南京航空航天大學(xué)MatrixTheory雙語(yǔ)矩陣論期末考試2015

1、第1頁(yè)（共3頁(yè)）PartI(必做題，共5題，70分)第1題（15分）得分LetP1,1denotethesetofallrealpolynomialsofdegreelessthan3withdomain（定義域）1,1.Theadditionandscalarmultiplicationaredefinedintheusualway.DefineaninnerproductonP1,1byp,q1p(t)q(t)dt.1(1)ConstructanorthonormalbasisforP1,1fromthebasis1,x,x2byusingtheGram-Schmidtorthogonal

2、izationprocess.x2(2)Letf(x)1P1,1.Findtheprojectionoffontothesubspacespannedby1,x.Solution:(1)1,2u11,11,1dx12p1x,111110,u2xp13222xdxxp1x,122p2x211x2,33x1u3x2p210(3x21),2x2,x2p24223(2)projx21,u1u1x21,u2u2x21,11x21,3x3x2222第2題（15分）得分LetbethelineartransformationonP3(thevectorspaceofrealpolynomialsofdegr

3、eelessthan3)definedby(p(x)xp(x)p(x).(1)FindthematrixArepresentingwithrespecttotheorderedbasis2for3.1,x,xP(2)FindabasisforP3suchthatwithrespecttothisbasis,thematrixBrepresentingisdiagonal.Findthekernel（核）andrange（值域）ofthistransformation.Solution:(1)(2)101T010(ThecolumnvectorsofTaretheeigenvectorsofA)

4、001ThecorrespondingeigenvectorsinP3are000T1AT010(TdiagonalizesA)0021,x,x211,x,x2T.Withrespecttothisnewbasis1,x,x21,therepresentingmatrixofisdiagonal.isthesubspaceconsistingofallconstantpolynomials.(3)ThekernelTherangeisthesubspacespannedbythevectorsx,x21第3題（20分）得分110LetA020.012(1)Findalldeterminantd

5、ivisorsandelementarydivisorsofA.(2)FindaJordancanonicalformofA.(3)ComputeeAt.(Givethedetailsofyourcomputations.)Solution:第2頁(yè)（共3頁(yè)）(1)110IA020,(特點(diǎn)多項(xiàng)式p()(1)(2)2.Eigenvaluesare1,2,2.)012DeterminantdivisoroforderD1()1,D2()1,D3()p()(1)(2)2Elementarydivisorsare(1)and(2)2.(2)TheJordancanonicalformis010(3)Fo

6、reigenvalue1,IA010，Aneigenvectorisp1(1,0,0)T011110Foreigenvalue2,2IA000,Aneigenvectorisp2(0,0,1)T0101100Solve(A2I)p3p2,(A2I)p3000p30weobtainthat0101101110P001,P1001010010第4題（10分）得分SupposethatAR33andA25A6IO.A?(1)WhatarethepossibleminimalpolynomialsofExplain.(2)Ineachcaseofpart(1),whatarethepossiblech

7、aracteristicpolynomialsofA?Explain.Solution:x2(1)AnannihilatingpolynomialofAis5x6.TheminimalpolynomialofAdividesanyannihilatingpolynomialofA.Thepossibleminimalpolynomialsare6,x1,andx25x6.TheminimalpolynomialofAdividesthecharacteristicpolynomialofA.SinceAisamatrixoforder3,thecharacteristicpolynomialo

8、fAisofdegree3.TheminimalpolynomialofAandthecharacteristicpolynomialofAhavethesamelinearfactors.Casex6,thecharacteristicpolynomialis(x6)3Casex1,thecharacteristicpolynomialis(x1)3Casex25x6,thecharacteristicpolynomialis(x1)2(x6)or(x6)2(x1)第5題（10分）得分LetA120AofA.00.FindtheMoore-Penroseinverse0Solution:1P

9、T1T,TT)11(PP)P(1,0)GG(GG250也能夠用SVD求.第3頁(yè)（共3頁(yè)）PartII(選做題,每題10分)請(qǐng)?jiān)谝韵骂}目中（第6至第9題）選擇三題解答.假如你做了四題，請(qǐng)?jiān)陬}號(hào)上畫(huà)圈注明需要批閱的三題.否則，閱卷者會(huì)任意精選三題批閱，這可能影響你的成績(jī).第6題LetP4bethevectorspaceconsistingofallrealpolynomialsofdegreelessthan4withusualadditionandscalarmultiplication.Letx1,x2,x3bethreedistinctrealnumbers.Foreachpairofpol

10、ynomialsfandginP4,define3f,gf(xi)g(xi).i1Determinewhetherf,gdefinesaninnerproducton4ornot.Explain.P第7題LetARnn.Showthatif(x)AxistheorthogonalprojectionfromRntoR(A),thenAissymmetricandtheeigenvaluesofAareall1sand0s.第8題LetACnn.ShowthatxHAxisreal-valuedforallxCnifandonlyifAisHermitian.第9題LetA，BCnnbeHerm

11、itianmatrices,andAbepositivedefinite.ShowthatABissimilartoBA,andissimilartoarealdiagonalmatrix.選做題得分若正面不夠書(shū)寫(xiě)，請(qǐng)寫(xiě)在反面.第6題解答Letf(x)(xx1)(xx2)(xx3).Thenf,f0.Butf0.Thisdoesnotdefineaninnerproduct.第7題解答Foranyx,AxxR(A)N(AT),AT(Axx)0.Hence,ATAAT.Thus.AAT.Fromabove,wehaveA2A.Thiswillimplythat2isanannihilatingpolynomialofA.TheeigenvalueofAmustbetherootsof第8題解答20.Thus,theeigenvaluesofAare1sand0s.SeeThm7.1.1,page182.也能夠用其余方法.第9題解答SinceAisnonsingular,ABA(BA)A1.Hence,AissimilartoBASinceAispositivedefinite,thereisanonsingularher