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Volume 14, Issue 1 (Jan 2016)

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Bound for the largest singular value of nonnegative rectangular tensors

Jun He
  • Corresponding author
  • School of mathematics, Zunyi Normal College, Zunyi, Guizhou, 563002, China
  • Email:
/ Yan-Min Liu
  • School of mathematics, Zunyi Normal College, Zunyi, Guizhou, 563002, China
/ Hua Ke
  • School of mathematics, Zunyi Normal College, Zunyi, Guizhou, 563002, China
/ Jun-Kang Tian
  • School of mathematics, Zunyi Normal College, Zunyi, Guizhou, 563002, China
/ Xiang Li
  • School of mathematics, Zunyi Normal College, Zunyi, Guizhou, 563002, China
Published Online: 2016-10-19 | DOI: https://doi.org/10.1515/math-2016-0068

Abstract

In this paper, we give a new bound for the largest singular value of nonnegative rectangular tensors when m = n, which is tighter than the bound provided by Yang and Yang in “Singular values of nonnegative rectangular tensors”, Front. Math. China, 2011, 6, 363-378.

Keywords: Bound; Rectangular tensor; Singular value

MSC 2010: 15A18; 15A69; 65F15; 65F10

1 Introduction

Let ℝ be the real field. An m-th order n dimensional square tensor ℬ consists of nm entries in ℝ, which is defined as follows: B=(bi1i2...im),bi1i2...im,1i1,i2,...imn

ℬ is called nonnegative if bi1i2im ≥ 0. To an n-vector x, real or complex, we define the n-vector: Bxm1=(nbii2...imxi2...xim)1in.

and x[m1]=(xim1)1in.

If ℬxm−1 = λx[m−1, x and λ are all real, then λ is called an H-eigenvalue of ℬ and x an H-eigenvector of ℬ associated with λ. If ℬxm−1 = λx, x and λ are all real, then λ is called a Z-eigenvalue of ℬ and x a Z-eigenvector of ℬ associated with λ [2, 3].

Assume that p, q, m and n are positive integers, and m, n ≥ 2. In this paper we consider a nonnegative (p, q)-th order m × n dimensional rectangular tensor. For any vector x and any real number α, denote x[α]=[x1α,,xnα]T.

Let Axp−1 yq be a vector in ℝm such that (Axp1yq)i=i2,,ip=1mj1,,jq=1naii2ipj1jqxipxj1yjq,

where i = 1,..., m. Let Axp yq−1 be a vector in ℝn such that (Axp1yq)i=i2,,ip=1mj1,,jq=1nai1i2ipjj2jqxipyj2yjq,

where j = 1,..., n. Throughout this paper, we denote M = p + q. If there are a number λ ∈ ℂ, vectors x ∈ ℂm\{0}, and y ∈ ℂn\{0} such that {Axp1yq=λx[M1]Axpyq1=λy[M1],

then λ is called the singular value of A, and (x, y) is the left and right eigenvectors pair of A, associated with λ. If λ ∈ ℝ, x ∈ ℝm, and y ∈ ℝn, then we say that λ is an H-singular value of A, and (x,y) is the left and right H-eigenvectors pair associated with λ [1,4].

We denote the cone {xn|xi(resp.>)0,i=1,...,n}by+n(resp.++n). The Perron-Frobenius Theorem for nonnegative irreducible rectangular tensors was given in [1, 4].

Theorem 1.1: Let A be a (p, q)-th order m × n dimensional nonnegative tensor. Then λ0(A) is the largest singular value with nonnegative left and right eigenvectors pair (x,y)+m×+n corresponding to it.Recently, there are so many results about the properties of square tensor [59], especially the upper bounds for the Z-spectral radius and H-spectral radius of a nonnegative square tensor [1018]. However, there are no results about the upper bounds for the largest singular value of a nonnegative rectangular tensor except the following one [1].

Theorem 1.2: Suppose that A ≥ 0 is a (p, q)-th order m × n dimensional rectangular tensor. Then λ0(A)max1im,1jn{ri(A),cj(A)},where ri(A)=i2,,ip=1mj1,,jq=1nai1i2ipj1jq,cj(A)=i1,,ip=1mj2,,jq=1nai1ipj2jq.Our goal in this paper is to show a tighter upper bound for the largest singular value of a nonnegative rectangular tensor when m = n. Let si (A) = max{ci, (A), ri (A)}, and rij(A)=ri(A)aijcij(A)=ci(A)ajjijj,ij.In particular, we give our main result as follows.

Theorem 1.3: Suppose that A ≥ 0 is a (p, q)-th order n × n dimensional rectangular tensor. Then λ0(A)max{Θ1(A),Θ2(A),Θ3(A),Θ4(A)},where Θ1(A)=maxi,jN,ji12{aiiii+ajjjj+rij(A)+Δ112(A)},Θ2(A)=maxi,jN,ji12{aiiii+ajjjj+cij(A)+Δ212(A)},Θ3(A)=maxi,jN,ji12{aiiii+ajjjj+rij(A)+Δ312(A)},Θ4(A)=maxi,jN,ji12{aiiii+ajjjj+cij(A)+Δ412(A)},and Δ1(A)=(aiiiiajjjj+rij(A))2+4aijjjjrj(A),Δ2(A)=(aiiiiajjjj+cij(A))2+4ajjijjcj(A),Δ3(A)=(aiiiiajjjj+rij(A))2+4aijjjjcj(A),Δ4(A)=(aiiiiajjjj+cij(A))2+4ajjijjrj(A).

