资源描述：
《calculus of vector- and matrix-valued functions》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。
1、CHAPTER9CalculusofVector-andMatrix-ValuedFunctionsInSection1ofthischapterwedevelopthecalculusofvector-andmatrix-valuedfunctions.Therearetwowaysofgoingaboutit:byrepresentingvectorsandmatricesintermsoftheircomponentsandentrieswithrespecttosomebasisandusingthecalculuso
2、fnumber-valuedfunctionsorbyredoingthetheoryinthecontextoflinearspaces.Hereweoptforthesecondapproach,becauseofitssimplicityandbecauseitistheconceptualwaytothinkaboutthesubject;butwereservetherighttogotocomponentswhennecessary.Inwhatfollows,thefieldofscalarsistherealo
3、rcomplexnumbers.InChapter7wedefinedthelengthofvectorsandthenormofmatrices;see(1)and(32).Thismadeitpossibletodefineconvergenceofsequencesasfollows.(i)AsequencexkofvectorsinR'1convergestothevectorxiflimllxk-x1l=0.kx(ii)AsequenceAkofnxnmatricesconvergestoAiflimIlAk-All
4、=0.kxWecouldhavedefinedconvergenceofsequencesofvectorsandmatrices,withoutintroducingthenotionofsize,byrequiringthateachcomponentofxktendtothecorrespondingcomponentofxand,inthecaseofmatrices,thateachentryofAktendtothecorrespondingentryofA.Butusingthenotionofsizeintro
5、ducesasimplificationinnotationandthinking,andisanaidinproof.ThereismoreaboutsizeinChapter14and15.LinearAlgebraandItsApplications.SecondEdition,byPeterD.LaxCopyright2007JohnWiley&Sons,Inc.121122LINEARALGEBRAANDITSAPPLICATIONS1.THECALCULUSOFVECTOR-ANDMATRIX-VALUEDFUNC
6、TIONSLetx(t)beavector-valuedfunctionoftherealvariablet,defined,say,fortin(0,1).Wesaythatx(t)iscontinuousattoiflimIIx(t)-x(to)II=0.(1)i-r0Wesaythatxisdifferentiableatto,withderivativei(to),ifx(to+h)-x(to)lim_X(to)=0.h-0hHerewehaveabbreviatedthederivativebyadot:x(t)=d
7、tx(t).Thenotionofcontinuityanddifferentiabilityofmatrix-valuedfunctionsisdefinedsimilarly.Thefundamentallemmaofdifferentiationholdsforvector-andmatrix-valuedfunctions.Theorem1.Ifk(t)=0foralltin(0,1),thenx(t)isconstant.EXERCISEI.Provethefundamentallemmaforvectorvalue
8、dfunctions.(Hint:Showthatforeveryvectory,(x(t),y)isconstant.)Weturntotherulesofdifferentiation.Linearity.(i)Thesumoftwodifferentiablefunct
