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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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# Bound for the largest singular value of nonnegative rectangular tensors

Jun He
/ Yan-Min Liu
/ Hua Ke
/ Jun-Kang Tian
/ Xiang Li
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: $B=(bi1i2...im),bi1i2...im∈ℝ,1≤i1,i2,...im≤n$

ℬ is called nonnegative if bi1i2im ≥ 0. To an n-vector x, real or complex, we define the n-vector: $Bxm−1=(∑nbii2...imxi2...xim)1≤i≤n.$

and $x[m−1]=(xim−1)1≤i≤n.$

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 ${x}^{\left[\alpha \right]}={\left[{x}_{1}^{\alpha },\dots ,{x}_{n}^{\alpha }\right]}^{T}$.

Let Axp−1 yq be a vector in ℝm such that $(Axp−1yq)i=∑i2,…,ip=1m∑j1,…,jq=1naii2…ipj1…jq…xipxj1…yjq,$

where i = 1,..., m. Let Axp yq−1 be a vector in ℝn such that $(Axp−1yq)i=∑i2,…,ip=1m∑j1,…,jq=1nai1i2…ip j​j2…jq…xipyj2…yjq,$

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 ${Axp−1yq=λx[M−1]Axpyq−1=λy[M−1],$

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 $\left\{x\in {ℝ}^{n}|{x}_{i}\ge \left(resp.>\right)0,i=1,...,n\right\}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}{ℝ}_{+}^{n}\left(\text{resp}\text{.}\text{\hspace{0.17em}}{ℝ}_{++}^{n}\right)$. The Perron-Frobenius Theorem for nonnegative irreducible rectangular tensors was given in [1, 4].

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 $\left(x,y\right)\in \text{\hspace{0.17em}}{ℝ}_{+}^{m}×\text{\hspace{0.17em}}{ℝ}_{+}^{n}$ 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].

Suppose that A ≥ 0 is a (p, q)-th order m × n dimensional rectangular tensor. Then $λ0(A)≤max⁡1≤i≤m, 1≤j≤n{ri(A),cj(A)},$

where $ri(A)=∑i2,…,ip=1m∑j1,…,jq=1nai1i2…ipj1…jq,cj(A)=∑i1,…,ip=1m∑j2,…,jq=1nai1…ipj2…jq.$

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 $rij(A)=ri(A)−aij cij(A)=ci(A)−aj…jij…j, i≠j.$

In particular, we give our main result as follows.

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 $Θ1(A)= max⁡i,j∈N,j≠i12{ai…i i…i+ aj…j j…j+rij(A)+Δ112(A)},Θ2(A)= max⁡i,j∈N,j≠i12{ai…i i…i+ aj…j j…j+cij(A)+Δ212(A)},Θ3(A)= max⁡i,j∈N,j≠i12{ai…i i…i+ aj…j j…j+rij(A)+Δ312(A)},Θ4(A)= max⁡i,j∈N,j≠i12{ai…i i…i+ aj…j j…j+cij(A)+Δ412(A)},$

and $Δ1(A)=(ai…i i…i−aj…j j…j+rij(A))2+4aij…jj…jrj(A),Δ2(A)=(ai…i i…i−aj…j j…j+cij(A))2+4aj…jij…jcj(A),Δ3(A)=(ai…i i…i−aj…j j…j+rij(A))2+4aij…jj…jcj(A),Δ4(A)=(ai…i i…i−aj…j j…j+cij(A))2+4aj…jij…jrj(A).$

## 2 Proof of Theorem 1.3

Suppose that A ≥ 0 is a (p, q)-th order n × n dimensional rectangular tensor. Then $λ0(A)≥ai…i i…i.$

Proof. Let ei be the n dimension real vector, whose i th entry is 1 and others 0. Hence, from Theorem 5 in [4], $λ0(A)≥ (Aeipeiq−1)i(ei)iM−1=ai…ii…i$

for 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 ${Axp−1yq=λ0(A)x[M−1]Axpyq−1=λ0(A)y[M−1].$(1)

