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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

Issues

An S-type upper bound for the largest singular value of nonnegative rectangular tensors

Jianxing Zhao
  • Corresponding author
  • College of Science, Guizhou Minzu University, Guiyang 550025, China
  • Email:
/ Caili Sang
  • College of Science, Guizhou Minzu University, Guiyang 550025, China
  • Email:
Published Online: 2016-11-12 | DOI: https://doi.org/10.1515/math-2016-0085

Abstract

An S-type upper bound for the largest singular value of a nonnegative rectangular tensor is given by breaking N = {1, 2, … n} into disjoint subsets S and its complement. It is shown that the new upper bound is smaller than that provided by Yang and Yang (2011). Numerical examples are given to verify the theoretical results.

Keywords: Nonnegative tensor; Rectangular tensor; Singular value

MSC 2010: 15A18; 15A42; 15A69

1 Introduction

Let ℝ(ℂ) be the real (complex) field, p, q, m, n be positive integers, m, n ≥ 2, N = {1, 2,…, n}, and R+n be the cone {x = (x1, x2, …, xn)T 𲈈 ℝn : xi ≥ 0, iN}. A real (p, q)-th order m × n dimensional rectangular tensor, or simply a real rectangular tensor A is defined as follows: A=(ai1ipj1jq);

where ai1ipj1jq ∈ ℝ for ik = 1,…, m,; k = 1,… p, and jk = 1,… n, k = 1,… q. When p = q = 1, A is simply a real m × n rectangular matrix. This justifies the word “rectangular”.

For any vector x and any real number α, denote x[α]=(x1α,x2α,,xnα)T. Let Axp1yq be a vector in ℝm such that (Axp1yq)i=i2,,ip=1mj1,,jq=1naii2ipj1jqxi2xipyj1yjq,

where i = 1,…, m. Similarly, let Axpyq1 be a vector in ℝn such that (Axpyq1)j=i1,,ip=1mj2,,jq=1nai1ipjj2jqxi1xipyj2yjq,

where j = 1,…, n. Let l = p + q. If there are a number λ ∈ ℂ, vectors x ∈ λm}\{0}, and y ∈ ℂn\{0} such that Axp1yq=λx[l1],Axpyq1=λy[l1],

then λ is called the singular value of A, and (x, y) is the left and right eigenvectors pair of A, associated with λ, respectively. 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, respectively. If a singular value is not an ℍ-singular value, we call it an ℕ-singular value of A. If p = q = 1, then this is just the usual definition of singular values for a rectangular matrix [1]. We call λ0=max{|χ|:λisasingularvalueofA}

is the largest singular value [2].

Note here that the notion of singular values for tensors was first proposed by Lim in [3]. When l is even, the definition in [1] is the same as in [3]. When l is odd, the definition in [1] is slightly different from that in [3], but parallel to the definition of eigenvalues of square matrices [4]; see [1] for details. For the sake of simplicity, the definition of singular values in this paper is the definition in [1].

We recall the weak Perron-Frobenius theorem for nonnegative rectangular tensors, which was given in [2].

Theorem 1.1: Let A be a(p, q)-th order m × n dimensional nonnegative tensor Then λ0 is the largest singular value with nonnegative left and right eigenvectors pair (x,y)R+m{0}×R+n{0} corresponding to it.

The largest singular value of a nonnegative rectangular tensor has a wide range of practical applications in the strong ellipticity condition problem in solid mechanics [5, 6] and the entanglement problem in quantum physics [7, 8]. Recently, there are many results about the properties of square tensors, especially the upper bounds for the ℤ-spectral radius and ℍ-spectral radius of a nonnegative square tensor [913]. However, there are no results about the upper bounds for the largest singular value of a nonnegative rectangular tensor except the following one [2].

Theorem 1.2: Let Abe a(p, q)-th order m × n dimensional nonnegative rectangular tensor Then min1im,1jn{Rj(A),Cj(A)}λ0max1im,1jn{Rj(A),Cj(A)}, where Rj(A)=i2,,ip=1mj1,,jq=1naii2ipj1jq,Cj(A)=i1,,ip=1mj2,,jq=1nai1ipjj2jq.

