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

Jianxing Zhao; Caili Sang

doi:10.1515/math-2016-0085

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

Jianxing Zhao und Caili Sang

Veröffentlicht/Copyright: 12. November 2016

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Aus der Zeitschrift Open Mathematics Band 14 Heft 1

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, …, x_n)^T 𲈈 ℝⁿ : x_i ≥ 0, i ∈ N}. A real (p, q)-th order m × n dimensional rectangular tensor, or simply a real rectangular tensor A is defined as follows:

A=(ai1⋯ipj1⋯jq);

where a_{i₁…i_pj₁…j_q} ∈ ℝ for i_k = 1,…, m,; k = 1,… p, and j_k = 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 Axp−1yq be a vector in ℝ^m such that

(Axp−1yq)i=∑i2,⋯,ip=1m∑j1,⋯,jq=1naii2⋯ipj1⋯jqxi2⋯xipyj1⋯yjq,

where i = 1,…, m. Similarly, let Axpyq−1 be a vector in ℝⁿ such that

(Axpyq−1)j=∑i1,⋯,ip=1m∑j2,⋯,jq=1nai1⋯ipjj2⋯jqxi1⋯xipyj2yjq,

where j = 1,…, n. Let l = p + q. If there are a number λ ∈ ℂ, vectors x ∈ λ^m}\{0}, and y ∈ ℂⁿ\{0} such that

Axp−1yq=λx[l−1],Axpyq−1=λy[l−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, respectively. 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, 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

Theorem 1.1 [2, Theorem 2]

LetAbe a(p, q)-th orderm × ndimensional nonnegative tensor Then λ₀is 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 [9–13]. 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

Theorem 1.2 see [2, Theorem 4]

LetAbe a(p, q)-th orderm × ndimensional nonnegative rectangular tensor Then

min1≤i≤m,1≤j≤n{Rj(A),Cj(A)}≤λ0≤max1≤i≤m,1≤j≤n{Rj(A),Cj(A)},

where

Rj(A)=∑i2,⋯,ip=1m∑j1,⋯,jq=1naii2⋯ipj1⋯jq,Cj(A)=∑i1,⋯,ip=1m∑j2,⋯,jq=1nai1⋯ipjj2⋯jq.

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,⋯,jq∈N},ΔS:={(i2,⋯ip,j1,⋯,jq):i2,⋯ip,j1,⋯,jq∈S},ΩN:={(i1,⋯ip,j2,⋯,jq):i1,⋯ip,j2,⋯,jq∈N},ΩS:={(i1,⋯ip,j2,⋯,jq):i1,⋯ip,j2,⋯,jq∈S},

and then

△S¯=△N∖△S,ΩS¯=ΩN∖ΩS

This implies that for a nonnegative rectangular tensor A=(ai1⋯ipj1⋯jq), we have that for i, j ∈ S,

rj(A)=∑i2.⋯.ip.j1,⋯.jq∈Nδii2⋯ipj1⋯jq=0aii2⋯ipj1⋯jq=riΔS(A)+riΔS¯(A),rij(A)=riΔS(A)+riΔS¯(A)−aij⋯jj⋯j,

cj(A)=∑i1.⋯,ip,j2…,jq∈Nδi1⋯ipj˙j2⋯jq=0ai1⋯ipjj2⋯jq=cjΩS(A)+cjΩS¯(A),cji(A)=cjΩS(A)+cjΩS¯(A)−aj⋯iji⋯i,

where

δi1⋯ipj1⋯jq=1,ifi1=⋯=ip=j1=⋯=jq,0,otherwise,

and

riΔS(A)=∑(i2,⋯,ip,j1.⋯jq)∈ΔSδii2⋯ipj1⋯jq=0aii2⋯ipj1⋯jq,riΔS¯(A)=∑(i2,⋯,ip,j1,⋅⋅,jq)∈ΔS¯aii2⋯ipj1⋯jq,

cjΩS(A)=∑(i1⋯,ip.j2…jq)∈ΩSδi1⋯ipjj⋯jq=0ai1⋯ipjj2⋯jq,cjΩS¯(A)=∑(i1,⋯,ip,j2,⋅⋅,jq)∈ΩS¯ai1⋯ipjj2⋯jq.

By Theorem 1.2, the following lemma is easily obtained.

Lemma 2.1

LetAbea(p, q)-th ordern × ndimensional nonnegative rectangular tensor Then

λ0≥ai⋯ii⋯i,i∈N.

