非奇异H-张量的新判定.pdf

Chin.Quart.J.of Math.2023,38(2):123133New Criteria for Nonsingular H-TensorsLIU Liang,WANG Ya-qiang(School of Mathematics and Information Science,Baoji University of Arts and Sciences,Baoji 721013,China)Abstract:H-tensor plays an important role in identifying positive definiteness of evenorder real symmetric tensors.In this paper,some definitions and theorems related toH-tensors are introduced firstly.Secondly,some new criteria for identifying nonsingularH-tensors are proposed,moreover,a new theorem for identifying positive definitenessof even order real symmetric tensors is obtained.Finally,some numerical examples aregiven to illustrate our results.Keywords:H-tensors;Generalized diagonally dominant;M-tensors2000 MR Subject Classification:15A39,15A69,15A57CLC number:O151.2Document code:AArticle ID:1002-0462(2023)02-0123-11DOI:10.13371/ki.chin.q.j.m.2023.02.0021.IntroductionLetC(R)be the complex(real)field andN=1,2,.,n.A tensorA=(ai1i2im)Cm,nis called a complex(real)ordermdimensionntensor,ifai1i2imC(R),whereijNforj=1,2,.,m.Ifn1=n2=.nm=n,thenAis called a square tensor,otherwise it is called arectangular tensor.For a tensorA=(ai1i2im)Cm,nand vectorx=(x1,x2,.,xn)TCn,Axm1is a vectorin Cn,whose ith component is(Axm1)i=Xi2,.,imNaii2imxi2xim,ijN,j=2,3,.,m.A tensorI=(i1i2im)Cm,n(m,n2)is called the identity tensor,if its elements satisfyi1i2im=1,i1=i2=im,0,otherwise.Received date:2022-06-26Foundation item:This work is partly supported by the National Natural Science Foundations of China(GrantNo.31600299);The Natural Science Foundation of Shaanxi province(Grant No.2020JM-622).Biographies:LIU Liang(1997-),male,native of Baoji,Shaanxi,graduate student of Baoji University of Artsand Sciences,engages in numerical algebra;WANG Ya-qiang(1983-),male,native of Baoji,Shaanxi,associateprofessor of Baoji University of Arts and Sciences,engages in numerical algebra.Corresponding author WANGYa-qiang:.123124CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38For a realm-ordern-dimensional tensorAand a scalarC,if there exists nonzero vectorxCnsuch thatAxm1=xm1,wherexm1Cnwith(xm1)ixm1i,thenis said to be an eigenvalue of tensorAandxan eigenvector associated with eigenvalue.In particular,xis real,thenis also real,and(;x)is said to be anH-eigenvalue pair of tensorA.The largest modulus of eigenvalue of tensorAiscalled the spectral radius of tensorAand denotes it by(A).Motivated by the characteristicsof nonsingular matrices and say a square tensor is nonsingular if its all eigenvalues are nonzero.An m-th degree homogeneous polynomial of n variables f(x)can be denoted asf(x)=Xi1,.,imNai1i2imxi1xi2xim,(1.1)wherexRn.The homogeneous polynomialf(x)in(1.1)can be expressed as the tensor productof a symmetric tensor A with order m dimension n and xmdefined byf(x)Axm=Xi1,.,imNai1i2imxi1xi2xim,where x=(x1,x2,.,xn)Rn.When m is even,f(x)is called positive definite iff(x)0,for any xRn,x6=0.The positive definiteness of multivariate polynomialf(x)plays an important role in thestability study of nonlinear autonomous systems.However,it is not easy to identify the positivedefiniteness of such a multivariate form.In 2007,Professor Qi showed thatf(x)is positivedefinite if and only if the real even-order symmetric tensorAis