Throughout this paper, we assume m = n. Our goal in this paper is to give a new upper bound for the largest singular value of a nonnegative rectangular tensor, and prove that the new upper bound is smaller than that in Theorem 1.2.

2 Main results

We begin with some notation. Given a nonempty proper subset S of N, we denote ΔN:={(i2,ip,j1,,jq):i2,ip,j1,,jqN},ΔS:={(i2,ip,j1,,jq):i2,ip,j1,,jqS},ΩN:={(i1,ip,j2,,jq):i1,ip,j2,,jqN},ΩS:={(i1,ip,j2,,jq):i1,ip,j2,,jqS},

and then S¯=NS,ΩS¯=ΩNΩS

This implies that for a nonnegative rectangular tensor A=(ai1ipj1jq), we have that for i, jS, rj(A)=i2..ip.j1,.jqNδii2ipj1jq=0aii2ipj1jq=riΔS(A)+riΔS¯(A),rij(A)=riΔS(A)+riΔS¯(A)aijjjj, cj(A)=i1.,ip,j2,jqNδi1ipj˙j2jq=0ai1ipjj2jq=cjΩS(A)+cjΩS¯(A),cji(A)=cjΩS(A)+cjΩS¯(A)ajijii,

where δi1ipj1jq=1,ifi1==ip=j1==jq,0,otherwise,

and riΔS(A)=(i2,,ip,j1.jq)ΔSδii2ipj1jq=0aii2ipj1jq,riΔS¯(A)=(i2,,ip,j1,,jq)ΔS¯aii2ipj1jq, cjΩS(A)=(i1,ip.j2jq)ΩSδi1ipjjjq=0ai1ipjj2jq,cjΩS¯(A)=(i1,,ip,j2,,jq)ΩS¯ai1ipjj2jq.

By Theorem 1.2, the following lemma is easily obtained.

Lemma 2.1: Let A be a(p, q)-th order n × n dimensional nonnegative rectangular tensor Then λ0aiiii,iN.

Theorem 2.2: Let A be a(p, q)-th order n × n dimensional nonnegative rectangular tensor, S be a nonempty proper subset of N, S¯ be the complement of S in N. Then λ0US(A)=max{U1S(A),U1S¯(A),U2S(A),U2S¯(A)},where U1S(A)=maxiS,jS¯12{aiiii+ajjjj+rjΔS¯(A)+[(aiiijajjjjrjΔS¯(A))2+4max{rj(A),ci(A)}rjΔS(A)]12},U1S¯(A)=maxis¯,jS12{aiiii+ajjjj+rjΔS¯¯(A)+[(aiiijajjjjrjΔS¯¯(A))2+4max{ri(A),ci(A)}rjΔS¯(A)]12},U2S(A)=maxiS,jS¯12{aiiii+ajjjj+cjΩS¯(A)+[(aiiiiajjjjcjΩS¯(A))2+4max{ri(A),ci(A)}cjΩS(A)]12},U2S¯(A)=maxis¯,jS12{aiiii+ajjjj+cjΩS¯¯(A)+[(aiiiiajjjjcjΩS¯¯(A))2+4max{ri(A),ci(A)}cjΩS¯(A)]12}.

