A new S type upper bound for the largest singular value of nonnegative rectangular tensors

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R E S E A R C H

Open Access

A new

S

-type upper bound for the largest

singular value of nonnegative rectangular

tensors

Jianxing Zhao

*

_{and Caili Sang}

*_{Correspondence:}

[email protected] College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, P.R. China

Abstract

By breakingN={1, 2,. . .,n}into disjoint subsetsSand its complement, a newS-type upper bound for the largest singular value of nonnegative rectangular tensors is given and proved to be better than some existing ones. Numerical examples are given to verify the theoretical results.

MSC: 15A18; 15A42; 15A69

Keywords: nonnegative tensor; rectangular tensor; singular value

1 Introduction

Singular value problems of rectangular tensors have become an important topic in applied mathematics and numerical multilinear algebra, and it has a wide range of practical appli-cations, such as the strong ellipticity condition problem in solid mechanics [, ] and the entanglement problem in quantum physics [, ].

LetR(respectively,C) be the real (respectively, complex) ﬁeld. Assume thatp,q,m,n are positive integers,m,n≥,l=p+q, andN={, , . . . ,n}. A real (p,q)th orderm× ndimensional rectangular tensor (or simply a real rectangular tensor)Ais deﬁned as follows:

A= (ai···ipj···jq), ai···ipj···jq∈R, ≤i, . . . ,ip≤m, ≤j, . . . ,jq≤n.

A real rectangular tensorAis called nonnegative if ai···ipj···jq ≥ forik= , . . . ,m,k=

, . . . ,p, andjv= , . . . ,n,v= , . . . ,q.

For vectorsx= (x, . . . ,xm)T,y= (y, . . . ,yn)Tand a real numberα, letx[α]= (xα,xα, . . . , xα

m)T,y[α]= (yα,yα, . . . ,yαn)T,Axp–yqbe anmdimension real vector whoseith component

is

Axp–_yq i=

m

i,...,ip=

n

j,...,jq=

aii···ipj···jqxi· · ·xipyj· · ·yjq,

(2)

andAxp_yq–_{be an}_n_{dimension real vector whose}_{jth component is}

Axp_yq–

j= m

i,...,ip=

n

j,...,jq=

ai···ipjj···jqxi· · ·xipyj· · ·yjq.

Ifλ∈C,x∈Cm_\{__}_{, and}_y_∈_Cn_\{__}_{are solutions of}

⎧ ⎨ ⎩

Axp–_yq₌_λ_x[l–]_, Axp_yq–₌_λ_y[l–]_,

then we say thatλis a singular value ofA,xandyare a left and a right eigenvectors ofA, associated withλ. Ifλ∈R,x∈Rm_{, and}_y_∈_Rn_{, then we say that}_λ_{is an H-singular value}

ofA,xandyare a left and a right H-eigenvectors ofA, associated with H-singular value

λ[]. Here,

λ=max

|λ|:λis a singular value ofA

is called the largest singular value [].

The definition of singular values for tensors was first introduced in []. Note here that whenlis even, the definitions in [] is the same as in [], and whenlis odd, the definition in [] is slightly different from that in [], but parallel to the definition of eigenvalues of square matrices []; see [] for details.

Recently, many people focus on bounding the largest singular value for nonnegative rect-angular tensors [, , ]. For convenience, we ﬁrst give some notation. Given a nonempty proper subsetSofN, we denote

N:=(i, . . . ,ip,j, . . . ,jq) :i, . . . ,ip,j, . . . ,jq∈N ,

S:=(i, . . . ,ip,j, . . . ,jq) :i, . . . ,ip,j, . . . ,jq∈S ,

N:=(i, . . . ,ip,j, . . . ,jq) :i, . . . ,ip,j, . . . ,jq∈N ,

S:=(i, . . . ,ip,j, . . . ,jq) :i, . . . ,ip,j, . . . ,jq∈S ,

and then

S₌N_\S_, S₌N_\S_.