positive definite 9.In 2014,Li et al.showed that if a real even-order symmetric tensorAwith|aiii|ofor alliNis anH-tensor,thenAis positive definite 6.Therefore,for a given tensor,a interestingquestion arises as to whether it is anH-tensor or not.Many criteria for judgingH-tensorshave been widely proposed,see 15,7,8,1016.In this paper,we still focus on the problemof determining theH-tensor,and some new criteria in identifying nonsingularH-tensors areestablished.Moreover,a new theorem for identifying positive definiteness of even order realsymmetric tensors is obtained.Throughout this paper,we will use the following definitions.Definition 1.1.6 Let tensorACm,n,its comparison tensor denoted byMA,is defined asMA=|ai1i2im|,if i1=i2=im,|ai1i2im|,otherwise.Definition 1.2.6 Let tensorACm,n,Ais said to be aZ-tensor if it can be written asA=cIB,wherec0 andBis a nonnegative tensor.Furthermore,ifc(B),thenAis saidto be an M-tensor,and if c(B),then A is said to be a nonsingular M-tensor.No.2LIU Liang et al:New Criteria for Nonsingular H-Tensors125Definition 1.3.6 Let tensorACm,n,if comparison tensorMAof tensorAis anM-tensor,then tensorAis called anH-tensor,and if comparison tensorMAis a nonsingularM-tensor,then tensor A is called a nonsingular H-tensor.Definition 1.4.10 Let ACm,n,A is called diagonally dominant if|aiii|Xi2imNii2im=0|aii2im|,iN,(1.2)and tensorAis called strictly diagonally dominant if all the inequalities hold with strict inequality.Definition 1.5.8 LetACm,n,Ais said to be generalized strictly diagonally dominant ifthere exists positive diagonal matrix D such that ADm1is strictly diagonally dominant.2.Criteria for nonsingular H-tensorsIn this section,some new criteria forH-tensors are given as follows.Before that somenotations and lemmas are given firstly.LetSbe a subset ofNandS=NS,=i2i3im:ikS,k=2,3,.,m,=i2i3im:ikS,k=2,3,.,m,based on the sets we denote thatri(A)=Xi2im6=iii2im|aii2im|,ri(A)=Xi2im6=iii2im|aii2im|,then ri(A)=ri(A)+ri(A).Obviously,whenS=N,then|aiii|ri(A)0,we obtain tensorAis aH-tensor,therefore,we always assume that S6=and S6=N.Lemma 2.1.6 TensorAis a nonsingularH-tensor if and only ifAis generalized strictlydiagonally dominant.Lemma 2.2.6 For square tensorA,if there exists a positively diagonal matrixDsuch thatADm1is a nonsingular H-tensor,then A is a nonsingular H-tensor.Lemma 2.3.6 If tensorAis irreducible and diagonally dominant with at least one strictinequality holding in(1.2),then it is generalized diagonally dominant.Lemma 2.4.12 LetACm,nbe an even order real symmetric tensor,andaiii0 for alliN,if A is an H-tensor,then A is positive definite.Theorem 2.1.For tensorA=(ai1i2im)Cm,n,if there exists a partition(S;S)of the indexset N such that|appp|rp(A)0,pS,(2.1)|aqqq|rq(A)0,qS,(2.2)126CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1,and if 0,12,then A is a nonsingular H-tensor.Proof.From(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1,then(|appp|rp(A)rp(A)(rq(A)|aqqq|rq(A)1.From the inequality(2.1),we obtain|appp|rp(A)rp(A)1.If 0,12 holds,then(|appp|rp(A)rp(A)1(|appp|rp(A)rp(A)(rq(A)|aqqq|rq(A)1,so|appp|rp(A)rp(A)rq(A)|aqqq|rq(A).Hence the paper defines the following positive diagonal matrixD=diag(d1,dn)withdiagonal entriesdi=1,if iS,d,if iS,where d1 is such