p Norm SDD tensors and eigenvalue localization

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R E S E A R C H

Open Access

p

-Norm SDD tensors and eigenvalue

localization

Qilong Liu and Yaotang Li

*

*_{Correspondence:}

[email protected] School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, P.R. China

Abstract

We present a new class of nonsingular tensors (p-norm strictly diagonally dominant

tensors), which is a subclass of strongH-tensors. As applications of the results, we

give a new eigenvalue inclusion set, which is tighter than those provided by Liet al.

(Linear Multilinear Algebra 64:727-736, 2016) in some case. Based on this set, we give a checkable suﬃcient condition for the positive (semi)deﬁniteness of an even-order symmetric tensor.

Keywords: p-norm SDD tensor; strongH-tensor; positive (semi)deﬁniteness; eigenvalue localization

1 Introduction

LetC(R) denote the set of all complex (real) numbers, and [n] :={, , . . . ,n}. Anmth-order

n-dimensional complex (real) tensor, denoted byA∈C[m,n](R[m,n]), is a multidimensional array ofnm_{elements of the form}

A= (aii···im), ai···im∈C(R),ij∈[n],j∈[m].

Whenm= ,Ais ann-by-nmatrix. A tensorA= (ai···im)∈R[m,n]is called nonnegative if each its entry is nonnegative, and it is called symmetric [, ] if

ai···im=aπ(i···im), ∀π∈m,

wheremis the permutation group ofmindices. Moreover, anmth-ordern-dimensional

tensorI= (δii···im) is called the identity tensor [] if

δii···im=

 ifi=i=· · ·=im,

 otherwise.

For ann-dimensional vectorx= (x,x, . . . ,xn)T, real or complex, we deﬁne then

-dimen-sional vector

Axm–:=

i,...,im∈[n]

aii···imxi· · ·xim

≤i≤n

,

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and then-dimensional vector

x[m–]_:=_xm– i

≤i≤n.

The following deﬁnition related to eigenvalues of tensors was ﬁrst introduced and stud-ied by Qi [] and Lim [].

Deﬁnition [, ] LetA= (aii···im)∈C[m,n]. A pair (λ,x)∈C×(Cn\ {}) is called an eigenvalue-eigenvector (or simply eigenpair) ofAif satisﬁes the equation

Axm–=λx[m–]. ()

We call (λ,x) an H-eigenpair if they are both real.

In addition, the spectral radius of a tensorAis deﬁned as

ρ(A) =max|λ|:λis an eigenvalue ofA .

Deﬁnition  A tensorA∈C[m,n] _{is said to be nonsingular if zero is not an eigenvalue}

ofA. Otherwise, it is called singular.

Tensor eigenvalue problems have gained special attention in the realm of numerical multilinear algebra, and they have a wide range in practice; see [, , –]. For in-stance, we can use the smallest H-eigenvalues of tensors to determine their positive (semi)definiteness, that is, for an even-order real symmetric tensorA, if its smallest H-eigenvalue is positive (nonnegative), thenAis positive (semi)definite; consequently, the multivariate homogeneous polynomialf(x) determined byAis positive (semi)definite []. Most often, it is difficult to compute the smallest H-eigenvalue. Therefore, we always try to give a distribution range of eigenvalues of a given tensor in the complex plane. In particular, if this range is in the right-half complex plane, which means that the smallest H-eigenvalue is positive, then the corresponding tensor is positive definite.

Qi [] generalized the Geršgorin eigenvalue inclusion theorem from matrices to real symmetric tensors, which can be easily extended to generic tensors; see [, ].

Theorem  LetA= (aii···im)∈C[m,n].Then

σ(A)⊆(A) =

i∈[n]

i(A),

whereσ(A)is the set of all the eigenvalues ofA,and

i(A) =

z∈C:|z–aii···i| ≤ri(A) , ri(A) =

i,...,im∈[n], δii_···im=

|aii···im|.

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of Theorem . We list the latest Brauer-type eigenvalue localization set as follows. For convenience, we denote

i=

(i,i, . . . ,im) :ij=ifor somej∈ {, , . . . ,m}, wherei,i, . . . ,im∈[n] ,

i=

(i,i, . . . ,im) :ij=ifor anyj∈ {, , . . . ,m}, wherei,i, . . . ,im∈[n] ,

and

r i

i (A) =

(i,...,im)∈ i, δii···im=

|aii···im|, r i

i (A) =

(i,...,im)∈ i

|aii···im|.