Theorem 2.2

LetAbea(p, q)-th order n × ndimensional nonnegative rectangular tensor, Sbe a nonempty proper subset ofN, S¯be the complement ofSin N. Then

λ0≤US(A)=max{U1S(A),U1S¯(A),U2S(A),U2S¯(A)},

where

U1S(A)=maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯j−aj⋯jj⋯j−rjΔS¯(A))2+4max{rj(A),ci(A)}rjΔS(A)]12},U1S¯(A)=maxi∈s¯,j∈S⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯¯(A)+[(ai⋯ii⋯j−aj⋯jj⋯j−rjΔS¯¯(A))2+4max{ri(A),ci(A)}rjΔS¯(A)]12},U2S(A)=maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+cjΩS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−cjΩS¯(A))2+4max{ri(A),ci(A)}cjΩS(A)]12},U2S¯(A)=maxi∈s¯,j∈S⁡12{ai⋯ii⋯i+aj⋯jj⋯j+cjΩS¯¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−cjΩS¯¯(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,…, x_n)^T and y = (y₁, y₂,…, y_n)^T such that

Axp−1yq=λ0x[l−1],Axpyq−1=λ0y[l−1].(1)(2)

Let

xt=max{xi:i∈S},xh=max{xi:i∈S¯};yf=max{yi:i∈S},yg=max{yi:i∈S¯};wi=max{xi,yi},i∈N,ws=max{wi:i∈S},wS¯=max{wi:i∈S¯}.

Then, at least one of x_t and x_h is nonzero, and at least of y_f and y_g is nonzero. Next, we present four cases to prove that.

Case I: Suppose that ws=xt,wS¯=xh, then x_t ≥ y_t, x_h ≥ y_h.

(i) If x_h ≥ x_t, then x_h = max{w_j : i ∈ N}. By the h-th equality in (1), we have

(λ0−ah⋯hh⋯h)xhl−1≤λ0xhl−1−ah⋯hh⋯hxhp−1yhq=∑(i2,⋯,ip,j1,⋅⋅,jq)∈ΔSahi2⋯ipj1⋯jqxi2⋯xipyj1⋯yjq+∑(i2,⋯.ip,j1,⋯,jq)∈ΔS¯δhi2⋯ipj1⋯jq=0ahi2⋯ipj1⋯jqxi2⋯xipyj1⋯yjq≤∑(i2,⋯,ip,j1,⋅⋅,jq)∈ΔSahi2⋯ipj1⋯jqxtl−1+∑(i2.⋯,ip,j1,⋯jq)∈ΔS¯δhi2⋯ipj1⋯jq=0ahi2⋯ipj1⋯jqxhl−1=rhΔS(A)xtl−1+rhΔS¯(A)xhl−1.

Hence,

(λ0−ah⋯hh⋯h−rhΔS¯(A))xhl−1≤rhΔS(A)xtl−1.(3)

If x_t = 0, then λ0−ah⋯hh⋯h−rhΔS¯(A)≤0 as x_h > 0, and it is obvious that λ0≤U1S(A) . Otherwise, x_t > 0. From the t-th equality in (1), we have

(λ0−at⋯tt⋯t)xtl−1≤λ0xtl−1−at⋯tt⋯txrp−1ytq=∑i2,⋯,ipj˙.⋅⋅,jq∈Nδti2⋯ipj1⋯jq=0ati2⋯ipj1⋯jqxi2…xjpyj1...yjq≤∑i2.⋯,ip,j.⋯.jq1∈N,δti2⋯ipj1⋯jq=0ati2⋯ipj1⋯jqxhl−1=rt(A)xhl−1,

i.e.,

(λ0−at⋯tt⋯t)xtl−1≤rt(A)xhl−1.(4)

From Lemma 2.1, we have λ₀-a_t… tt… t ≥ 0. Multiplying (3) with (4), we have

(λ0−at⋯tt⋯t)(λ0−ah⋯hh⋯h−rhΔS¯(A))xtl−1xhl−1≤rt(A)rhΔS(A)xtl−1xhl−1.

Note that xtl−1xhl−1>0. Then

(λ0−at⋯tt⋯t)(λ0−ah⋯hh⋯h−rhΔS¯(A))≤rt(A)rhΔS(A).(5)

Solving (5) gives

λ0≤12{at⋯tt⋯t+ah⋯hh⋯h+rhΔS¯(A)+[(at⋯tt⋯t−ah⋯hh⋯h−rhΔS¯(A))2+4rt(A)rhΔS(A)]12}≤maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ri(A)rjΔS(A)]12}≤U1S(A).

(ii) If x_t ≥ x_h, then x_t = max{w_j : i ∈ N}. Similarly to the proof of (i), we can obtain that

(λ0−ah⋯hh⋯h)(λ0−at⋯tt⋯t−rtΔS¯¯(A))≤rh(A)rtΔS¯(A).

This gives

λ0≤12{ah⋯hh⋯h+at⋯tt⋯t+rtΔS¯¯(A)+[(ah⋯hh⋯h−at⋯tt⋯t−rtΔS¯¯(A))2+4rh(A)rtΔS¯(A)]12}≤maxi∈s¯,j∈S⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯¯(A))2+4ri(A)rjΔS¯(A)]12}≤U1S¯(A).

