南京航空航天大学Matrix-Theory双语矩阵论期末考试2015

第1页（共3页）PartI(必做题，共5题，70分)第1题（15分）得分Let[1,1]Pdenotethesetofallrealpolynomialsofdegreelessthan3withdomain（定义域）[1,1].Theadditionandscalarmultiplicationaredefinedintheusualway.Defineaninnerproducton[1,1]Pby11,()()pqptqtdt.(1)Constructanorthonormalbasisfor[1,1]Pfromthebasis21,,xxbyusingtheGram-Schmidtorthogonalizationprocess.(2)Let2()1fxx[1,1]P.Findtheprojectionoffontothesubspacespannedby{1,x}.Solution:(1)1111,12dx,112u,1111111,[]02222pxxdx,12132xpuxxp,22211331,,22322pxxxx,22232210(31)4xpuxxp-------------------------------------------------------------------------------------------(2)2211221,1,projxuuxuu2211331,1,2222xxxx----------------------------------------------------------------------------------------------------------------第2题（15分）得分Letbethelineartransformationon3P(thevectorspaceofrealpolynomialsofdegreelessthan3)definedby(())'()''()pxxpxpx.(1)FindthematrixArepresentingwithrespecttotheorderedbasis[21,,xx]for3P.(2)Findabasisfor3Psuchthatwithrespecttothisbasis,thematrixBrepresentingisdiagonal.(3)Findthekernel（核）andrange（值域）ofthistransformation.Solution:(1)-----------------------------------------------------------------------------------------------------------------(2)101010001T(ThecolumnvectorsofTaretheeigenvectorsofA)Thecorrespondingeigenvectorsin3Pare1000010002TAT(TdiagonalizesA)22[1,,1][1,,]xxxxT.Withrespecttothisnewbasis2[1,,1]xx,therepresentingmatrixofisdiagonal.-------------------------------------------------------------------------------------------------------------------(3)Thekernelisthesubspaceconsistingofallconstantpolynomials.Therangeisthesubspacespannedbythevectors2,1xx-----------------------------------------------------------------------------------------------------------------------第3题（20分）得分Let110020012A.(1)FindalldeterminantdivisorsandelementarydivisorsofA.(2)FindaJordancanonicalformofA.(3)ComputeAte.(Givethedetailsofyourcomputations.)Solution:第2页（共3页）(1)110020012IA,(特征多项式2()(1)(2)p.Eigenvaluesare1,2,2.)Determinantdivisoroforder1()1D,2()1D,23()()(1)(2)DpElementarydivisorsare2(1)and(2).----------------------------------------------------------------------------------------------------------------------(2)TheJordancanonicalformis--------------------------------------------------------------------------------------------------------------------------(3)Foreigenvalue1,010010011IA，Aneigenvectoris1(1,0,0)TpForeigenvalue2,1102000010IA,Aneigenvectoris2(0,0,1)TpSolve32(2)AIpp,331100(2)00000101AIppweobtainthat101001010P,1110001010P--------------------------------------------------------------------------------------------------------------------第4题（10分）得分Supposethat33RAandOIAA652.(1)WhatarethepossibleminimalpolynomialsofA?Explain.(2)Ineachcaseofpart(1),whatarethepossiblecharacteristicpolynomialsofA?Explain.Solution:(1)AnannihilatingpolynomialofAis256xx.TheminimalpolynomialofAdividesanyannihilatingpolynomialofA.Thepossibleminimalpolynomialsare6x,1x,and256xx.---------------------------------------------------------------------------------------------------------------(2)TheminimalpolynomialofAdividesthecharacteristicpolynomialofA.SinceAisamatrixoforder3,thecharacteristicpolynomialofAisofdegree3.TheminimalpolynomialofAandthecharacteristicpolynomialofAhavethesamelinearfactors.Case6x,thecharacteristicpolynomialis3(6)xCase1x,thecharacteristicpolynomialis3(1)xCase256xx,thecharacteristicpolynomialis2(1)(6)xxor2(6)(1)xx-------------------------------------------------------------------------------------------------------------------第5题（10分）得分Let120000A.FindtheMoore-PenroseinverseAofA.Solution:1()(1,0)TTPPPP,111()250TTGGGG也可以用SVD求.------------------------------------------------------------------------------------------------------------------第3页（共3页）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Let4Pbethevectorspaceconsistingofallrealpolynomialsofdegreelessthan4withusualadditionandscalarmultiplication.Let123,,xxxbethreedistinctrealnumbers.Foreachpairofpolynomialsfandgin4P,define31,()()iiifgfxgx.Determinewhether,fgdefinesaninnerproducton4Pornot.Explain.第7题LetnnAR.ShowthatifxxA)(istheorthogonalprojectionfromnRto)(AR,thenAissymmetricandtheeigenvaluesofAareall1’sand0’s.第8题LetnnAC.ShowthatxxAHisreal-valuedforallnCxifandonlyifAisHermitian.第9题LetnnBAC，beHermitianmatrices,andAbepositivedefinite.ShowthatABissimilartoBA,andissimilartoarealdiagonalmatrix.若正面不够书写，请写在反面.第6题解答Let123()()()()fxxxxxxx.Then,0ff.But0f.Thisdoesnotdefineaninnerproduct.第7题解答Foranyx,()()xxTARANA,()xx0TAA.Hence,TTAAA.Thus.TAA.Fromabove,wehave2AA.Thiswillimplythat2isanannihilatingpolynomialofA.TheeigenvalueofAmustbetherootsof02.Thus,theeigenvaluesofAare1’sand0’s.第8题解答SeeThm7.1.1,page182.