This implies that, for a nonnegative rectangular tensorA= (ai···ipj···jq), we have, fori,j∈S,

ri(A) =

i,...,ip,j,...,jq∈N δii···ipj···jq=

aii···ipj···jq=r S

i (A) +r

S

i (A), r j

i(A) =ri(A) –aij···jj···j,

cj(A) =

i,...,ip,j,...,jq∈N δi···ipjj···jq=

ai···ipjj···jq=c S

j (A) +c

S

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where

δi···ipj···jq=

⎧ ⎨ ⎩

, ifi=· · ·=ip=j=· · ·=jq,

, otherwise,

and

rS

i (A) =

(i_,...,ip,j_,...,jq)∈S δ_ii

···ipj···jq=

aii···ipj···jq, r S

i (A) =

(i,...,ip,j,...,jq)∈S

aii···ipj···jq,

c_jS(A) = (i,...,ip,j,...,jq)∈S

δi_···ipjj···jq=

ai···ipjj···jq, c S

j (A) =

(i,...,ip,j,...,jq)∈S

ai···ipjj···jq.

In [], Yang and Yang gave the following bound for the largest singular value of a non-negative rectangular tensorA.

Theorem ([], Theorem ) LetAbe a(p,q)th order m×n dimensional nonnegative rectangular tensor.Then

λ≤ max ≤i≤m,≤j≤n

Ri(A),Cj(A) ,

where

Ri(A) = m

i,...,ip=

n

j,...,jq=

aii···ipj···jq, Cj(A) =

m

i,...,ip=

n

j,...,jq=

ai···ipjj···jq.

Whenm=n, Heet al.[] have given an upper bound which is lower than that in Theo-rem .

Theorem ([], Theorem .) LetAbe a(p,q)th order n×n dimensional nonnegative rectangular tensor.Then

λ≤(A) =max

(A),(A),(A),(A) ,

where

(A) = max

i,j∈N,i=j

 

ai···ii···i+aj···jj···j+rji(A)

+ai···ii···i–aj···jj···j+rji(A)



+ aij···jj···jrj(A)

  _,

(A) = max

i,j∈N,i=j

 

ai···ii···i+aj···jj···j+cji(A)

+ai···ii···i–aj···jj···j+cji(A)



+ aj···jij···jcj(A)

  _,

(A) = max

i,j∈N,i=j

 

ai···ii···i+aj···jj···j+rji(A)

+ai···ii···i–aj···jj···j+rji(A)



+ aij···jj···jcj(A)

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(A) = max

i,j∈N,i=j

 

ai···ii···i+aj···jj···j+cji(A)

+ai···ii···i–aj···jj···j+cji(A)



+ aj···jij···jrj(A)

  _.

Similarly, under the condition ofm=n, by breakingN={, , . . . ,n}into disjoint subsets S and its complement S, Zhao and Sang [] provided an¯ S-type upper bound for the largest singular value of nonnegative rectangular tensors.

Theorem ([], Theorem .) LetAbe 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

λ≤US(A) =max

U_S(A),U_S¯(A),U_S(A),U_¯S(A) , where

U_S(A) = max

i∈S,j∈¯S

 

ai···ii···i+aj···jj···j+r

S

j (A)

+ai···ii···i–aj···jj···j–r

S

j (A)



+ maxri(A),ci(A) r

S

j (A)

  _,

U_S¯(A) = max

i∈¯S,j∈S

 

¯

S

j (A)

¯

S

j (A)



+ maxri(A),ci(A) r

¯

S

j (A)

  _,

US

(A) = max

i∈S,j∈¯S

 

ai···ii···i+aj···jj···j+c

S

j (A)

+ai···ii···i–aj···jj···j–c

S

j (A)



+ maxri(A),ci(A) c

S

j (A)

  _,

US¯

(A) = max

i∈¯S,j∈S

 

¯

S

j (A)

+ai···ii···i–aj···jj···j–c

¯

S

j (A)



+ maxri(A),ci(A) c

¯

S

j (A)

  _.