Proof: Let λ0 be the largest singular value of A. According to Theorem 1.1, there exist two nonzero nonnegative vectors x = (x1, x2,…, xn)T and y = (y1, y2,…, yn)T such that Axp1yq=λ0x[l1],Axpyq1=λ0y[l1].(1)(2)Let xt=max{xi:iS},xh=max{xi:iS¯};yf=max{yi:iS},yg=max{yi:iS¯};wi=max{xi,yi},iN,ws=max{wi:iS},wS¯=max{wi:iS¯}.Then, at least one of xt and xh is nonzero, and at least of yf and yg is nonzero. Next, we present four cases to prove that.Case I: Suppose that ws=xt,wS¯=xh, then xtyt, xhyh.(i) If xhxt, then xh = max{wj : iN}. By the h-th equality in (1), we have (λ0ahhhh)xhl1λ0xhl1ahhhhxhp1yhq=(i2,,ip,j1,,jq)ΔSahi2ipj1jqxi2xipyj1yjq+(i2,.ip,j1,,jq)ΔS¯δhi2ipj1jq=0ahi2ipj1jqxi2xipyj1yjq(i2,,ip,j1,,jq)ΔSahi2ipj1jqxtl1+(i2.,ip,j1,jq)ΔS¯δhi2ipj1jq=0ahi2ipj1jqxhl1=rhΔS(A)xtl1+rhΔS¯(A)xhl1.Hence, (λ0ahhhhrhΔS¯(A))xhl1rhΔS(A)xtl1.(3)If xt = 0, then λ0ahhhhrhΔS¯(A)0 as xh > 0, and it is obvious that λ0U1S(A) . Otherwise, xt > 0. From the t-th equality in (1), we have (λ0atttt)xtl1λ0xtl1attttxrp1ytq=i2,,ipj˙.,jqNδti2ipj1jq=0ati2ipj1jqxi2xjpyj1...yjqi2.,ip,j..jq1N,δti2ipj1jq=0ati2ipj1jqxhl1=rt(A)xhl1,i.e., (λ0atttt)xtl1rt(A)xhl1.(4)From Lemma 2.1, we have λ0-atttt ≥ 0. Multiplying (3) with (4), we have (λ0atttt)(λ0ahhhhrhΔS¯(A))xtl1xhl1rt(A)rhΔS(A)xtl1xhl1.Note that xtl1xhl1>0. Then (λ0atttt)(λ0ahhhhrhΔS¯(A))rt(A)rhΔS(A).(5)Solving (5) gives λ012{atttt+ahhhh+rhΔS¯(A)+[(attttahhhhrhΔS¯(A))2+4rt(A)rhΔS(A)]12}maxiS,jS¯12{aiiii+ajjjj+rjΔS¯(A)+[(aiiiiajjjjrjΔS¯(A))2+4ri(A)rjΔS(A)]12}U1S(A).(ii) If xtxh, then xt = max{wj : iN}. Similarly to the proof of (i), we can obtain that (λ0ahhhh)(λ0attttrtΔS¯¯(A))rh(A)rtΔS¯(A).This gives λ012{ahhhh+atttt+rtΔS¯¯(A)+[(ahhhhattttrtΔS¯¯(A))2+4rh(A)rtΔS¯(A)]12}maxis¯,jS12{aiiii+ajjjj+rjΔS¯¯(A)+[(aiiiiajjjjrjΔS¯¯(A))2+4ri(A)rjΔS¯(A)]12}U1S¯(A).Case II: Suppose that ws = yf, wS¯=yg, then yfxf, ygxg. If ygyf, then yg = max{wi : i 𢈈 N}. Similarly to the proof of (i) in Case I, we have (λ0affff)(λ0aggggcgΩS¯(A))cf(A)cgΩS(A).This gives λ012{affff+agggg+cgΩS¯(A)+[(affffaggggcgΩS¯(A))2+4cf(A)cgΩS(A)]12}maxiS,jS¯12{aiiii+ajjjj+cjΩS¯(A)+[(aiiiiajjjjcjΩS¯(A))2+4ci(A)cjΩS(A)]12}U2S(A).If yfyg, then yf = max{wi : iN}. Similarly to the proof of (ii) in Case I, we have (λ0agggg)(λ0affffcfΩS¯¯(A))cg(A)cfΩS¯(A).This gives λ012{agggg+affff+cfΩS¯¯(A)+[(agggaffffcfΩS¯¯(A))2+4cg(A)cfΩS¯(A)]12}maxis¯,jS12{aiiii+ajjjj+cjΩS¯¯(A)+[(aiiiiajjjjcjΩS¯¯(A))2+4ci(A)cjΩS¯(A)]12}U2S¯(A).Case III: Suppose that ws = xt, wS¯=yg, then xtyt, ygxg. If ygxt, then yg = max{wi : iN}. Similarly to the proof of (i) in Case I, we have (λ0atttt)(λ0aggggcgΩS¯(A))rt(A)cgΩS(A).This gives λ012{atttt+agggg+cgΩS¯(A)+[(attttaggggcgΩS¯(A))2+4rt(A)cgΩS(A)]12}maxiS,jS¯12{aiiii+ajjjj+cjΩS¯(A)+[(aiiijajjjjcjΩS¯(A))2+4ri(A)cjΩS(A)]12}U2S(A).If xtyg, then xt = max{wi : iN}. Similarly to the proof of (ii) in Case I, we can obtain that (λ0agggg)(λ0attttrtΔS¯¯(A))cg(A)rtΔS¯(A).This gives λ012{agggg+atttt+rtΔS¯¯(A)+[(aggggattttrtΔS¯¯(A))2+4cg(A)rtΔS¯(A)]12}maxis¯,jS12{aiiii+ajjjj+rjΔS¯¯(A)+[(aiiiiajjjjrjΔS¯¯(A))2+4ci(A)rjΔS¯(A)]12}U1S¯(A).Case IV: Suppose that ws=yf,wS¯=xh, then yfxf, xhyh. If xhyf, then xh = max{wj : iN}. Similarly to the proof of (i) in Case I, we have (λ0affff)(λ0ahhhhrhΔS¯(A))cf(A)rhΔS(A).This gives λ012{affff+ahhhh+rhΔS¯(A)+[(affffahhhhrhΔS¯(A))2+4cf(A)rhΔS(A)]12}maxiS,jS¯12{aiiii+ajjjj+rjΔS¯(A)+[(aiiiiajjjjrjΔS¯(A))2+4ci(A)rjΔS(A)]12}U1S(A).If yfxh, then yf = max{wi} : iN}. Similarly to the proof of (ii) in Case I, we can obtain that (λ0ahhhh)(λ0affffcfΩS¯¯(A))rh(A)cfΩS¯(A).This gives λ012{ahhhh+affff+cfΩS¯¯(A)+[(ahhhhaffffcfΩS¯¯(A))2+4rh(A)cfΩS¯(A)]12}maxis¯,jS12{aiiii+ajjjj+cjΩS¯¯(A)+[(aiiiiajjjjcjΩS¯¯(A))2+4ri(A)cjΩS¯(A)]12}U2S¯(A).The conclusion follows from Case I, II, III and IV. □