Theorem [] LetA= (aii···im)∈C[m,n].Then

σ(A)⊆(A) =

i∈[n] ˆ

i(A)

∪ i,j∈[n],

i=j

ˆ

i,j(A)∩i(A)

,

where

ˆ

i(A) =

z∈C:|z–ai···i| ≤ri i(A)

and

ˆ

i,j(A) =

z∈C:|z–ai···i|–rii(A)

|z–aj···j|–rji(A)

≤r i

i (A)rj i(A) .

Liet al.[] proved that the set(A) in Theorem  is tighter than(A) in [] andK(A) in []; for details, see Theorem . in [].

In this paper, we continue this research on the eigenvalue localization problem for ten-sors. A class of strictly diagonally dominant tensors that involve a parameterpin the in-terval [,∞], denoted byp-norm SDD tensor, is introduced in Section . In Section ,

we discuss the relationships betweenp-norm SDD tensors and strongH-tensors. A new

eigenvalue inclusion set for tensors based on p-norm SDD tensors is obtained in

Sec-tion , and numerical results show that the new set is tighter than(A) in Theorem  in some case. Finally, in Section , we give a checkable suﬃcient condition for the positive (semi)deﬁniteness of even-order symmetric tensors.

2 p-Norm SDD tensors

In this section, we propose a new class of nonsingular tensors, namely p-norm strictly

diagonally dominant tensors. First, some notation and the deﬁnition of strictly diagonally dominant tensors are given.

Given a tensorA= (aii···im)∈C[m,n]and a real numberp∈[,∞], denote

rp_i(A) :=

i,...,im∈[n], δii_···im=

|aii···im|

p 

p

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In particular, if p= , then r

i(A) = ri(A) for all i ∈ [n]. If p =∞, then ri∞(A) = maxi,...,im∈[n],

δii···im=

|aii···im|for alli∈[n]. For a vectorx= (x,x, . . . ,xn)T∈Cn, thelq-norm on

Cn_is

xq:=

i∈[n] |xi|q



q .

Deﬁnition [] A tensorA= (aii···im)∈C[m,n]is diagonally dominant if

|aii···i| ≥ri(A) for alli∈[n], ()

andAis strictly diagonally dominant if the strict inequality holds in () for alli.

Remark  A= (aii···im)∈C

[m,n]_{is strictly diagonally dominant if and only if}

max i∈[n]

ri(A) |aii···i|

< .

It is well known that strictly diagonally dominant tensors are nonsingular. An interesting problem arises: for a tensorA= (aii···im)∈C

[m,n]_satisfying

max i∈[n]

r_ip(A)

|aii···i|

< ,

isAnonsingular or not? Certainly, whenp= ,Ais a strictly diagonally dominant tensor, which means thatAis nonsingular, but whenp> ,Amay be singular as the following simple example shows.

Example  LetA= (aijk)∈R[,], where

a=a= –, and the remaining aijk= .

Then, sinceAe_{= , where}_e_{= (, , )}T_{, this implies }_∈_σ₍_A_{). However, for every}_p_{> ,}

we have

max i∈[]

rp_i(A)

|aiii| = 

–p p _{< .}

Therefore, something needs to be added in order to obtain a nonsingularAfor a real

numberp∈(,∞]. We provide an answer further, but we ﬁrst introduce a class of strictly diagonally dominant tensors that involve a parameterpin the interval [,∞].

Deﬁnition  LetA= (aii···im)∈C[m,n]andp∈[,∞],Ais called ap-norm strictly diag-onally dominant tensor (or, shortly,p-norm SDD tensor) if

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where

δp(A) := (δ,δ, . . . ,δn)T, δi:=

rp_i(A)

|ai···i|



m–

for alli∈[n],

andqis Hölder’s complement ofp, that is,_p+_q= .

Remark  Deﬁnition  extends the concept ofSDD(p) matrix given in [] to tensors. Clearly, theSDD(p) matrix is a nd-orderp-norm SDD tensor.

Remark  Takingp= ,Ais a -norm SDD tensor if and only if

δ(A)_∞=max i∈[n]



m–

< ,

that is,

max i∈[n]

< ,

which is equivalent to the fact thatAis a strictly diagonally dominant tensor. The other extreme case isp=∞.Ais a∞-norm SDD tensor if and only if

δ_∞(A)_=

i∈[n]

maxi,...,im∈[n],

δii···im=

|aii···im|

|aii···i|



m–

< .