2 Proof of Theorem 1.3

Lemma 2.1: Suppose that A ≥ 0 is a (p, q)-th order n × n dimensional rectangular tensor. Then λ0(A)aiiii.Proof. Let ei be the n dimension real vector, whose i th entry is 1 and others 0. Hence, from Theorem 5 in [4], λ0(A)(Aeipeiq1)i(ei)iM1=aiiiifor any i. Thus, we complete the proof. □Proof of Theorem 1.3. Let λ0(A) be the largest singular value of A. Then there exist two nonnegative vectors x = {x1, x2, …, xn)T and y = {y1, y2, …, yn)T such that {Axp1yq=λ0(A)x[M1]Axpyq1=λ0(A)y[M1].(1)Denote xt=max{xi,1in},ys=max{yi,1in}.and ωi = max{xi, yi}. Let g be an index such that ωg = max{ωi, iN}, N = {1,..., n}. Obviously, ωg ≠ 0. Let h be an index such that ωh, = max{ωi, iN, ig}.Case I: we suppose ωg = xt, ωh = xs, then, the t-th equations in (1) imply λ0(A)xtattttxtp1ytq=ti2ipj1jqttttati2ipj1jqxi2xipyj1yjq.Then, we can get λ0(A)atttt(ytxt)qxtM1ti2ipj1jqttttti2ipj1jqtssssati2ipj1jqxtM1+atssssxsM1.By using ytxt1 and Lemma 2.1, we have (λ0(A)atttt)xtM1(λ0(A)atttt(ytxt)q)xtM1rts(A)xtM1+atssssxsM1.Then, we get (λ0(A)attttrts(A))xtM1atssssxsM1.(2)Similarly, the s-th equations in (1) imply (λ0(A)assss)xsM1rs(A)xtM1.(3)Multiplying inequalities (2) with (3), we have (λ0(A)attttrts(A))(λ0(A)assss)atssssrs(A).Then, we can get λ0(A)maxi,jN,ji12aiiii+ajjjj+rij(A)+Δ112..(A)=Θ1(A).where Δ1(A)=(aiiiiajjjj+rij(A))2+4aijjjjrj(A).Case II: we suppose ωg = yt, ωh = ys, then, the t-th equations in (1) imply λ0(A)ytattttxtpytq1=i1iptj2jqttttai1ipj2jqxi1xipyj2yjq.Similar to the proof of Case I, we get (λ0(A)attttcts(A))ytM1asstssysM1.(4)(λ0(A)assss)ysM1cs(A)ytM1.(5)Multiplying inequalities (4) with (5), we have (λ0(A)attttcts(A))(λ0(A)assss)atsssscs(A).Then, we can get λ0(A)maxi,jN,ji12aiiii+ajjjj+cij(A)+Δ212..(A)=Θ2(A),where Δ2(A)=(aiiiiajjjjcji(A))2+4ajjijjcj(A).Case III: we suppose ωg = xt, ωh = ys, then, the t-th equations in (1) imply (λ0(A)attttrts(A))xtM1atssssysM1.(6)the s-th equations in (1) imply (λ0(A)assss)ysM1cs(A)xtM1.(7)Multiplying inequalities (6) with (7), we have (λ0(A)attttrts(A))(λ0(A)assss)atsssscs(A).then, we can get λ0(A)maxi,jN,ji12aiiii+ajjjj+rij(A)+Δ312..(A)=Θ3(A),where Δ3(A)=(aiiiiajjjj+rij(A))2+4aijjjjcj(A),Case IV: we suppose ωg = yt, ωh = ys, similar to the proof of Case III, we can get λ0(A)maxi,jN,ji12{aiiii+ajjjj+cij(A)+Δ412(A)}=Θ4(A).where Δ4(A)=(aiiiiajjjj+cij(A))2+4ajjijjrj(A).Thus, we complete the proof.Remark 2.2. If si(A) = max{ci(A), ri(A)} = rj(A), then cj(A) ≤ ri(A), from Theorem 3.5 in [13], we know that Θ3(A)Θ1(A)ri(A).Similarly, if sj (A) = max{ci, (A), ri, (A)} = ci (A), then ri (A) ≤ ci (A), from Theorem 3.5 in [13], we know that Θ4(A)Θ2(A)ci(A).that is to say, our new bound in Theorem 1.3 is always better than the bound in Theorem 4 in [1].Example 2.3. Consider the nonnegative (2, 2)-th order 2 × 2 dimensional rectangular tensor with entries defined as follows: a1111=12,a2222=3,andaijkl=13elsewhereBy Theorem 1.2, we have λ0(A)5.3333.By Theorem 1.3, we have λ0(A)5.1667.Hence, the bound in Theorem 1.3 is tight and sharper.

Acknowledgement

This work was supported by the Doctoral Scientific Research Foundation of Zunyi Normal College (No.BS[2015]09); Science and technology Foundation of Guizhou province (Qian ke he Ji Chu [2016]1161). Liu is supported by National Natural Science Foundations of China (Grants nos. 71461027); Science and technology talent training object of Guizhou province outstanding youth (Qian ke he ren zi [2015]06); Guizhou province natural science foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15851 talents elite project funding; Zhunyi innovative talent team (Zunyi KH(2015)38). Tian is supported by the Science and Technology fund Project of GZ (Qian ke he J Zi [2015]2147, Qian jiao he KY[2015]451). Ke is supported by the Science and Technology fund Project of GZ (Qian Ke He J Zi LKZS [2012]08). Li is supported by the Science and Technology fund Project of GZ (Qian Ke He LH[2015]7047).

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About the article

Received: 2016-05-07

Accepted: 2016-08-24

Published Online: 2016-10-19

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0068. Export Citation

© 2016 Jia et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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