半正陳式跡不等式v的推廣

半正陳式跡不等式v的推廣

1“8.5”要求下的表..............................................................禮賓部，非分離性特征的所有接受者，其本質(zhì)隨機抽樣的接受者，以聲明的形式存在的，并且a是b（h（n）），a是a（h（n）），a是a（h（n）），a是a（h（n）），a是a（），m是a（a）。當(dāng)所有進入的量子表中的a類和ii類時，母性特征的數(shù)量是確定的，并且是可接近于任何國家的特征。有序。1（a）2（a）。RajinikantPatelandMisubikoTodaconcludedandprovedaseriesoftraceinequalitiesoftheHermitianmatrices,wecallasthePatel-Todainequalityofthematrixtrace.ThetraceinequalityVisRecentlyBo-yingWang,ectgeneralizedtheinequality(1)fromtracetoeigenvaluesofthecorrespondingmatrices,Theorem5]asthefollowingwhereA,B,C∈(n),B≥CandA+C∈H+(n).Inand,SomePatel-Todamatrixtraceinequalitiesweregeneralizedtothequaternionmatrices.Butthecommonpropertyinliesin:thediscussedmatricesareallpositivedefiniteorpositivesemi-definitematrices.Example1.1LetObviously,A∈(2),(2),andB≥C,butTheexample1.1showsthatthecondition“bothBandCarepositivesemi-definite”isnotnecessaryonesuchthatthePatel-TodamatrixtraceinequalityV(i.e.,theinequality(1))holds.Inthispaper,wewillgeneralizetheconclusionsofandinthemoreweakerconditionthanthatof“positivesemi-definiteproperty”.2各要求下的主要內(nèi)容Lemma2.1LetA∈H(n),B∈(n).ThentheeigenvaluesofI+ABareallreal,andProofBy,Theorem1.3.20],weseethatandABhavethesameeigenvalues,thenalltheeigenvaluesofABarereal.Sowegettheequation(3)by[6,Problem3.2.6].Lemma2.2LetA∈(n),B,C∈H(n)andB≥C.IfA+Cisinvertible,thentheeigenvaluesof(A+C)-1(A+B)areallreal.ProofSinceA+B≥A+C,thereisD∈(n)suchthatA+B=(A+C)+D.Fromthelemma2.1,weobtainthat(A+C)-1(A+B)=I+(A+C)-1Dand(∈H(n))havethesameeigenvalues,therebyalltheeigenvaluesof(A+C)-1(A+B)arereal.Thefollowingresult(see,Corollary7.7.4(a)]or[6,Theorem6.8])iswellknownLemma2.3(see,Lemma7.2.3andTheorem7.2.7])LetA,B(∈H(n))beallinvertible,theeigenvaluesofB-1Aarerealandλn(B-1A)>0.ThenThusweobtaintheresultbythelemmas2.2and2.3asfollowsLemma2.4LetA∈(n),B,C∈H(n)andB≥C.IfA+Cisinvertibleandλn[(A+C)-1(A+B))>0,thenA+Bisinvertibleand(A+B)-1≤(A+C)-1.By[5,Observation7.7.2],[6,Theorem7.9]and[7,Theorems7.2.1,7.2.2],wehaveLemma2.5LetA,B∈H(n).IfA≥B,then1)PAP*≥PBP*,P*istheconjugatetransposematrixofP;2)λi(A)≥λi(B),i=1,2,…,n.3國際習(xí)慣法第....3日非織造條件..........................................................3.3.3Theorem3.1LetA∈(n),B,C∈H(n)andB≥C.IfA+Cisinvertibleandλi((A+C)-1(A+B))>0,thenA+Bisinvertible,theeigenvaluesof(A+B)-1Band(A+C)-1Careallreal,andtheinequality(2)holds.ProofFromthelemma2.4and1)inthelemma2.5,wegetA+Bisinvertible,andWeknowthat(A+B)-1B=I-(A+B)-1AandAandhavethesameeigenvaluesbythelemma2.1,hencealltheeigenvaluesof(A+B)-1B((A+C)-1C)areallreal.Thuswecangettheinequality(2)by(3),(4)and2)ofLemma2.5.Example3.2Let∈H(n),thenwegettheeigenvaluesofABarenotallrealnumbersbysimplecomputation.Theexample3.2showsthattheeigenvaluesoftheproductoftheHermitianmatricesmaynotbereal.Relativetoand,thethetheorem3.1abandonstherequire“positivesemi-definiteproperty”for“B,C∈(n),A+C∈H+(n)”,thereforeourconclusion“theeigenvaluesofareallreal”isnecessary,whichisgivenintheproofofthetheorem3.1.Therebytheresultisstraightforwardin[2,Theorem5].Fromthetheorem3.1,weknowthatTheorem3.3LetA∈(n),B,C∈H(n)andB≥C.IfA+Cisinvertibleandλn[(A+C)-1(A+B))>0,thenA+BisinvertibleandIntheexample1.1,wehaveλ2,andthesematricesnaturallysatisfytheconditionsofthetheorem3.3,thePatel-TodamatrixtraceinequalityVholds.Example3.4LetBytheexample1.1,weknowB≥CandA+Cisinvertible.Atthemoment,wehavesimultaneously,weobtainRelativetoand,wegiveuptherequirefor“B,C∈(n)”and“A+C∈H+(n)”,inordertogettheresultsofand,weaddthecondition“λn((A+C)-1(A+B))>0”inthetheorems3.1and3.3.Throughtheaboveanalysis,theresultsofandnaturallysatisfytheaddedcondition;theexample3.4indicatesthatthecorrespondingconclusionsofandarenotboundtoholdwhenthe.addedconditionisnotsatisfiedandthepremiseof“positivesemi-definite”isabandoned.henceouraddedconditionisreasonable.,Theorem6]provedthemoreordinaryresultthan(2)asthefollowingwhereA+istheMoore-PenroseinverseofA.ThenB≥C.FromA+B,A+C∈(2),wegetTheexample3.5indicatesthatthecondition“B≥C∈(n)”(henceA+C∈(n))assumedby[2,Theorem6]isweakenedto“B≥CandA+C∈(n)”,atthesametime,theaddedconditionofthetheorem3.1iscorrespondinglychangedas“λn((A+C)+(A+B))≥0”.Generally,wecannotobtaintheresult(5),whichisthegeneralizationof(2).Itillustratesthat:usingthesamemethodofthetheorem3.1,abandoningtheconditi