on an eigenvalue inequality involving the hadamard product 2017 fumio hiai参考.pdf

该【on an eigenvalue inequality involving the hadamard product 2017 fumio hiai参考 】是由【小舍儿】上传分享，文档一共【10】页，该文档可以免费在线阅读，需要了解更多关于【on an eigenvalue inequality involving the hadamard product 2017 fumio hiai参考 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。AcceptedManuscriptOnaneigenvalueinequalityinvolvingtheHadamardproductFumioHiai,MinghuaLinPII:S0024-3795(16)30543-2DOI:http://dx./.:LAA13934Toappearin:LinearAlgebraanditsApplicationsReceiveddate:10October2016Accepteddate:12November2016Pleasecitethisarticleinpressas:,,OnaneigenvalueinequalityinvolvingtheHadamardproduct,LinearAlgebraAppl.(2017),http://dx./.,typesetting,andreviewoftheresultingproofbeforeitispublishedinits?,$FumioHiai1TohokuUniversity(Emeritus),Hakusan3-8-16-303,Abiko270-1154,JapanMinghuaLin2DepartmentofMathematics,ShanghaiUniversity,Shanghai,200444,ChinaAbstractLetA,Bben×npositivede?≤t≤1nnnλi(A?B)≥λi((AtB)(A1?tB))≥λi(AB),k=1,...,=ki=ki=kThisgivesaweightedextensionofaresultofAndo[1].:inequality,majorization,eignevalue,singularvalue2010MSC:15A45,×:(A)(A)istheithlargestsingularvalueofA.$:hiai.******@2Correspondingauthorat:DepartmentofMathematics,ShanghaiUniversity,Shanghai200444,:******@.;URL:./en/mlinPreprintsubmittedtoLinearAlgebraanditsApplicationsNovember23,??tBistheweightedgeometricmean,.,1/2?1/2?1/2t1/2AtB=A(ABA)A,0≤t≤1forpositivede?nitematricesAandB;whent=1/2,wesimplyputABforA1/≥B(orB≤A)meansthatA?Bispositivesemide?.AistheoperatornormofA,.,A=σ1(A).??,BapatandSunder[3,Theorem3]provednnλi(A?B)≥λi(A)λi(B),k=1,...,n.(1)i=ki=kThissettlesaconjectureraisedbyMarshallandOlkin[9,].TheHorntheorem[9,]saysthatnnλi(AB)≥λi(A)λi(B),k=1,...,=ki=kWiththisinmind,BapatandJohnson[8]conjecturedastrongerinequalitythan(1)nnλi(A?B)≥λi(AB),k=1,...,n.(2)i=ki=kInequality(2)wasindependentlycon?rmedbyAndo[1]andVisick[10].Asitisknown[2]thatkk2λi(AB)≤λi(AB),k=1,...,ni=1i=1whichisequivalenttonn2λi(AB)≥λi(AB),k=1,...,n,(3)i=ki=k2Ando[1,Theorem3]provedthefollowingstrengtheningof(2)nn2λi(A?B)≥λi(AB),k=1,...,n.(4)i=ki=kThewaythatAndoproved(2)isoriginalandhisproofof(4)reliesonaproofof(2)?,wepresentaweightedextensionof(3)and(4).Ourideaofproofisdi?erentfromthatofAndo’s[1]sinceourexcursionstartswiththeweakestinequality(1)andthenderives(4)sothat(2),Bbepositivede?(1?t)A+tB≥A-geometricmeaninequality;see[5,].TheproofofnextlemmaheavilyusesbasicoperationsoftheKroneckerproduct([4,]).,Bbepositivede?≤t≤1(AtB)?(A1?tB)≤A??BisaprincipalsubmatrixofA?B,itsu?cestoshow(AtB)?(A1?tB)≤(1?t)A?B+tB?A.(5)3Compute(AtB)?(A1?tB)=(AtB)?(BtA)=(A1/2(A?1/2BA?1/2)tA1/2)?(B1/2(B?1/2AB?1/2)tB1/2)=(A1/2?B1/2)(A?1/2BA?1/2)t?(B?1/2AB?1/2)t(A1/2?B1/2)t=(A?B)1/2(A?1/2BA?1/2)?(B?1/2AB?1/2)(A?B)1/2t=(A?B)1/2(A?B)?1/2(B?A)(A?B)?1/2(A?B)1/2=(A?B)t(B?A).Hence,(5)[11,Corollary10]whichsays(AtB)?(A1?tB)+(A1?tB)?(AtB)≤A?B+B?A.(6)However,(5)isclearlystrongerthan(6).Toillustratethedi?erenceofourproofideafrom[1],weuseBapatandSunder’sinequality(1)toderiveAndo’sinequality(4).Taket=1/2,byamonotonicitypropertyfortheeigenvaluesofHermitianmatrices[7,],(A?B)≥λi((AB)?(AB)),i=1,...,n,andhencennλi(A?B)≥λi((AB)?(AB)),k=1,...,n.(7)i=ki=kNowby(1),weobtainnn2λi((AB)?(AB))≥λi(AB),k=1,...,n.