Next, we compare the upper bound in Theorem 2.2 with that in Theorem 1.2.

Theorem 2.3: Let A be a(p, q)-th order n × n dimensional nonnegative rectangular tensor, S be a nonempty proper subset of N,S¯ be the complement of S in N. Then US(A)maxi,jN{Ri(A),Cj(A)}.

3 Numerical examples

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

Example 3.1: Let A=(aijkl) be a (2, 2)-th order 3 × 3 dimensional nonnegative rectangular tensor with entries defined as follows: A(:,:,1,1)=210000212,A(:,:,2,1)=110200221,A(:,:,3,1)=9101001122,A(:,:,1,2)=021002220,A(:,:,2,2)=111120222,A(:,:,3,2)=100002221,A(:,:,1,3)=101110121,A(:,:,2,3)=001101111,A(:,:,3,3)=110201222.Let S ={1, 2}. Obviously S¯={3}. By Theorem 1.2, we have λ045. By Theorem 2.2, we have λ038.4165. In fact, λ0 = 31.3781. This example shows that the upper bound in Theorem 2.2 is smaller than that in Theorem 1.2.

Example 3.2: Let A=(aijkl) be a(2, 2)-th order 2 × 2 dimensional nonnegative rectangular tensor with entries defined as follows: a1111=a1112=a1222=a2112=a2121=a2221=1,other aijkl = 0. By Theorem 2.2, we have λ03.In fact, λ0 = 3. This example shows that the upper bound in Theorem 2.2 is sharp.

4 Conclusions

In this paper, by breaking N into disjoint subsets S and its complement, an S-type upper bound US(A) for the largest singular value of a nonnegative rectangular tensor A with m = n is obtained, which improves the upper bound in [2]. Then an interesting problem is how to pick S to make US(A) as small as possible. But this is difficult when n is large. In the future, we will research this problem.

Acknowledgement

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), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073) and Natural Science Programs of Education Department of Guizhou Province (Grant No. [2016]066).

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About the article

Received: 2016-08-13

Accepted: 2016-10-11

Published Online: 2016-11-12

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0085. Export Citation

© 2016 Zhao and Sang, 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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