In this paper, we continue this research, and give a newS-type upper bound for the largest singular value of nonnegative rectangular tensors. It is proved that the new upper bound is better than those in Theorems -.

2 Main results

Theorem  LetAbe 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

λ≤S(A) =max

_S(A),_S¯(A),_S(A),_S¯(A),_S(A),_¯S(A),_S(A),_S¯(A) , where

_S(A) = max

i∈S,j∈¯S

 

S

i (A) +r

S

j (A)

+ai···ii···i–aj···jj···j+r

S

i (A) –r

S

j (A)



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_¯S(A) = max

i∈¯S,j∈S

 

¯

S

i (A) +r

¯S

j (A)

¯

S

i (A) –r

S¯

j (A)



+ r_iS¯(A)r_jS¯(A) _,

_S(A) = max

i∈S,j∈¯S

 

S

i (A) +c

S

j (A)

+ai···ii···i–aj···jj···j+c

S

i (A) –c

S

j (A)



+ c_iS(A)c_jS(A) _,

_¯S(A) = max

i∈¯S,j∈S

 

¯

S

i (A) +c

S¯

j (A)

¯

S

i (A) –c

S¯

j (A)



+ c_iS¯(A)c_jS¯(A)   _,

_S(A) = max

i∈S,j∈¯S

 

S

i (A) +c

S

j (A)

S

i (A) –c

S

j (A)



+ r_iS(A)c_jS(A)   _,

_¯S(A) = max

i∈¯S,j∈S

 

¯

S

j (A) +c

S¯

i (A)

¯

S

j (A) +c

S¯

i (A)



+ r_jS¯(A)c_i¯S(A)   _,

_S(A) = max

i∈S,j∈¯S

 

S

j (A) +c

S

i (A)

S

j (A) +c

S

i (A)



+ r_jS(A)c_iS(A)   _,

_¯S(A) = max

i∈¯S,j∈S

 

¯

S

i (A) +c

¯

S

j (A)

¯

S

i (A) –c

S¯

j (A)

 + rS¯

i (A)c

¯S

j (A)

  _.

Proof Becauseλis the largest singular value ofA, from Theorem  in [], there are non-negative nonzero vectorsx= (x,x, . . . ,xn)Tandy= (y,y, . . . ,yn)T, such that

Axp–_yq₌_λ

x[l–], ()

Axp_yq–₌_λ

y[l–]. ()

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 ofxtandxhis nonzero, and at least one ofyf andygis nonzero. We next

divide into four cases to prove.

Case I: IfwS=xt,wS¯ =xh, thenxt≥yt,xh≥yh.

(i) Ifxh≥xt, thenxh=max{wi:i∈N}. From () of Theorem . in [], we have

λ–ah···hh···h–r

S

h (A)

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Ifxt= , byxh> , we haveλ–ah···hh···h–r

S

h (A)≤. Thenλ≤ah···hh···h+r

S

h (A)≤

S

(A). Otherwise,xt> . From (), we have

(λ–at···tt···t)xlt–≤λxlt––at···tt···txpt–y q t

=

(i,...,ip,j,...,jq)∈S δ_ti

···ipj···jq=

ati···ipj···jqxi· · ·xipyj· · ·yjq

+

(i,...,ip,j,...,jq)∈S

ati···ipj···jqxi· · ·xipyj· · ·yjq

≤

(i_,...,ip,j_,...,jq)∈S δti_···ipj···jq=

ati···ipj···jqx

l–

t +

(i,...,ip,j,...,jq)∈S

ati···ipj···jqx

l–

h

=r_tS(A)xl_t–+r_tS(A)xl_h–, i.e.,

λ–at···tt···t–r

S

t (A)

xl_t–≤r_t S(A)xl_h–. () Ifλ–at···tt···t–r

S

t (A)≤, thenλ≤at···tt···t+r

S

t (A)≤S(A). Ifλ–at···tt···t–r

S

t (A) > ,

multiplying () with () and noting thatxl–

t xlh–> , we have

S

t (A)

S

h (A)

≤r_t S(A)r_hS(A). ()

Solvingλin () gives

λ≤  

at···tt···t+ah···hh···h+r

S

t (A) +r

S

h (A)

+at···tt···t+r

S

t (A) –ah···hh···h–r

S

h (A)



+ r_tS(A)r_hS(A)  

≤ max

i∈S,j∈¯S

 

S

i (A) +r

S

j (A)

S

i (A) –r

S

j (A)



+ r_iS(A)r_jS(A)  

=_S(A).

