南京航空航天大学Matrix-Theory双语矩阵论期末考试2015

1、NUAA第1页（共6页）MatrixTheory,Final,TestDate:2015年12月28日矩阵论班号：学号姓名必做题（70分）选做题（30分）总分题号12345得分PartI(必做题，共5题，70分）第1题（15分）得分Letr1,1denotethesetofallrealpolynomialsofdegreelessthan3withdomain(定义域)1,1.Theadditionandscalarmultiplicationaredefinedintheusualway.DefineaninnerproductonP1,1by1P,q1P(t)q(t)dt.(1)Cons

2、tructanorthonormalbasisforGram-Schmidtorthogonalizationprocess.P1,1fromthebasis1,x,x2byusingthe(2)LetSolution:f(x)2x1P1,1.Findtheprojectionoffontothesubspacespannedby1,x.(2)1|dx2,1PiP21X，2x2,J|x0,xP3xxp2x,_2projx1,u1u131-x一23U32xP22xP2.102(3x21)42x1,u2u22.11x1,-2-2第2题（15分）得分Letbethelineartransformat

3、iononP3(thevectorspaceofrealpolynomialsofdegreelessthan3)definedby(1)Findthematrix(P(x)Arepresentingxp'(x)P''(x).withrespecttotheorderedbasisi,x,x2forP3.(2) Findabasisforisdiagonal.(3) FindthekernelSolution:(1)(10(xx(x2)2P3suchthatwithrespecttothisbasis,thematrix(核)andrange(值域)ofthistran

4、sformation.002A0102x2002Brepresenting(2)101T010(ThecolumnvectorsofTaretheeigenvectorsofA)001Thecorrespondingeigenvectorsinp3are000T1AT010(TdiagonalizesA)00221,x,x211,x,x2T.Withrespecttothisnewbasis,x,x,therepresentingmatrixofisdiagonal.(3)Thekernelallconstantpolynomials.Therangeisthesubspacespannedb

5、ythevectorsisthesubspaceconsisting2x,x1of第3题（20分）得分110Leta020012(1) Findalldeterminantdivisorsandelementarydivisorsof(2) FindaJordancanonicalformofA.(3) ComputeeAt.(Givethedetailsofyourcomputations.)Solution:A.1020,(特征多项式p()(121)(一、22).Eigenvaluesare1,2,2.)DeterminantdivisoroforderElementarydivisors

6、are(_一2D()1,D2()1,D3()p()(1)(2)21)and(2)2.(2)TheJordancanonicalformis100J021002010(3)Foreigenvalue1,IA010,Aneigenvectorisp1(1,0,0)T011110Foreigenvalue2,2IA000,Aneigenvectorisp2(0,0,1)T0101100Solve(A2I)p3p2,(A2I)p3000p30weobtainthat0101p3(1,1,0)T1011110P001,P001010010101te00110tet2t-ee0AtePeJP100102t

7、e,2tte00102te0010002t0100,2t2tetee第4题（10分）得分332SupposethatARandA5A6IO.(1) WhatarethepossibleminimalpolynomialsofA?Explain.polynomialsofA?(2) Ineachcaseofpart(1),whatarethepossiblecharacteristicExplain.Solution:2(1)AnannihilatingpolynomialofAisx5x6TheminimalpolynomialofAdividesanyannihilatingpolynomi

8、alofA.Thepossibleminimalpolynomialsare2x6,x1,andx5x6.(2)TheminimalpolynomialofAdividesthecharacteristicpolynomialofA.SinceAisamatrixoforder3,thecharacteristicpolynomialofAisofdegree3.TheminimalpolynomialofAandthecharacteristicpolynomialofAhavethesamelinearfactors.Casex6,thecharacteristicpolynomialis

9、(x6)3Casex1,thecharacteristicpolynomialis(x1)3Case2x5x6,thecharacteristicpolynomialis(x1)2(x6)or(x6)2(x1)第5题（10分）得分Leta.FindtheMoore-PenroseinverseAofA.Solution:APG_T_1_T_TT1P(PP)P(1,0),GG(GG)110-r1,、1AGP2(1,0)2055000也可以用SVD求.11-250PartII（选做题，每题10分）请在以下题目中（第6至第9题）选择三题解答.如果你做了四题，请在题号上画圈标明需要批改白三题.否则，阅

10、卷者会随意挑选三题批改，这可能影响你的成绩第6题LetP4bethevectorspaceconsistingofallrealpolynomialsofdegreelessthan4withusualadditionandscalarmultiplication.Letdistinctrealnumbers.Foreachpairofpolynomials3f,gf(x)g(x).i1X,X2,%bethreefandginP4,defineDeterminewhetherf,gdefinesaninnerproductonp4ornot.Explain.第7题LetARnn.Showtha

11、tif(x)AxistheorthogonalprojectionfromRntoR(A),thenAissymmetricandtheeigenvaluesofAareall1'sand0's.第8题LetACnn.ShowthatxHAxisreal-valuedforallxCnifandonlyifAisHermitian.第9题LetA,BCnnbeHermitianmatrices,andAbepositivedefinite.ShowthatABissimilartoBA,andissimilartoarealdiagonalmatrix.选做题得分若正面不够书写

12、，请写在反面第6题解答Letf(x)(xx1)(xinnerproduct.第7题解答Foranyx,AxxThus.AAT.Fromabove,wehaveofA.Theeigenvalueofx2)(xxa).ThenR(A)f,f0.Butf0.ThisdoesnotdefineanN(At),AT(Axx)0.Hence,ATAAT.2AA.ThiswillimplythatAmustbetherootsof2isanannihilating0.Thus,theeigenvaluespolynomialofAare1'sand0's.第8题解答SeeThm7.1.1,page182.也可以用其它方法.第9题解答1-、-1_SinceAisnonsingular,ABA(BA)A.Hence,AissimilartoBASinceAispositivedefinite,thereisanons