Thep-norm SDD tensors can also be characterized in the following way.

Proposition  LetA∈C[m,n]and p∈[,∞].ThenAis a p-norm SDD tensor if and only if there exists an entrywise positive vector x= (x,x, . . . ,xn)T∈Rnsuch thatxq≤,where

q is Hölder’s complement of p such that

x_im–|ai···i|>rip(A) for all i∈[n]. ()

Proof Necessity. Suppose thatAis ap-norm SDD tensor. It follows from inequality () of Deﬁnition  that there exists a suﬃciently smallε>  such that, forxi:=δi+ε> , where

i∈[n],xq≤. Thus,xm–

i >δim–= rp_i(A)

|ai···i|, which implies inequality ().

Suﬃciency. Suppose that there exists an entrywise positive vectorx>  such thatxq≤

 and inequality () holds. By inequality () we have

xm–_i >r

p i(A) |ai···i| =δ

m–

i for alli∈[n],

which impliesxi>δifor alli∈[n], which, together withxq≤, yields δp(A)_q<xq≤.

Thus,Ais ap-norm SDD tensor. The proof is completed.

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Theorem  LetA∈C[m,n]_{be a p-norm SDD tensor}_._Then_A_{is nonsingular}_.

Proof Suppose thatAis singular, that is, ∈σ(A). It follows from equality () that there existsy∈Cn_{\ {}__}_{, such that}

Aym–= . ()

Without the loss of generality, we can assume thatyq= . Then, equality () yields

ai···iym–i = –

i,...,im∈[n], δii···im=

aii···imyi· · ·yim for alli∈[n],

which implies that

|ai···i||yi|m–=

i,...,im∈[n], δii_···im=

aii···imyi· · ·yim

for alli∈[n]. ()

Then, applying the Hölder inequality to the right-hand side of equality (), we obtain

|ai···i||yi|m–≤

i,...,im∈[n],

δii···im=

|aii···im|p



p

i,...,im∈[n],

δii···im=

|yi· · ·yim|q



q

=rp_i(A)

_n

j= |yj|q

m–

–|yi|(m–)q



q

=rp_i(A)y(m–)q_q –|yi|(m–)q



q

≤rp_i(A)ym–_q

=rp_i(A) for alli∈[n]. ()

SinceAis ap-norm SDD tenor, there exists an entrywise positive vectorx>  such that

xq≤ and inequality () holds. Combining inequality () with (), we obtain

|ai···i||yi|m–≤rpi(A) <xm–i |ai···i| for alli∈[n],

which means that

|yi|<xi for alli∈[n].

Thus,xq>yq= , which contradictsxq≤. The proof is completed.

3 Relationships betweenp-norm SDD tensors and strongH-tensors

The following lemma shows that the strongH-tensors play an important role in identify-ing the positive deﬁniteness of even-order real symmetric tensors.

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It is known that the strictly diagonally dominant tensors are a subclass of strong

H-tensors. An interesting problem arises: whether the class ofp-norm SDD tensors is

a subclass of strongH-tensors for an arbitraryp∈[,∞]. In this section, we discuss this problem. We ﬁrst recall the deﬁnition of strongH-tensors.

Deﬁnition  [] A tenorA= (aii···im)∈R[m,n] is called anM-tensor if there exist a nonnegative tensorBand a positive real numberη≥ρ(B) such thatA=ηI-B. Ifη>ρ(B), thenAis called a strongM-tensor.

Deﬁnition [] LetA= (aii···im)∈C[m,n]. We call another tensorM(A) = (mii···im) the

comparison tensor ofAif

mii···im=

+|aii···im| if (i,i, . . . ,im) = (i,i, . . . ,i),

–|aii···im| if (i,i, . . . ,im)= (i,i, . . . ,i).

Deﬁnition [] We call a tensor anH-tensor if its comparison tensor is anM-tensor. We call it a strongH-tensor if its comparison tensor is a strongM-tensor.

Note that Liet al.[] also provided an equivalent deﬁnition of strongH-tensors; for details, see [].

In [], the multiplication of matrices has been extended to tensors. In the following, we state these results for reference.

Deﬁnition [] LetA= (aii···im) andB= (bii···ik) ben-dimensional tensors of orders m≥ andk≥, respectively. The productABis the followingn-dimensional tensorCof order (m– )(k– ) +  with entries

ciαα···αm–=

i,...,im∈[n]

aii···imbiα· · ·bimαm–,

wherei∈[n] andα, . . . ,αm–∈ {jj· · ·jk:jl∈[n],l= , , . . . ,k}.