(ii) Ifxt≥xh, similar to the proof of (i), we have

¯

S

h (A)

¯

S

t (A)

≤r_h¯S(A)r_tS¯(A),

and

λ≤  

ah···hh···h+at···tt···t+r

¯

S

h (A) +r

¯

S

t (A)

+ah···hh···h–at···tt···t+r

¯

S

h (A) –r

¯

S

t (A)

 + rS¯

h (A)r

¯

S

t (A)

 

≤ max

i∈¯S,j∈S

 

¯

S

i (A) +r

¯

S

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¯

S

i (A) –r

S¯

j (A)



+ r_iS¯(A)r_jS¯(A)  

=_S¯(A).

Case II: Assume thatwS=yf,wS¯=yg. Ifyg≥yf, similar to the proof of (i), we have

λ–af···ﬀ···f –c

S

f (A)

λ–ag···gg···g–c

S

g (A)

≤c_fS(A)c_gS(A),

and

λ≤  

af···ﬀ···f+ag···gg···g+c

S

f (A) +c

S

g (A)

+af···ﬀ···f–ag···gg···g+c

S

f (A) –c

S

g (A)



+ c_fS(A)c_gS(A)  

≤ max

i∈S,j∈¯S

 

S

i (A) +c

S

j (A)

S

i (A) –c

S

j (A)



+ c_iS(A)c_jS(A)  

=_S(A).

Ifyf≥yg, similarly, we have

¯

S

g (A)

λ–af···ﬀ···f–c

¯

S

f (A)

≤cS¯

g (A)c

¯

S

f (A)

and

λ≤  

ag···gg···g+af···ﬀ···f+c

¯

S

g (A) +c

¯

S

f (A)

+ag···g···g–af···ﬀ···f+c

¯

S

g (A) –c

S¯

f (A)



+ c_gS¯(A)c_f¯S(A)

≤ max

i∈¯S,j∈S

 

¯

S

i (A) +c

S¯

j (A)

¯

S

i (A) –c

S¯

j (A)



+ c_iS¯(A)c_jS¯(A)  

=_S¯(A).

Case III: Assume thatwS=xt,wS¯=yg. Ifyg≥xt, similar to the proof of (i), we have

S

t (A)

S

g (A)

≤r_t S(A)c_gS(A)

and

λ≤  

at···tt···t+ag···gg···g+r

S

t (A) +c

S

g (A)

+at···tt···t–ag···gg···g+r

S

t (A) –c

S

g (A)



+ r_tS(A)c_gS(A)  

≤ max

i∈S,j∈¯S

 

S

i (A) +c

S

j (A)

S

i (A) –c

S

j (A)



+ r_iS(A)c_jS(A)  

(8)

Ifxt≥yg, similarly, we have

¯

S

g (A)

¯

S

t (A)

≤c_gS¯(A)r_tS¯(A)

and

λ≤  

at···tt···t+ag···gg···g+r

¯

S

t (A) +c

¯S

g (A)

+at···tt···t–ag···gg···g+r

¯

S

t (A) –c

S¯

g (A)



+ r_tS¯(A)c_g¯S(A)  

≤ max

i∈¯S,j∈S

 

¯

S

j (A) +c

¯

S

i (A)

¯

S

j (A) +c

S¯

i (A)



+ r_jS¯(A)c_i¯S(A)  

=_S¯(A).

