MIT英文版线性代数试卷

1、18.06Quiz3SolutionssorStrangMay8,2010Profes-YourPRINTEDnameis:1.Yourrecitationnumberis2.3.(40points)SupposeuisaunitvectorinRn,sovTu=1.ThisproblemisaboutthenbynTsymmetricmatrixH=I2uuShowdirectlythatH2=I.SinceH=H、wenowknowthatHisnotonlySolutionsymmetricbutalsoExplicitly,wefindH2=(/2uuT)2=I24uuT+4uuuTu

2、uT(2points):sinceuTu=1,H2=I(3points).SinceH=HT、wealsohaveHTH=1,implyingthatHisanorthogonal(orunitary)matrix.OneeigenvectorofHisuitself.Findthecorrespondingeigenvalue.SolutionSinceHu=(I2izuT)w=u2uuTu=u2u=u.X=1.Ifvisanyvectorperpendiculartou,showthatvisaneigenvectorofHandfindtheeigenvalue.Withallthese

3、eigenvectorsthateigenvaluemustberepeatedhowmanytimes?IsHdiagonalizable?Whyorwhynot?SolutionForanyvectorvorthogonaltou(i.e.uTv=0)、wehaveHv=(/2izuT)v=v2uuTv=qsotheassociatedAis1Theorthogonalcomplementtothespacespannedbyuhasdimensionn1,sothereisabasisof(n1)orthonormaleigenvectorswiththiseigenvalue.Addi

4、ngintheeigenvectoru.wefindthatHisdiagonalizable.FindthediagonalentriesHuandHuintermsof,unAddupHu+HnnandseparatelyadduptheeigenvaluesofH.SolutionSincezthdiagonalentryofizuTisuf,theidiagonalentryofHisH辽=12?(3points).Summingthesetogethergives刀;Ha=n2刀uj=n2(3points).AddinguptheeigenvaluesofHalsogives刀人=(

5、1)1+(n1)(1)=n2(4points).(30points)SupposeAisapositivedefinitesymmetricnbynmatrix.HowdoyouknowthatX-1isalsopositivedefinite?(WeknowA-1issymmetricIjusthadane-mailfromtheInternationalMonetaryFundwiththisquestion.)SolutionSinceamatrixispositive-definiteifandonlyifallitseigenvaluesarepositive(5points),an

6、dsincetheeigenvaluesof.41aresimplytheinversesoftheeigenvaluesofA,人一1isalsopositivedefinite(theinverseofapositivenumberispositive)(5points).SupposeQisanyorthogonalnbynmatrixHowdoyouknowthatQAQT=QAQXispositivedefinite?WritedownwhichtestyouareusingSolutionUsingtheenergytext(xTAx0fornonzerowefindthatxTQ

7、AQTx=(QTx)TA(QTx)0forallnonzeroxaswell(sinceQisinvertible).Usingthepositiveeigenvaluetest，sinceAissimilartoQAQ1andsimilarmatriceshavethesameeigenvalues,QAQ1alsohasallpositiveeigenvalues(5pointsfortest,5pointsforapplication)ShowthattheblockmatrixA.A.B=A.A.ispositivesemidefiniteHowdoyouknowBisnotposit

8、ivedefinite?SolutionFirst,sinceBissingular,itcannotbepositivedefinite(ithaseigenvaluesof0).However,thepivotsofBarethepivotsofAinthefirstnrowsfollowedbyOsintheremainingrows,sobythepivottest,Bisstillsemi-definite.Similarly,thefirstnupper-leftdeterminantsofBarethesameasthoseofA.whiletheremainingonesare

9、Os,givinganotherproof.Finally,givenanonzerovectorXu=ywherexandyarevectorsinRn.onehasuTBu=(忑+切了出+卩)whichisnonnegative(andzerowhen+y=0).(30points)Thisquestionisaboutthematrix0-1A=40Finditseigenvaluesandeigenvectors.2Writethevectorw(0)=asacombinationofthoseeigenvectors0SolutionSincedet(AXI)=A2+4,theeig

10、envaluesare2i.2i(4points)Twoassociatedeigenvectorsare12zT.12tT5thoughtherearemanyotherchoices(4points).w(0)isjustthesumofthesetwovectors(2points).Solvetheequation=Austartingwiththesamevectoru(0)attimet=0.CLLInotherwords:thesolutionu(t)iswhatcombinationoftheeigenvectorsofA?SolutionOnesimplyaddsinfact

11、orsofeXitoeachterm,giving11u(t=砂+严-2i2iFindthe3matricesintheSingularValueDecompositionA=USVTintwosteps.-First,computeVandSusingthematrixA1A-Second,findthe(orthonormal)columnsofU.SolutionNotethatA=VY7UtUHVt=V2VTsothediagonalentriesofEaresimplythepositiverootsoftheeigenvaluesof04-100-14016001i.e.=4,cr2=1Sinceisalreadydiagonal.VistheidentitymatrixThecolumnsofUshouldsatisfyAui=(tV,Au2