(8)i=ki=kAndo’sinequality(4)followsfrom(7)and(8).,BbeHermitianmatricessuchthatA≥?nitematrixC,λi(AC)≥λi(BC),i=1,...,≥C1/2BC1/2,itfollows1/21/21/21/2λi(AC)=λi(CAC)≥λi(CBC)=λi(BC)fori=1,...,-valuesarereplacedwithsingularvaluesevenifBispositivede?,we?rst?ndanexamplesuchthatA≥B>0butthatA2?B2isnotpositivesemide?nite(thiscanbeeasilydone;.,[4,]).Thatmeanssuchthatv?A2v<v?=vv?toseethat√√2?2?22σ1(AC)=λ1(CAC)=vAv<vBv=λ1(CBC)=σ1(BC).Ourmainresultisthefollowingtheorem,whichisaweightedextensionof(3)and(4).,Bbepositivede?≤t≤1nnnλi(A?B)≥λi((AtB)(A1?tB))≥λi(AB),k=1,...,n.(9)i=ki=ki=,wegetλi(A?B)≥λi((AtB)?(A1?tB)),i=1,...,n,andsonnλi(A?B)≥λi((AtB)?(A1?tB)),k=1,...,n.(10)i=ki=kThe?rstinequalityof(9)nowfollowsby(10)and(2).5Weproceedtoprovethesecondinequalityof(9).Asnλi((AtB)(A1?tB))=det((AtB)(A1?tB))i=1=det(AtB)det(A1?tB)=(detA)1?t(detB)t(detA)t(detB)1?tn=(detA)(detB)=detAB=λi(AB),i=1Thesecondinequalityof(9)isequivalenttokkλi(AB)≥λi((AtB)(A1?tB)),k=1,...,n.(11)i=1i=1Usingastandardargumentviatheanti-symmetrictensorpower(.,[4,]),toshow(11)itsu?cestoshowλ1(AB)≥λ1((AtB)(A1?tB)).Thiswouldfollowoncethefollowingimplicationisveri?edλ1(AB)≤1=?λ1((AtB)(A1?tB))≤1.(12)Therestisdevotedtoaproofof(12).Itisknownthatλ1(AB)≤1isequivalentto0≤B≤A?puteλ1((AtB)(A1?tB))1/2?1/2?1/2t1/21/2?1/2?1/21?t1/2=λ1(A(ABA)A)(A(ABA)A)?1/2?1/2t?1/2?1/21?t=λ1(A(ABA)A)(ABA)?1/2?1/2t?1/2?1?1/21?t≤λ1(A(ABA)A)(AAA)?1/2?1/2t2t=λ1(ABA)A?1/2?1?1/2t2t≤λ1(AAA)A=1,-Heinzinequality([4,])(.,[4,])that(11)impliesthefollowingtraceinequalitytr(AB)≥tr((AtB)(A1?tB)),whichwasrecentlyobservedbyBhatia,LimandYamazakiin[6,()].nλAB≥nσABk,...,nλA?BσA?BAsi=ki()i=ki(),=1,andi()=i()forpositivede?nitematricesA,B,itfollowsfromthe?rstinequalityof(9)thatnnσi(A?B)≥σi((AtB)(A1?tB)),k=1,...,=ki=kItisnaturaltoaskwhetheronecouldreplaceeigenvalueswithsingularvaluesinthesecondinequalityof(9).,Bbepositivede?≤t≤1nnσi((AtB)(A1?tB))≥σi(AB),k=1,...,=ki=kEquivalently,thefollowingnorminequalityholdsAB≥(AtB)(A1?tB).≤t≤3/,Cbepositivede?ACA≤A3/2CA1/2.3/21/23/21/ACA=λ1(ACA)=λ1(ACA)≤ACA.,Cbepositivede?≤t≤3/4A1/2CtAC1?tA1/2≤ACA.?edACA≤1=?A1/2CtAC1?tA1/2≤,ACA≤1impliesC≤A?2andsoA2≤C?(A1/2CtAC1?tA1/2)?A1/2CtAC1?tA1/2=A1/2CtAC1?tAC1?tACtA1/2≤A1/2CtAC1?tC?1/2C1?tACtA1/2=A1/2CtAC3/2?2tACtA1/2≤A1/2CtAA?3+4tACtA1/2as0≤3/2?2t≤1=A1/2Ct(A2)?1/2+2tCtA1/2≤A1/2CtC1/2?2tCtA1/2as0≤?1/2+2t≤1=A1/2C1/2A1/2≤,A1/2CtAC1?tA1/2≤=A?1/2BA?1/≤t≤3/4,?t1/23/21/2(AtB)(A1?tB)=ACACA≤ACA=AB,(11)withequalityfork=nisrewritten,inthenotationoflog-majorization([2]),as1/21/21/21/2|(AtB)(A1?tB)|?log|AB|for0≤t≤,theinequalityprovedabovesays|(AtB)(A1?tB)|?log|AB|for1/4≤t≤3/?nitematricesA,-in-AidforScienti?cRe-search(C)[1],MajorizationrelationsforHadamardproducts,(1995)57-64.[2],plementaryGolden-Thompsontypeinequalities,(1994)113-131.[3],OnmajorizationandSchurproducts,(1985)107-117.[4],MatrixAnalysis,GTM169,Springer-Verlag,NewYork,1997.[5],PositiveDe?niteMatrices,PrincetonUniversityPress,Prince-ton,2007.[6],,,Somenorminequalitiesformatrixmeans,(2016)112-122.[7],,MatrixAnalysis,2nded.,CambridgeUniversityPress,2013.[8],Aweakmultiplicativemajorizationcon-jectureforHadamardproducts,(1988)246-247.[9],Inequalities:TheoryofMajorizationandItsApplications,AcademicPress,.(,),Springer,NewYork,2011.[10],AweakmajorizationinvolvingthematricesA?BandAB,(1995)731-744.[11],Onsomere?nementoftheCauchy-Schwarzinequality,LinearAlgebraAppl.,420(2007)433-