Case IV: Assume thatwS=yf,wS¯=xh. Ifxh≥yf, similar to the proof of (i), we have

λ–af···ﬀ···f –c

S

f (A)

S

h (A)

≤c_fS(A)r_hS(A)

and

λ≤  

af···ﬀ···f+ah···hh···h+r

S

h (A) +c

S

f (A)

+af···ﬀ···f–ah···hh···h–r

S

h (A) +c

S

f (A)

 + cS

f (A)r S h (A)   ≤ max

i∈S,j∈¯S

 

S

j (A) +c

S

i (A)

S

j (A) +c

S

i (A)

 + cS

i (A)r

S

j (A)

 

=_S(A).

Ifyf≥xh, similarly, we have

¯

S

h (A)

λ–af···ﬀ···f–c

¯

S

f (A)

≤r_h¯S(A)cS¯

f (A)

and

λ≤  

ah···hh···h+af···ﬀ···f+r

¯

S

h (A) +c

S¯

f (A)

+ah···hh···h–af···ﬀ···f+r

¯

S

h (A) –c

S¯

f (A)



+ r_hS¯(A)c_fS¯(A)  

≤ max

i∈¯S,j∈S

 

¯

S

i (A) +c

S¯

j (A)

¯

S

i (A) –c

¯

S

j (A)

 + rS¯

i (A)c ¯ S j (A)  

=_S¯(A).

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We next give the following comparison theorem for these upper bounds in Theorems -.

Theorem  LetAbe 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

S(A)≤US(A)≤(A)≤max

i,j∈N

Ri(A),Cj(A) .

Proof I. By Remark . in [],(A)≤maxi,j∈N{Ri(A),Cj(A)}holds.

II. Next, we proveUS_(A)_≤_{(A). Here, we only prove}_US

(A)≤(A). Similarly, we can proveUS¯

(A),US(A),U

¯

S

(A)≤(A), respectively. (i) Suppose that

U_S(A) = max

i∈S,j∈¯S

 

S

j (A)

S

j (A)



+ ri(A)r

S

j (A)

  _.

From the proof of Theorem . in [], we can see that the boundU_S(A) is obtained by solvingλfrom

(λ–ai···ii···i)

λ–aj···jj···j–r

S

j (A)

≤ri(A)r

S

j (A). ()

From the proof of Theorem . in [], we can see that the bound

(A) = max

i,j∈N,i=j

 

ai···ii···i+aj···jj···j+rij(A)

+ai···ii···i–aj···jj···j–rij(A)



+ aji···ii···iri(A)

 

is obtained by solvingλfrom

(λ–ai···ii···i)

λ–aj···jj···j–rij(A)

≤aji···ii···iri(A). ()

Takingi∈S,j∈ ¯Sin (), by the proof of Theorem  in [], we know that ifλsatisﬁes (), thenλsatisﬁes (), which implies that

(A)≥ max

i∈S,j∈¯S

 

ai···ii···i+aj···jj···j+rij(A)

+ai···ii···i–aj···jj···j–rji(A)



+ aji···ii···iri(A)

 

≥U_S(A).

Obviously,US

(A)≤(A). (ii) Suppose that

U_S(A) = max

i∈S,j∈¯S

 

S

j (A)

S

j (A)



+ ci(A)r

S

j (A)

(10)

Similar to the proof of (i), we can obtainU_S(A)≤(A)≤(A).

III. Finally, we prove thatS(A)≤US(A). Here, we only prove_S(A)≤US(A). Simi-larly, we can prove_S¯(A),_S(A),_S¯(A),_S(A),_S¯(A),_S(A),_¯S(A)≤US₍_A_),

respec-tively.

Leti∈Sandj∈ ¯S. From the proof of Theorem , we can see that the boundS

(A) is obtained by solvingλfrom

λ–ai···ii···i–r

S

i (A)

S

j (A)

≤r_iS(A)r_jS(A). ()

(i) Suppose thatrS

i (A)r

S

j (A) = . Ifλ–ai···ii···i–r

S

i (A) > ,i.e.,λ>ai···ii···i+r

S

i (A),

thenλ–aj···jj···j–r

S

j (A)≤, and for anyi∈S,

(λ–ai···ii···i)

S

j (A)

≤≤ri(A)r

S

j (A).