that|appp|rp(A)rp(A)dm1,pS,dm1rq(A)|aqqq|rq(A),qS.Let maxpS(|appp|rp(A)rp(A)=P,minqS(rq(A)|aqqq|rq(A)=Q,thenQdm10,thenrp(B)=rp(B)+rp(B)rp(A)+dm1rp(A)rp(A)+|appp|rp(A)rp(A)rp(A)=|appp|=|bppp|.If rp(A)=0,then from the inequality(2.1),rp(B)=rp(B)+rp(B)rp(A)+dm1rp(A)=rp(A)rq(A)|aqqq|rq(A)(|aqqq|rp(A)rq(A)=0.This means that tensorADm1is strictly diagonally dominant,andAis generalized strictlydiagonally dominant,hence it is a nonsingular H-tensor by Lemma 2.1.Theorem 2.2.For irreducible tensorA=(ai1i2im)Cm,n,if there exists a partition(S;S)of the index set N such that|appp|rp(A)0,pS,(2.3)|aqqq|rq(A)0,qS,(2.4)(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1,(2.5)in addition,the strict inequality holds for at least onep0Sandq0S,and if0,12,thenAis a nonsingular H-tensor.Proof.From(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1,then(|appp|rp(A)rp(A)(rq(A)|aqqq|rq(A)1.128CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38From the inequality(2.3),we obtain|appp|rp(A)rp(A)1.If 0,12 holds,then(|appp|rp(A)rp(A)1(|appp|rp(A)rp(A)(rq(A)|aqqq|rq(A)1.So|appp|rp(A)rp(A)rq(A)|aqqq|rq(A),and|ap0p0p0|rp0(A)rp0(A)rq0(A)|aq0q0q0|rq0(A).Hence the paper defines the following positive diagonal matrixD=diag(d1,dn)withdiagonal entriesdi=1,if iS,d,if iS,where d1 is such that|appp|rp(A)rp(A)dm1,pS,dm1rq(A)|aqqq|rq(A),qS.Let maxpS(|appp|rp(A)rp(A)=P,minqS(rq(A)|aqqq|rq(A)=Q,thenQdm1P.Now,consider tensorB=ADm1.TheBremains irreducible asDis positively diagonal.Itis easy to see that for any iN,ri(B)=Xi2im6=iii2im|bii2im|=Xi2im6=iii2im|aii2im|=ri(A),ri(B)=Xi2im6=iii2im|bii2im|dm1ri(A),and|bppp|=|appp|,|bqqq|=dm1|aqqq|.Thus for p0S,if rp0(A)0,thenrp0(B)=rp0(B)+rp0(B)rp0(A)+dm1rp0(A)No.2LIU Liang et al:New Criteria for Nonsingular H-Tensors129rp0(A)+|ap0p0p0|rp0(A)rp0(A)rp0(A)=|ap0p0p0|.If rp0(A)=0,then from the inequality(2.5),rp0(B)=rp0(B)+rp0(B)rp0(A)+dm1rp(A)=rp0(A)0,thenrp(B)=rp(B)+rp(B)rp(A)+dm1rp(A)rp(A)+|appp|rp(A)rp(A)rp(A)=|appp|=|bppp|.If rp(A)=0,then from the inequality(2.7),rp(B)=rp(B)+rp(B)rp(A)+dm1rp(A)=rp(A)|appp|=|bppp|.For qS,from the inequality(2.4),we obtain|bqqq|rq(B)=dm1|aqqq|rq(B)rq(B)dm1|aqqq|rq(A)dm1rq(A)=dm1(|aqqq|rp(A)rq(A)rq(A)|aqqq|rq(A)(|aqqq|rp(A)rq(A)=0.Thus,tensorADm1is diagonally dominant with at least one strict inequality.SinceADm1is irreducible,and also know thatADm1is generalized diagonally dominant by Lemma2.3.So A is generalized diagonally dominant and it is a nonsingular H-tensor.Based on the criteria forH-tensors,it is easy to get a new condition for the positivedefiniteness of an even order real symmetric tensor.Theorem 2.3.LetA=(ai1i2im)be an even-order real symmetric tensor with ordermanddimensionn,andaiii0,for alliN,ifAsatisfies one of the following conditions,thenAis positive definite:(1)all of the conditions of Theorem 2.1.