Bound for the largest singular value of nonnegative rectangular tensors

Jun He; Yan-Min Liu; Hua Ke; Jun-Kang Tian; Xiang Li

doi:10.1515/math-2016-0068

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Bound for the largest singular value of nonnegative rectangular tensors

Jun He , Yan-Min Liu , Hua Ke , Jun-Kang Tian and Xiang Li

Published/Copyright: October 19, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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 n^m entries in ℝ, which is defined as follows:

B=(bi1i2...im),bi1i2...im∈ℝ,1≤i1,i2,...im≤n

ℬ is called nonnegative if b_{i₁i₂…i_m} ≥ 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 ℬx^m−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 ℬx^m−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 Ax^p−1y^q be a vector in ℝ^m such that

(Axp−1yq)i=∑i2,…,ip=1m∑j1,…,jq=1naii2…ipj1…jq…xipxj1…yjq,

where i = 1,..., m. Let Ax^py^q−1 be a vector in ℝⁿ such that

(Axp−1yq)i=∑i2,…,ip=1m∑j1,…,jq=1nai1i2…ip jj2…jq…xipyj2…yjq,

where j = 1,..., n. Throughout this paper, we denote M = p + q. If there are a number λ ∈ ℂ, vectors x ∈ ℂ^m\{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 ∈ ℝ^m, 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 {x∈ℝn|xi≥(resp.>)0,i=1,...,n} by ℝ+n(resp. ℝ++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)∈ ℝ+m× ℝ+ncorresponding to it.

Recently, there are so many results about the properties of square tensor [5–9], especially the upper bounds for the Z-spectral radius and H-spectral radius of a nonnegative square tensor [10–18]. 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)≤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 s_i (A) = max{c_i, (A), r_i (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.

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)= 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

Lemma 2.1

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

λ0(A)≥ai…i i…i.

Proof. Let e_i 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, …, x_n)^T and y = {y₁, y₂, …, y_n)^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 ω_i = max{x_i, y_i}. Let g be an index such that ω_g = max{ω_i, i ∈ N}, N = {1,..., n}. Obviously, ω_g ≠ 0. Let h be an index such that ω_h, = max{ω_i, i ∈ N, i ≠ g}.

Case I: we suppose ω_g = x_t, ω_h = x_s, 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 ytxt≤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 ω_g = y_t, ω_h = y_s, 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 ω_g = x_t, ω_h = y_s, 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 ω_g = y_t, ω_h = y_s, 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 s_i(A) = max{c_i(A), r_i(A)} = r_j(A), then c_j(A) ≤ r_i(A), from Theorem 3.5 in [13], we know that

Θ3(A)≤Θ1(A)≤ri(A).

Similarly, if s_j (A) = max{c_i, (A), r_i, (A)} = c_i (A), thenr_i (A) ≤ c_i (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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Received: 2016-5-7

Accepted: 2016-8-24

Published Online: 2016-10-19

Published in Print: 2016-1-1

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