Matrix-Theory-final-2017(印刷版)

NUAA第1页（共7页）MatrixTheory,FinalTestDate：2017年1月5日矩阵论班号及班内序号: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（10分）得分Let114142142110A.(a)FindanorthonormalbasisforthecolumnspaceofAusingGram-Schmidtorthogonalizationprocess.(b)FindamatrixQandRsuchthatAQR,whereQisamatrixwhosecolumnvectorsformanorthonormalsetandRisanuppertriangularmatrixwithpositivediagonalelements.Solution:第2页（共7页）第2题（10分）得分Let3Pbethevectorspaceconsistingofallrealpolynomialsofdegreelessthan3.Letbethelinearmappingfrom3Pto2Rdefinedby11()(())(0)pxdxpxp(1)Findtherangeandkernelofthismapping.(2)FindamatrixAsuchthat2()aabxcxAbcSolution:第3页（共7页）第3题（20分）得分Let262011043A(1)FindtheSmithnormalformandallelementarydivisorsofA.(2)FindaJordancanonicalformJofA,andfindanonsingularmatrixPsuchthat1PAPJ.(3)ComputeJte.(Givethedetailsofyourcomputations.)Solution:第4页（共7页）第4题（10分）得分SupposethatAR33andtheminimalpolynomialofAis()(1)(2)m.(1)WhatarethepossiblecharacteristicpolynomialsofA?Explain.(2)WhatarethepossibleJordancanonicalformsofA?Explain.Solution:第5页（共7页）第5题（20分）得分Let32102111AR(1)FindtheMoore-PenroseinverseAofmatrixA.(2)Findallleast-squaressolutionsofthesystemxbA,where(0,1,0)Tb.(3)FindtheorthogonalprojectionmatrixPfromR3tothecolumnspaceofA.Solution:第6页（共7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LetnnAOOBR,whereAandBaresquarematrices.ShowthatifAOOBisdiagonalizable,thenbothAandBarediagonalizable.第7题LetAbeaskew-Hermitianmatrix,i.e.,HAA.Showthat(a)IAandIAareinvertible.(b)1()()IAIAisaunitarymatrixwitheigenvaluesnotequalto1第8题LetAandBbennHermitianmatrices.ShowthatAandBaresimilariftheyhavethesamecharacteristicpolynomial.第9题LetRmnA.ShowthatnAAIifandonlyifrank()An若正面不够书写，请写在反面.选做题解答：选做题得分第7页（共7页）