem 1.3.

4 Conclusions

In this paper, we obtain a lower bound and an upper bound for the minimum eigenvalue of an 𝓜-tensors, which improved the known bounds obtained by He and Huang [8], and Zhao and Sang [9].

Acknowledgements

The authors are very indebted to the reviewers for their valuable comments and corrections, which improved the original manuscript of this paper. This work is supported by the National Natural Science Foundation of China (Nos. 11361074, 11501141), the Natural Science Programs of Education Department of Guizhou Province (Grant No.[2016]066), and the Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073).

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Accepted: 2017-01-16

Published Online: 2017-03-16

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 296–303, ISSN (Online) 2391-5455,

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