一般矩阵函数的变差_英文_

一般矩阵函数的变差_英文_ ------------------------------------------------------------------------------------------------ 一般矩阵函数的变差_英文_ 第23卷第5期 2007年10月大 学 数 学 COLLEGEMATHEMATICSVol.23,l.5Oct.2007 VariationofGeneralMatrixFunction LIUXiu-sheng (SchoolofMath.andPhysics,HuangshiInstituteofTechnology,Huangshi,Hubei435003,China) Abstract:LetSndenotethesymmetricgroupofdegreenandletGbeasubgroupofSn,andletVbeacharacterofdegree1onG.IfAisannbyncomplexmatrix,wedefinethegeneralizedmatrixfunctiondGVbyd(A)=GV RIGEV(R)i=1FaniR(i).Itisthenusethelp-operatornorm((1[p[])toobtaintwoextendinequalities inthevariationforgeneralmatrixfunction. Keywords:lp-operatornorm;tensorproduct;nonnegativematrix;generalmatrixfunction;projecture CLCNumber:O151.21 DocumentCode:C —————————————————————————————————————— ------------------------------------------------------------------------------------------------ ArticleID:1672-1454(2007)05-0134-03 1 Introduction In[1],[2]and[3],FriedlandS.,BhatiaR.andLiuXiu-shengprovedforanytwomatricesA,Bonthesetn,nofnbyncomplexmatricestheinequalities |det(A)-det(B)|[n+A-B+max(+A+,+B+)n-1, 1[p[],(1.1) (1.2) (1.3)|per(A)-per(B)|[n+A-B+max(+A+,+B+)n-1, 1[p[],andGn-1|dGV(A)-dV(B)|[n+A-B+max(+A+,+B+), p=1,] holdsrespectively.Here+#+=+#+p,thelp-operatornorm. Thepurposeofthisnoteistoproveanimprovesresultforthegeneralmatrixfunctionandananalogousresultforthegeneralnonnegativematrixfunction.Werecallthefollowing: LetSndenotethesymmetricgroupofdegreenandletGbeasubgroupofSnoforderg.LetVbeacharacterofdegree1onG,i.e.,anontrivialhomomorphismofGintothecomplexnumbers.IfAisannbyncomplexmatrix,wedefinethegeneralizedmatrixfunctiondGVby d(A)=GV RIGnEV(R)i=1FaiR(i).(1.4) —————————————————————————————————————— ------------------------------------------------------------------------------------------------ Thisgeneralmatrixfunctioninducesthepermanent(G=Sn,V(R)=1),thedeterminant(G=Sn,V(R)=signR),andotherassortedinterestingfunction Forx=(x1,,,xn)I i=1n[4].and1[p[]wedefineasusualthelp-operatornorm+x+p=En|xi| pandtheassociatedoperatornorm +A+p=max{+Ax+pB+x+p[1}(1.5) Receiveddate:2005-12-19 Foundationitem:TheNSFofHubeiEducationCommittee(No:2004X157) 第5期 LIUXiu-sheng:VariationofGeneralMatrixFunction135 2 TheoremandProof Westartwithfourlemmas. Lemma1[2] ForAiIni,ni,i=1,2,,,s, +A1á,áAs+p= LetAI n,n n i=1 F s +Ai+p.(2.1) ,anddenotethenthtensorpowerofAbyánA,i.e., —————————————————————————————————————— ------------------------------------------------------------------------------------------------ áA=n-times Lemma2 [2] ForA,BIn,n +ánA-ánB+p[n+A-B+max(+A++B+)n-1. (2.2) Lemma3 IfVisanycharacterofafinitegroupG,then|V(R)|[V(e)foreachRIG.Lemma 4 LetA,BIn,nandA,B,A-Bbenonnegativematrices.Then ánA-ánB isalsoanonnegativematrix. Proof LetCi=á(A-B)áNotethatA,B,A-Barenonnegativematrices, n-i n i-1 n [5] thenCiisanonnegativematrix.HenceáA-áB= Theorem1 LetA,BI n,n i=1 