Remark  Whenm=  andA= (aij) is a matrix of dimensionn, thenABis anmth-order

n-dimensional tensor, and we have

(AB)ii···im=

l∈[n]

ailbli···im, ij∈[n],j∈[m].

In particular, the product of a diagonal matrixX=diag(x,x, . . . ,xn) and the tensorAis

given by

(XA)ii···im=xiaii···im, ij∈[n],j∈[m].

Remark  Given ann-by-nmatrixXand twomth ordern-dimensional tensorsA,B, we have the right distributive law for tensors [], that is,

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Based on this multiplication of tensors, Kannan, Shaked-Monderer, and Berman [] established a necessary and suﬃcient condition for a tensor to be a strongH-tensor.

Lemma [] LetA∈C[m,n].ThenAis a strongH-tensor if and only if ai···i= for all

i∈[n]and

ρI–D–_M₍_A₎M(A)< ,

where D_M(A)is the diagonal matrix with the same diagonal entries asM(A).

The following lemma is given by Qi [].

Lemma [] LetA∈C[m,n]_and_B₌_a₍_A₊_b_I_),_{where a and b are two complex numbers}_.

Thenμis an eigenvalue ofBif and only ifμ=a(λ+b)andλis an eigenvalue ofA.In this case,they have the same eigenvectors.

Next, we present an equivalence condition for singular tensors.

Lemma  Let A∈C[m,n].Then Ais singular if and only if DAis singular, where D=

diag(d,d, . . . ,dn)is a positive diagonal matrix.

Proof Suppose thatDAis singular, that is, ∈σ(DA). Then there exists a vectorx= (x,x, . . . ,xn)T=  such that

i,...,im∈[n]

diaii···imxi· · ·xim=  for alli∈[n],

which is equivalent to

i,...,im∈[n]

aii···imxi· · ·xim=  for alli∈[n],

which implies ∈σ(A), that is,Ais singular. The proof is completed.

By applying Lemmas , , and , we can now reveal the relationship ofp-norm SDD

tensors and strongH-tensors.

Theorem  LetA∈C[m,n].IfAis a p-norm SDD tensor,thenAis a strongH-tensor.

Proof By Lemma  the theorem will be proved if we can show thatρ(I–D–

M(A)M(A)) < .

Assume, on the contrary, that there existsλ∈σ(I–D–

M(A)M(A)) such that|λ| ≥. Then,

by Lemma ,

∈σλI–I+D–_M₍_A)M(A).

According to the right distributive law for tensors, we have

λI–I+D–_M₍_A₎M(A) =D_M–₍_A₎(λ– )DM(A)I+M(A)

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which, together with Lemma , yields

∈σ(λ– )DM(A)I+M(A)

. ()

However, there exists an entrywise positive vectorx>  such thatxq≤, and inequality () holds becauseAis ap-norm SDD tensor. By inequality () we have

xm–_i (λ– )|ai···i|+|ai···i|=xm–i |λ||ai···i|

≥xm–_i |ai···i|

>rp_i(A)

=rp_i(λ– )DM(A)I+M(A)

,

which implies that (λ– )DM(A)I+M(A) is ap-norm SDD tensor. By Theorem  we have

 /∈σ(λ– )DM(A)I+M(A)

,

which contradicts (). The proof is completed.

4 Eigenvalue localization

Similarly to matrices, a nonsingular class of tensors can lead to an eigenvalue localization result. In this section, we illustrate this fact with the class ofp-norm SDD tensors.

Theorem  LetA∈C[m,n]and p∈[,∞].Then

σ(A)⊆p(A).

When p= ,(A) =(A).When p> ,

p(A) =

z∈C:

i∈[n]

r_ip(A)

|z–ai···i|

p

(m–)(p–)

≥

.

Proof Clearly, ifp= ,σ(A)⊆(A) can be easily obtained from Theorem . Ifp> , sup-pose that there existsλ∈σ(A) such thatλ∈/p(A), that is,

i∈[n]

r_ip(A)

|λ–ai···i|

p

(m–)(p–)

< . ()

LetB:=λI–A= (bii···im). Sinceλ∈σ(A), this, together with Lemma , yields thatB is surely singular. On the other hand, by the deﬁnition ofBwe obtainr_ip(B) =rp_i(A) and

|bi···i|=|λ–ai···i|for alli∈[n], so that () becomes

i∈[n]

r_ip(B)

|bi···i|

p

(m–)(p–)

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which implies

δp(B)_q< ,

whereq is Hölder’s complement ofp, which means thatBis ap-norm SDD tensor. By

Theorem ,Bis nonsingular. This leads to a contradiction.