Case II: Suppose that w_s = y_f, wS¯=yg, then y_f ≥ x_f, y_g ≥ x_g. If y_g ≥ y_f, then y_g = max{w_i : i 𢈈 N}. Similarly to the proof of (i) in Case I, we have

(λ0−af⋯ff⋯f)(λ0−ag⋯gg⋯g−cgΩS¯(A))≤cf(A)cgΩS(A).

This gives

λ0≤12{af⋯ff⋯f+ag⋯gg⋯g+cgΩS¯(A)+[(af⋯ff⋯f−ag⋯gg⋯g−cgΩS¯(A))2+4cf(A)cgΩS(A)]12}≤maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+cjΩS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−cjΩS¯(A))2+4ci(A)cjΩS(A)]12}≤U2S(A).

If y_f ≥ y_g, then y_f = max{w_i : i ∈ N}. Similarly to the proof of (ii) in Case I, we have

(λ0−ag⋯gg⋯g)(λ0−af⋯ff⋯f−cfΩS¯¯(A))≤cg(A)cfΩS¯(A).

This gives

λ0≤12{ag⋯gg⋯g+af⋯ff⋯f+cfΩS¯¯(A)+[(ag⋯g⋯g−af⋯ff⋯f−cfΩS¯¯(A))2+4cg(A)cfΩS¯(A)]12}≤maxi∈s¯,j∈S⁡12{ai⋯ii⋯i+aj⋯jj⋯j+cjΩS¯¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−cjΩS¯¯(A))2+4ci(A)cjΩS¯(A)]12}≤U2S¯(A).

Case III: Suppose that w_s = x_t, wS¯=yg, then x_t ≥ y_t, y_g ≥ x_g. If y_g ≥ x_t, then y_g = max{w_i : i ∈ N}. Similarly to the proof of (i) in Case I, we have

(λ0−at⋯tt⋯t)(λ0−ag⋯gg⋯g−cgΩS¯(A))≤rt(A)cgΩS(A).

This gives

λ0≤12{at⋯tt⋯t+ag⋯gg⋯g+cgΩS¯(A)+[(at⋯tt⋯t−ag⋯gg⋯g−cgΩS¯(A))2+4rt(A)cgΩS(A)]12}≤maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+cjΩS¯(A)+[(ai⋯ii⋯j−aj⋯jj⋯j−cjΩS¯(A))2+4ri(A)cjΩS(A)]12}≤U2S(A).

If x_t ≥ y_g, then x_t = max{w_i : i ∈ N}. Similarly to the proof of (ii) in Case I, we can obtain that

(λ0−ag⋯gg⋯g)(λ0−at⋯tt⋯t−rtΔS¯¯(A))≤cg(A)rtΔS¯(A).

This gives

λ0≤12{ag⋯gg⋯g+at⋯tt⋯t+rtΔS¯¯(A)+[(ag⋯gg⋯g−at⋯tt⋯t−rtΔS¯¯(A))2+4cg(A)rtΔS¯(A)]12}≤maxi∈s¯,j∈S⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯¯(A))2+4ci(A)rjΔS¯(A)]12}≤U1S¯(A).

Case IV: Suppose that ws=yf,wS¯=xh, then y_f ≥ x_f, x_h ≥ y_h. If x_h ≥ y_f, then x_h = max{w_j : i ∈ N}. Similarly to the proof of (i) in Case I, we have

(λ0−af⋯ff⋯f)(λ0−ah⋯hh⋯h−rhΔS¯(A))≤cf(A)rhΔS(A).

This gives

λ0≤12{af⋯ff⋯f+ah⋯hh⋯h+rhΔS¯(A)+[(af⋯ff⋯f−ah⋯hh⋯h−rhΔS¯(A))2+4cf(A)rhΔS(A)]12}≤maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ci(A)rjΔS(A)]12}≤U1S(A).

If y_f ≥ x_h, then y_f = max{w_i} : i ∈ N}. Similarly to the proof of (ii) in Case I, we can obtain that

(λ0−ah⋯hh⋯h)(λ0−af⋯ff⋯f−cfΩS¯¯(A))≤rh(A)cfΩS¯(A).

This gives

λ0≤12{ah⋯hh⋯h+af⋯ff⋯f+cfΩS¯¯(A)+[(ah⋯hh⋯h−af⋯ff⋯f−cfΩS¯¯(A))2+4rh(A)cfΩS¯(A)]12}≤maxi∈s¯,j∈S⁡12{ai⋯ii⋯i+aj⋯jj⋯j+cjΩS¯¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−cjΩS¯¯(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

LetAbea(p, q)-th order n × ndimensional nonnegative rectangular tensor, Sbe a nonempty proper subset ofN,S¯be the complement ofSin N. Then

US(A)≤maxi,j∈N⁡{Ri(A),Cj(A)}.