E —————————————————————————————————————— ------------------------------------------------------------------------------------------------ n Ciisalsoanonnegativematrix. .Then p Gn-1 |dGV(A)-dV(B)|[n##+A-B+max(+A+,+B+). (2.3) for+#+=+#+p,thelp-operatornorm(1[p[]). Proof LetA,Barefinitesequenceincomponentwithnaturalnumber.WerecallthatIfR IG,AI#n,n #n,n={A|A=(A(1),,,A(n)),1[A(i)[n,i=1,2,,,n}andR(i)=A(i),i=1,2,,,n,thenwedef ineR=A. KAB FaA(i)B(i) i=1n WeintroducetheprojecturePGby PG(áA)= n AI#n,nBI#n,n . —————————————————————————————————————— ------------------------------------------------------------------------------------------------ B=1orelseKAB=0.WhereifBIGandA=(1,2,,,n),thenKA LetB1,B2,,,Bj=(1,2,,,n),,,B#n,nbesequenceof#n,ninlexicographicorderandl etbiIG,thenbB=V(Bi)orelsebB=0,i=1,2,,=(bB1,bB2,,,bB#n,n).HereifB,#n,n.S upposeii T e=(0,,,eBj=1,,,0),itiseasytoseethat [PG(áA)]b=dV(A)e andobviously +[PG(ánA)]b+p[+(ánA)b+p. Hence,weget Gnnnn |dGV(A)-dV(B)|+e+p=+[PG(áA-áB)]b+p[+(áA-áB)b+p n G [+áA-áB+p+b+p=+áA-áB+p n n n n RIG E —————————————————————————————————————— ------------------------------------------------------------------------------------------------ |V(R)| p [+áA- áB+p n n RIG E V(e) p p =+ánA-ánB+p. p Thistogetherwith(2.2)gives(2.3). Theorem2 LetA,BI GV n,n andA,B,A-Bbenonnegativematrices.Then G n-1 |d(A)-dV(B)|[n+A-B+max(+A+,+B+) plpfo[p(2.4) —————————————————————————————————————— ------------------------------------------------------------------------------------------------ 136大 学 数 学 第23卷Proof WeintroducetheprojecturePGby PG(áA)=WhereifA,BIGandnKABFaA(i)B(i)i=1nAI#n,nBI#n,n. i=1FanA(i)B(i)=i=1FaniBc(i),thenKAB=V(Bc)orelseKAB=0. TLetB1,B2,,,B#n,nbesequenceof#n,ninlexicographicorderandletb=(bB1,bB2,,,bB#n,n).Here ifBiIG,thenbB=1orelsebB=0,i=1,2,,,#n,n.Itiseasytoseethatii [PG(ánA)b]=dGV(A)b andobviously +[PG(ánA)b]+p[+[(ánA)b]+p Hence,weget |dV(A)-dV(B)|+b+p=+[PG(áA-áB)]b+p[+(áA-áB)b]+p [+áA-áB+p+b+p. Thistogetherwith(2.2)gives(2.4). [References] [1] FriedlandS.Variationoftensorpowersandspectra[J].Lin.Multilin.Alg.1982,12:81-98. [2] BhatiaRandElsnerL.Onthevariationofpermanent[J].Lin.Multilin.Alg.,1990,27:105-110. —————————————————————————————————————— ------------------------------------------------------------------------------------------------ [3] LiuXiu-sheng.Inequalitiesonthevariationforgeneralmatrixfunctions[J].Act aMathematiaScientia,2004, 24A(5):623-640. [4] MarcusMandCholletJ.Constructionoforthonormalbasesinhighersymmetr yclassesoftensoes[J].Lin. Multilin.Alg.1986,19:105-110. [5] WangBo-ying.Multilinearalgebra[M].Beijing:BeijingNormalUniversityPres s,1985.nnGGnnnn 一般矩阵函数的变差 刘修生 (黄石理工学院数理学院,湖北黄石435003) [摘 要]设Sn是n次对称群,G为Sn的子群,V是G的次数为1的 特征标.如果A是一个n阶复变矩阵,定义一般矩阵函数dGV为 d(A)=GVn RIGEV(R)Fai=1iR(i). 本文用lp-算子范数(1[p[])的性质证明了一般矩阵函数变差的两 个不等式. [关键词]lp-算子范数;张量积;非负矩阵;一般矩阵函数;投影 ——————————————————————————————————————