南京航空航天大学Matrix-Theory双语矩阵论期末考试2015

乙=<九(1) ||1||=、：<1,1>==v2,u]=-2=工>[乙=<九(1) ||1||=、：<1,1>==v2,u]=-2=工>[=[>工xdx]X=0v-2 7r2 -121 21%一p 1%一p1II一丘%1 1p=<%2,-J2>石+<%2,13 .3 1-%>—%=一2丫2 3u=户4=①3%2-1)3 .2-p2|| 4(2) proj=<%2一1,u>u+<%2一1,u>u=<%2一1,3-%>213一%2NUAA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2V21八2 =+0=--372 3|第2题(15分).|得分LetobethelineartransformationonP^(thevectorspaceofrealpolynomialsofdegreelessthan3)definedbyo(p(%))=xp'(x)+p"(x).FindthematrixArepresentingowithrespecttotheorderedbasis[1,x,x2]forP.FindabasisforPsuchthatwithrespecttothisbasis,thematrixBrepresentingoisdiagonal.(3)Findthekernel(核)andrange(值域)ofthistransformation.Solution:(1)o(D=0(002)o(x)=xA=010o(x2)=2+2x2[0022/(2)'101)T=010(ThecolumnvectorsofTaretheeigenvectorsofA)[001)ThecorrespondingeigenvectorsinPare'000)T-1AT=010 (TdiagonalizesA)1002)[1,x,x2+1]=[1,x,x2]T.Withrespecttothisnewbasis-x,x2+1],therepresentingmatrix of o is diagonal.—Thekernelisthesubspaceconsistingofallconstantpolynomials.Therangeisthesubspacespannedbythevectorsx,x2+1

第3题（20分）得分LetALetA=2—1FindalldeterminantdivisorsandelementarydivisorsofAFindaJordancanonicalformofA(3)ComputeeAtSolution:(1)(Givethedetailsofyourcomputations.)「1—10<0(3)ComputeeAtSolution:(1)(Givethedetailsofyourcomputations.)「1—10<0001—2,（特征多项式p(九)=(九一1)(1—2)2.Eigenvaluesare1,2,2.)DeterminantdivisorElementarydivisorsoforderD(1)=1,1D(1)=1,D(1)=p(1)=(1—1)(1—2)2are(1—1)and(1—2)200、—10 ，Aneigenvectorisp=(1,0,0)t1 —1J 110、00,Aneigenvectorisp=(0,0,1)t210J(2)TheJordancanonicalformis(2)「1001J=021[002J「0Foreigenvalue1,I—A=010「1Foreigenvalue2,21—A=0、0

Solve(A-21)p=p,(A-21)p=03 2 310P=(IQ)T3(101A(110AP=00-1,P-1=001;010,、0-10,(101A(et00Al(110Aletet-e2t0A0lleAt=PejP—1=00-1lI0e21te21I001l=l0e2te2tl、010J100e21.10-10J10-te2tJ第4题（10分）得分0-10weobtainthatSupposethatAeR3x3andA2-5A-60-10weobtainthatWhatarethepossibleminimalpolynomialsofA?Explain.Ineachcaseofpart(1),whatarethepossiblecharacteristicpolynomialsofA?Explain.Solution:AnannihilatingpolynomialofAis%2—5%—6TheminimalpolynomialofAdividesanyannihilatingpolynomialofA.Thepossibleminimalpolynomialsare%—6,%+1,and%2—5%—6TheminimalpolynomialofAdividesthecharacteristicpolynomialofA.SinceAisamatrixoforder3,thecharacteristicpolynomialofAisofdegree3.TheminimalpolynomialofAandthecharacteristicpolynomialofAhavethesamelinearfactors.Case%-6,thecharacteristicpolynomialis(%-6)3Case%+1,thecharacteristicpolynomialis(%+1)3Case%2-5%-6,thecharacteristicpolynomialis(%+1)2(%-6)or(%-6)2(%+1)第5题（10分）得分Let(1A=1020A第5题（10分）得分Let(1A=1020A001Solution:(120)=PGPSolution:(120)=PGP+=(PtP)-iPt=(1,0)=Gt(GGt)-i2(1，0)=50J也可以用SVD求.PartII（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的三题.否则，阅卷者会随意挑选三题批改，这可能影响你的成绩.第6题LetPbethevectorspaceconsistingofallrealpolynomialsofdegreeless第6题4than4withusualadditionandscalarmultiplication.Let%,%,%bethreedistinctrealnumbers.ForeachpairofpolynomialsfandginP,define<f,g>=♦f(%)g(%)•i=1Determinewhether<f,g>definesaninnerproductonPornot.Explain.第7题 LetAgRnxn.Showthatifo(x)=AxistheorthogonalprojectionfromRntoR(A),thenAissymmetricandtheeigenvaluesofAareall1sandOs.第8题LetAgCnxn.ShowthatxHAxisreal-valuedforallxgCnifandonlyifAisHermitian.第9题LetA,BgCnxnbeHermitianmatrices,andAbepositivedefinite.ShowthatABissimilartoBA,andissimilartoarealdiagonalmatrix.选做题得分若正面不够书写，请写在反面.选做题得分若正面不够书写，请写在反面.第6题解答Then<f,f>=0.Butfw0.ThisdoesnotdefineanLetf(%)=(%-%)(Then<f,f>=0.Butfw0.Thisdoesnotdefinean第7题解答Foranyx,Ax-xgR(A)1=N(At),At(Ax-x)=0.Hence,AtA=AtThus.A=ATFromabove,wehaveA2=A.Thiswillimplythat九2一九isanannihilatingpolynomialofA.TheeigenvalueofAmustbetherootsof>2一九=0.Thus,theeigenvaluesofAare1’sand0’s.第8题解答SeeThm7.1.1,page182.也可以用其它方法.第9题解答SinceAisnonsingular,AB=A(BA)A-1.Hence,AissimilartoBASinceAispositivedefinite,thereisanonsingularhermitianmatrixPsuchthatA=PPh.AB=PPhB=P(PhBP)P-iSincePhBPisHermitian,itissimilartoarealdiagonalmatrix.ABissimilartoPHBP, PhBPissimilartoarealdiagonalmatrix.ThusABissimilartoarealdiagonalmatrix.LetPdenotethesetofallrealpolynomialsofdegreelessthan3withdomain(定义域)[一1,1].TheadditionandscalarmultiplicationaredefinedintheusualwaDefineaninnerp