(2)all of the conditions of Theorem 2.2.3.ExamplesExample 3.1.Consider a 4 order 4 dimensional tensor A with entriesa1111=a2222=a3333=a4444=3,a1222=13,a2111=a4111=a4222=2,a2444=43,130CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38and all other entries are zeros and let S=(1,3),S=(2,4)and=13.By calculation,we haver1(A)=13,r3(A)=0,r2(A)=43,r4(A)=2,|a1111|r1(A)=830,|a3333|r3(A)=30,|a2222|r2(A)=530,|a4444|r4(A)=10.For p=1 and q=2,(|appp|rp(A)(|aqqq|rq(A)1=(|a1111|r1(A)13(|a2222|r2(A)23=313(53)23,(rp(A)(rq(A)1=(r1(A)13(r2(A)23=(13)13223,we get(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.For p=1 and q=4,(|appp|rp(A)(|aqqq|rq(A)1=(|a1111|r1(A)13(|a4444|r4(A)23=313,(rp(A)(rq(A)1=(r1(A)13(r4(A)23=(13)13223,we get(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.For p=3 and q=2,(|appp|rp(A)(|aqqq|rq(A)1=(|a3333|r3(A)13(|a2222|r2(A)23=313(53)23,(rp(A)(rq(A)1=(r3(A)13(r2(A)23=0,we get(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.For p=3,q=4,(|appp|rp(A)(|aqqq|rq(A)1=(|a3333|r3(A)13(|a4444|r4(A)23=313,(rp(A)(rq(A)1=(r3(A)13(r4(A)23=0,we get(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.From the above calculation and Theorem 2.1,it easy to conclude thatAis a nonsingularH-tensor.Example 3.2.Consider a 4 order 4 dimensional tensor A with entriesa1111=a2222=a3333=a4444=10,a1333=10,a3111=a3222=a2444=a4222=5,No.2LIU Liang et al:New Criteria for Nonsingular H-Tensors131and all other entries are zeros and let S=(1,3),S=(2,4)and=13.By calculation,we haver1(A)=10,r3(A)=10,r2(A)=5,r4(A)=5,|a1111|r1(A)=0,|a3333|r3(A)=0,|a2222|r2(A)=50,|a4444|r4(A)=50.For p=1 and q=2,(|appp|rp(A)(|aqqq|rq(A)1=(|a1111|r1(A)13(|a2222|r2(A)23=013523=0,(rp(A)(rq(A)1=(r1(A)13(r2(A)23=0,we have(|appp|rp(A)(|aqqq|rq(A)1=(rp(A)(rq(A)1.For p=1 and q=4,(|appp|rp(A)(|aqqq|rq(A)1=(|a1111|r1(A)13(|a4444|r4(A)23=013523=0,(rp(A)(rq(A)1=(r1(A)13(r4(A)23=0,we have(|appp|rp(A)(|aqqq|rq(A)1=(rp(A)(rq(A)1.For p=3 and q=2,(|appp|rp(A)(|aqqq|rq(A)1=(|a3333|r3(A)13(|a2222|r2(A)23=513523=5,(rp(A)(rq(A)1=(r3(A)13(r2(A)23=513023,we have(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.For p=3 and q=4,(|appp|rp(A)(|aqqq|rq(A)1=(|a3333|r3(A)13(|a4444|r4(A)23=513523=5,(rp(A)(rq(A)1=(r3(A)13(r4(A)23=0,we have(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.From Theorem 2.2,it easy to obtain that A is a nonsingular H-tensor.Example 3.3.Letf(x)=Ax4=10 x41+10 x42+10 x43+4x1x32+4x1x33+4x31x2+4x31x3be a 4th-degree homogeneous polynomial,wherex=(x1,x2,x3)R.It is easy to get an 4-order 3-dimensional symmetric tensor A with entriesa1111=a2222=a3333=10,a1222=a2122=a2212=a2221=1,132CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38a1333=a3133=a3313=a3331=1,a2111=a1211=a1121=a1112=1,a3111=a1311=a1131=a1113=1,and all other entries are zeros and let S=(1),S=(2,3)and=13.By calculation,we haver1(A)=8,r1(A)=0,r1(A)=8,r2(A)=4,r2(A)=1,r2(A)=3,r3(A)=4,r3(A)=1,r3(A)=3,|a1111|r1(A)=20,|a2222|r2(A)=70,|a3333|r3(A)=70.For p=1 and q=2,(|appp|rp(A)(|aqqq|rq(A)1=(|a1111|r1(A)13(|a2222|r2(A)23=1013723,(rp(A)(rq(A)1=(r1(A)13(r2(A)23=813123,we have(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.For p=1 and q=3,(|appp|rp(A)(|aqqq|rq(A)1=(|a1111|r1(A)13(|a3333|r3(A)23=1013723,(rp(A)(rq(A)1=(r1(A)13(r3(A)23=813123,we have(|appp|rp(A)(|aqqq|rq(A)1(rp(A)(rq(A)1.It showed thatAsatisfies all the condition of Theorem 2.1.Thus,from Theorem 2.3,weobtain that A is positive definite,that is f(x)is positive definite.AcknowledgementsThe authors are grateful to the referee for careful reading of the paper and valuablesuggestions and comments.References1CUI J J,PENG G H,LU Q,HUANG Z G.New iterative criteria for strongH-tensors and an 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