That is to say, ifλsatisﬁes (), thenλsatisﬁes (), which implies thatS(A)≤US(A)≤ US(A).

Ifλ–ai···ii···i–r

S

i (A)≤, thenλ–aj···jj···j–r

S

j (A)≥,i.e.,λ≥aj···jj···j+r

S

j (A). From

(), we can obtainλ–aj···jj···j–r

S

j (A)≤r

S

j (A),i.e.,

λ–aj···jj···j≤rj(A). ()

Byλ–ai···ii···i–r

S

i (A)≤≤r

S

i (A),i.e.,λ–ai···ii···i≤ri(A), we have

¯

S

i (A)≤r

¯S

i (A). ()

Multiplying () with (), we can obtain

(λ–aj···jj···j)

¯

S

i (A)

≤r_iS¯(A)rj(A), ()

which implies that ifλsatisﬁes (), thenλsatisﬁes (), consequently,S(A)≤U

¯

S

(A)≤ US_(A).

(ii) Suppose thatrS

i (A)r

S

j (A) > . Then dividing () byr

S

i (A)r

S

j (A), we have

(λ–ai···i–r

S

i (A))

rS

i (A)

(λ–aj···j–r

S

j (A))

rS

j (A)

≤. ()

Furthermore, ifλ–ai···i–r S i (A)

r_iS(A) ≥

, then by Lemma . in [] and (), we have

(λ–ai···i)

ri(A)

S

j (A))

rS

j (A)

≤(λ–ai···i–r

S

i (A))

rS

i (A)

S

j (A))

rS

j (A)

≤.

Thus, () holds, which implies that ifλsatisﬁes (), thenλsatisﬁes (), consequently,

_S(A)≤U_S(A). And ifλ–ai···i–r S i (A)

riS(A)

≤, then () holds, which leads to () from (). This

implies that ifλsatisﬁes (), thenλsatisﬁes (), consequently,S(A)≤U

¯

S

(11)

3 Numerical examples

Example  LetA= (aijkl) be a (, )th order × dimensional nonnegative rectangular

tensor with entries deﬁned as follows:

A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥

⎦, A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥ ⎦,

A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥ ⎦,

A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥

⎦, A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥ ⎦,

A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥ ⎦,

A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥

⎦, A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥ ⎦,

A(:, :, , ) =

⎡ ⎢ ⎣

        

⎤ ⎥ ⎦.

By Theorem , we have

λ≤.

By Theorem , we have

λ≤..

TakingS={, },S¯={}, by Theorem , we have

λ≤.;

by Theorem , we have

λ≤..

In fact,λ= .. This example shows that the upper bound in Theorem  is smaller than those in Theorems -.

Example  LetA= (aijkl) be a (, )th order × dimensional nonnegative rectangular

tensor with entries deﬁned as follows:

(12)

the otheraijkl= . By Theorem , we have

λ≤.

In fact,λ= . This example shows that the upper bound in Theorem  is sharp.

4 Conclusions

In this paper, a newS-type upper boundS_{(A) of the largest singular value for a}

nonneg-ative rectangular tensorAwithm=nis obtained by breakingNinto disjoint subsetsS and its complement. It is proved that the boundS(A) is better than those in [, , ].

Note here that whenn= ,(A) =US(A) =S(A), and whenn≥,(A)≥US(A)≥

S_{(A) always holds. How to pick}_S_{to make}S_{(A) as small as possible is an interesting}

problem, but diﬃcult whennis large. We will research this problem in the future.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.

Acknowledgements

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 Natural Science Programs of Education Department of Guizhou Province (Grant No. [2016]066), Foundation of Guizhou Science and Technology Department (Grant No. [2015]2073) and National Natural Science Foundation of China (No. 11501141).

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 25 December 2016 Accepted: 25 April 2017

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