Proof

Here, we only prove U1S(A)≤maxi,j∈N⁡{Rj(A),Cj(A)}. Similarly, we can prove U1S¯(A),U2S(A),U2S¯(A)≤maxi,j∈N⁡{Ri(A),Cj(A)}, respectively. We next divide into two cases to prove.

Case I: Suppose that

U1S(A)=maxi∈S,j∈S¯⁡12(ai⋯ii⋯i+aj…jj…j+rjΔs¯(A)+[(ai…ii…i−aj…jj…j−rjΔs¯(A))2+4ri(A)rjΔs(A)]12.

(i) For any i∈S,j∈S¯, if Rj(A)≤Ri(A), then

0≤rj(A)≤aj⋯jj⋯j−ai⋯ii⋯i+rjΔS¯(A)+rjΔS(A).

Hence,

(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ri(A)rjΔS(A)≤(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4(aj⋯jj⋯j−ai⋯ii⋯i+rjΔS¯(A)+rjΔS(A))rjΔS(A)=(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4(aj⋯jj⋯j−ai⋯ii⋯i+rjΔS¯(A))rjΔS(A)+4(rjΔS(A))2=(aj⋯jj⋯j−ai⋯ii⋯i+rjΔS¯(A)+2rjΔS(A))2.

Furthermore,

ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4rj(A)rjΔS(A)]12≤ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[aj⋯jj⋯j−ai⋯ii⋯i+rjΔS¯(A)+2rjΔS(A)]=2aj⋯jj⋯j+2rjΔS¯(A)+2rjΔS(A)=2Rj(A),

which implies

U1S(A)=maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ri(A)rjΔS(A)]12}≤maxj∈s¯⁡{Rj(A)}≤maxi,j∈N⁡{Ri(A),Cj(A)}.

(ii) For any i∈S,j∈S¯, if Rj(A)≤Rj(A), then

0≤rjΔS(A)≤ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A)+ri(A).

Similarly to the proof of (i), we can obtain

ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ri(A)rjΔS(A)]12≤2Ri(A).

Hence,

U1S(A)=maxi∈S,j∈S¯⁡12{ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ri(A)rjΔS(A)]12}≤maxi∈S⁡{Ri(A)}≤maxi,j∈N⁡{Ri(A),Cj(A)}.

Case II: Suppose that

U1S(A)=maxi∈S,j∈S¯⁡12ai⋯ii⋯i+aj⋯jj…j+rjΔs¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔs¯(A))2+4ci(A)rjΔs(A)]12.

(iii) For any i∈S,j∈S¯, if Ci(A)≤Rj(A), then

0≤ci(A)≤aj⋯jj⋯j−ai⋯ii⋯i+rjΔS¯(A)+rjΔS(A).

Similarly to the proof of (i), we can obtain

ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ci(A)rjΔS(A)]12≤2ci(A).

Hence,

U1S(A)=maxi∈S,j∈S¯⁡12ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ci(A)rjΔS(A)]12≤maxi∈s⁡{ci(A)}≤maxi,j∈N⁡{Ri(A),Cj(A)}.

(iv) For any i∈S,j∈S¯, if Rj(A)≤Ci(A), then

0≤rjΔS(A)≤ai⋯jii⋯i−aj⋯jj⋯j−rjΔS¯(A)+ci(A).

Similarly to the proof of (ii), we can obtain

ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ci(A)rjΔS(A)]12≤2Ci(A).

Hence,

U1S(A)=maxi∈S,j∈S¯⁡12ai⋯ii⋯i+aj⋯jj⋯j+rjΔS¯(A)+[(ai⋯ii⋯i−aj⋯jj⋯j−rjΔS¯(A))2+4ci(A)rjΔS(A)]12≤maxi∈S⁡{Ci(A)}≤maxi,j∈N{Ri(A),Cj(A)}⋅

The conclusion follows from Cases I and II. □

3 Numerical examples

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

Example 3.1

LetA=(aijkl)bea (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.

LetS ={1, 2}. ObviouslyS¯={3}. By Theorem 1.2, we have

λ0≤45.

By Theorem 2.2, we have

λ0≤38.4165.

In fact, λ₀ = 31.3781. This example shows that the upper bound in Theorem 2.2 is smaller than that inTheorem 1.2.

Example 3.2

LetA=(aijkl)bea(2, 2)-th order 2 × 2 dimensional nonnegative rectangular tensor with entries defined as follows:

a1111=a1112=a1222=a2112=a2121=a2221=1,

othera_ijkl = 0. By Theorem 2.2, we have

λ0≤3.

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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Received: 2016-8-13

Accepted: 2016-10-11

Published Online: 2016-11-12

Published in Print: 2016-1-1

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