Denote $xt=max⁡{xi, 1≤i≤n}, ys=max⁡{yi, 1≤i≤n}.$

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 $λ0(A)xt−at…t t…txtp−1ytq=∑ti2…ipj1…jq≠t…tt…tati2…ipj1…jqxi2…xipyj1…yjq.$

Then, we can get $λ0(A)−at…t t…t(ytxt)qxtM−1≤∑ti2…ipj1…jq≠t…tt…tti2…ipj1…jq≠t s…ss…sati2…ipj1…jqxtM−1+ats…ss…sxsM−1.$

By using $\frac{{y}_{t}}{{x}_{t}}\le 1$ and Lemma 2.1, we have $(λ0(A)−at…t t…t)xtM−1≤(λ0(A)−at…t t…t(ytxt)q)xtM−1≤rts(A)xtM−1+ats…ss…sxsM−1.$

Then, we get $(λ0(A)−at…t t…t−rts(A))xtM−1≤ats…ss…sxsM−1.$(2)

Similarly, the s-th equations in (1) imply $(λ0(A)−as…s s…s)xsM−1≤rs(A)xtM−1.$(3)

Multiplying inequalities (2) with (3), we have $(λ0(A)−at…t t…t−rts(A))(λ0(A)−as…s s…s)≤ats…ss…srs(A).$

Then, we can get $λ0(A)≤maxi,j∈N,j≠i12ai…ii…i+aj…jj…j+rij(A)+Δ112..(A)=Θ1(A).$

where $Δ1(A)=(ai…i i…i−aj…j j…j+rij(A))2+4aij…jj…jrj(A).$

Case II: we suppose ωg = yt, ωh = ys, then, the t-th equations in (1) imply $λ0(A)yt−at…t t…txtpytq−1=∑i1…iptj2…jq≠t…tt…tai1…ipj2…jqxi1…xipyj2…yjq.$

Similar to the proof of Case I, we get $(λ0(A)−at…t t…tcts(A))ytM−1≤as…sts…sysM−1.$(4)

$(λ0(A)−as…s s…s)ysM−1≤cs(A)ytM−1.$(5)

Multiplying inequalities (4) with (5), we have $(λ0(A)−at…t t…t−cts(A))(λ0(A)−as…s s…s)≤ats…s s…scs(A).$

Then, we can get $λ0(A)≤maxi,j∈N,j≠i12ai…ii…i+aj…jj…j+cij(A)+Δ212..(A)=Θ2(A),$

where $Δ2(A)=(ai…i i…i−aj…j j…j−cji(A))2+4aj…jij…jcj(A).$

Case III: we suppose ωg = xt, ωh = ys, then, the t-th equations in (1) imply $(λ0(A)−at…t t…t−rts(A))xtM−1≤ats…s s…sysM−1.$(6)

the s-th equations in (1) imply $(λ0(A)−as…s s…s)ysM−1≤cs(A)xtM−1.$(7)

Multiplying inequalities (6) with (7), we have $(λ0(A)−at…t t…t−rts(A))(λ0(A)−as…s s…s)≤ats…s s…scs(A).$

then, we can get $λ0(A)≤maxi,j∈N,j≠i12ai…ii…i+aj…jj…j+rij(A)+Δ312..(A)=Θ3(A),$

where $Δ3(A)=(ai…i i…i−aj…j j…j+rij(A))2+4aij…jj…jcj(A),$

Case IV: we suppose ωg = yt, ωh = ys, similar to the proof of Case III, we can get $λ0(A)≤max⁡i,j∈N,j≠i12{ai…i i…i+ aj…j j…j+cij(A)+Δ412(A)}=Θ4(A).$

where $Δ4(A)=(ai…i i…i−aj…j j…j+cij(A))2+4aj…jij…jrj(A).$

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: $a1111=12, a2222=3, and aijkl=13 elsewhere$

By 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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Accepted: 2016-08-24

Published Online: 2016-10-19

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 761–766, ISSN (Online) 2391-5455,

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