Remark  Whenm= , Theorem  reduces to the result of [].

Remark  In particular, takingp=∞, we have

∞(A) =

z∈C:

i∈[n]

maxi,...,im∈[n],

δii···im=

|aii···im|

|z–ai···i|

 (m–)

≥

.

It follows from Theorem  that

σ(A)⊆p(A),

but since this conclusion holds for anyp∈[,∞], we immediately have the following the-orem.

Theorem  LetA∈C[m,n].Then

σ(A)⊆

p∈[,∞]

p(A),

wherep(A)is deﬁned as in Theorem.

The following example shows that∞(A) is tighter than(A) in some case.

Example  LetA= (aijk)∈R[,]andB= (bijk)∈R[,]with elements deﬁned as follows:

a= ., a= , and the remaining aijk= ,

b= , b= , and the remaining bijk= ,

respectively. The eigenvalue inclusion regions(A) ((B)),∞(A) (∞(B)) and the ex-act eigenvalues ofA(B) are drawn in Figure A (Figure B), where(A) ((B)),∞(A) (∞(B)) and the exact eigenvalues ofA(B) are respectively denoted by the blue area, the green area, and red asterisks. In addition, by Corollary . in [] we have

σ(A) ={., ., . + .i, . – .i}

and

σ(B) ={., ., . + .i, . – .i}.

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[image:11.595.116.480.79.241.2]

Figure 1 The eigenvalue inclusion sets of tensors

AandBfor Example 2.

5 Determining the positive (semi)deﬁniteness for an even-order real symmetric tensor

By applying the results obtained in Sections  and  we give a suﬃcient condition for the positive (semi)deﬁniteness of an even-order real symmetric tensor.

Theorem  LetA= (aii···im)∈R[m,n]be an even-order symmetric tensor with ai···i> for all i∈[n].IfAis a p-norm SDD tensor,thenAis positive deﬁnite.

Proof The theorem follows immediately from Lemma  and Theorem .

Theorem  LetA= (aii···im)∈R[m,n]be an even-order symmetric tensor with ai···i> for

all i∈[n],and p∈[,∞].If

δp(A)_q≤,

where q is Hölder’s complement of p.ThenAis positive semideﬁnite.

Proof Ifp= , then

δ(A)_∞=max i∈[n]



m–

≤,

which implies thatAis diagonally dominant. By Theorem  of [] it follows thatAis

positive semideﬁnite.

Ifp> , then letλbe an H-eigenvalue ofA, andλ< . By Theorem  we haveλ∈p(A), which implies that

i∈[n]

r_ip(A)

|λ–ai···i|

p

(m–)(p–)

≥.

However, it follows fromaii···i>  for alli∈[n] that

i∈[n]

r_ip(A)

|ai···i|

p

(m–)(p–)

>

i∈[n]

r_ip(A)

|λ–ai···i|

p

(m–)(p–)

(12)

which implies

δp(A)_q> ,

which contradictsδp(A)q≤. This completes the proof.

Example  LetA∈R[,]_{be a symmetric tensor with elements deﬁned as follows:}

a= , a=a=a=a= –, a= ,

and the remainingaiiii= . By computation,

|a|=  <  =r 

 (A) =

(i,i,i)∈ ,

δi_i_i_=

|aiii|,

which means that the statement (I) of Proposition . in [] does not hold, and hence we cannot use Proposition . in [] to determine the positive deﬁniteness ofA. However, it is easy to verify thatAis a∞-norm SDD tensor. By Theorem ,Ais positive deﬁnite.

6 Conclusions

In this paper, we proposed a new class of nonsingular tensors (p-norm SDD tensors) and proved that the class ofp-norm SDD tensors is a subclass of strongH-tensors. Further-more, we presented a new eigenvalue inclusion set, which is tighter than those provided by Liet al.[] in some case. Based on this set, we presented a checkable suﬃcient condition for the positive (semi)deﬁniteness of an even-order symmetric tensor.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This work was supported by the National Natural Science Foundations of China (11361074, 11501141), Natural Science Foundations of Yunnan Province (2013FD002) and Foundation of Guizhou Science and Technology Department ([2015]2073).

Received: 1 May 2016 Accepted: 23 June 2016

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