一类广义特征值反问题

一类广义特征值反问题 ? JournalofMathematlcalResearch&Exposition Vo1.12,No.2tM1992 TheInverseGeneralizedEigenvalueProblem’ y讥Qingziang (Dept.ofMath.,YanchengTeachers’College-Jiangsu,China) Abstract.Thispaperpresentsakindofinversegeneralizedeigenvalueproblemforreal symmetricbandmrix+andgivesaprooftotheexistenceofthesolutiontotheproblem forJacobiandordinarysymmetricmatrices. 1.Introduction Inthispaper,weconsiderakindofinversegeneralizedeigenvalueproblemasfollows: ProblemIGE:一 Givenarealsymmedeterminedbythemeasuredspectrum.In structuralmechanics,AandBarecalledmassmatrixandstiffnessmatrixrespectively, SOtheproblemiShowtodeterminethemassdistributionofastructurebyitssti ffness distributionanditsnaturalfrequenciesofvibrationunderboundaryconditions.(Becaflse ofthereciprocityofmatrixAand日 ingeneralizedeigenvalueproblemincertainsense, thisproblemcanalSObeinterpretedashowtodeterminethestiffnessdistributionofa structure,byitsmassdistributionanditsnaturalftequenciesofvibration).Inthefollowing twosections.wearegoingtodiscusstwospecialc~s:r=一landr=LForr= n—l,whichmeansthatAana日 arereaIsymmetrixmatrices,wefirstgivearesult abouttheinverseeigenvalueproblemofrealsymmetricmatrixobtainedbyFdedland andthentransformthei】<…=1,…,—l;k:n一..,n(2) andgivetheproofofexistenceofth . e , solutionoftheproblem,thiscallberegardedasa generaljati0n0ftheresultofHaldl1J. 2?Theinversegeneralizedelgenvalueproblemforrealsymmetricmatrices Ifr=n一1,theproblemIGEcanbest—1.Thenthen? m6ero/distinctmatrices whichsatisfytheconditionD,thetheoremisequalto21.where f=?h ?=1 ? l articularI司thereholdstrictinequalitiesin《i)|thenthennmberofdistinctmatrices? Aisequalto2(1)/2.- Nowwereturntotheca8ewhereBis&realdistinctsolutionAisthenumberof.disginctsolutionAJ. Theabovediscussioncanbestatedasatheoremabouttheinversegeneralizede igen- valueproblemforrealsymmetricm~trix: .L Theorem2Let)._l,k=1,…,,lbesequencesD,realnumberssatisfyingthein. equalitiesfblemforJacobimatrix Whenr=1,theproblemIGEiscalledtheinversegeneralizedeigenvalueproblemfor Jacobimatrix,aboutwhichwehavethef0l】0win5theoem: 一 271— Theorem3LetB5edpositivedefiniteJacobimatrix.Givenrealnumbers()1,( 陆)n-1 satislying el<bus : [『.1(H) wherec,especially Cn-1-- 器:鲁c0 Therefore,wehavelima— 卢1():+oo,itfollowsth%tthere冀xistaconstantl>—l suchthat(f)>c.Itisob..ri’oUSfrom(14),thatlima+?妒()=l,thisgiventhatthere existsaconstantu>f,suchthat妒)<c.Sincep()iscontinuouson, 叫,thereexists apointA0suchthat【A0):c.Theproofiscomplete. Lerglina3工et 日: .ldl dl’ ‘ .一 l dn一1cn bepositivedefinite.Let(A)and尸n—l(A)sati~$ytheconditioninLemmaand”(k= l,…,n)bedned6rj.Thenthereezistrealnumbersnand6s=ehthat tn(A):(cA—a)t一l—l(A)一(一lA一6)t一2一2(A),(15) where尸n一2(A)is4moniepolynomialo/degreen一2,whichhasn一2realrootsinteriased 妇n—lrealrootso/—l(A). ProofFirstweassumed,,-1?0.Let(A):tn.Pn(A)一(cnA—d)—.Pn一1(A).Nowwe determinetheconstant?SOthat(A)hasarealrootofmultiplicitytwo.Todothis,we onlyneedtosolvetheequationsaboutnandA: (A):t尸n(A)一fc—djt一lj一1()=0,,(16) ‘【)=c()一c.t一l尸一l()一(c一),一1一l()0.(17) Multiplying(16)by—l(A),andmultiplying(17)by—l(A)andsubstracting, we obtaintn(A)—l(A)一tn尸n(A)一l(A)一cnt.l【—l(A)】=0,i_e., (A)—l()一(A)一l(A):—Cn_tn-I【— l(A)】.(18) ?n From(n)一 :entn—l一一ltn一2(19) —— 273—— 一 一 一 一 ?l一 and一 1?0byassumption,tn-2>0tn>0,accordingtoLemma1,wehave — chin — - 1 :1+— d2 n_2— tn-2 >ntsinbothsidesof(23)gives Cl口2+C2口l一2dlbl=(Clc2一d{)(al+a2)(25) —— 274—— ? ? ala2—6i=(ClC2一di)al2.(26) From(23),wehaveal=c1.Substitutingin(25)and(26)andeliminatinga2give s — clc2卢f+2dl卢l6l一6i=clc2一di)【al2一卢l(l+2)l?i?e-,f .6i一2dl卢l6l+clc2卢}+(clc2一)【l2一卢l(l+2)l=0. Thisisaquadraticequationof61withdiscriminant ?=4di卢}一4clc2卢i一4(cic2一di)【al2一卢l(al+a2)】 =4(cic2一di)【一卢j+(l+a2)卢l—la2l = 4(clC2一d})(卢l—1)(a2—81). BecauseofthepositivedefinitenessofBandtheinterlacingcondition(9),weh ave ?>0.Sothequadraticequationhastwodistinctroots.Substitutingoneofth eminf 25),wecandetermine2.andwithal,6l,wec8J[1constructmatrixA. Weassumefhatthetheoremis,ruefor,lm一1.Because dl<卢l<?--<a竹t一1<卢m—l<m, wit:wit--1 (Friedland,The?Ar棚masymmetricmtrix加m埔.spectraldata,J Math.App1.,71(1o79),412—422.?.. 翻箍产,话I司题,咩短 26c『嘲6’一类广义特征值反闾l题 .. 殷庆祥.’0.1.J (江苏盐城师专敦学系), 摘要 本文提出了一个实对称带状矩阵的广义特征值反问题,并且证明了对于2acobi矩阵和一 般对称矩阵,问题的存在性.’ —— 276—— ?