lecture16, norm of vectors and matrices

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1、JimLambersMAT461/561SpringSemester2009-10Lecture16NotesThesenotescorrespondtoSection7.1inthetext.NormsofVectorsandMatricesInthenextlecture,wewillstudyiterativemethodsforsolvingsystemsoflinearequationsoftheformAx=b.Suchmethodsproduceasequenceofiteratesx(1),x(2),:::,thatwillhopefullyconvergetothes

2、olutionx.However,werstneedtodenewhatitmeansforaninnitesequenceofvectorstoconvergetosomevector.This,inturn,requiresanotionofdistance"betweenvectors.Givenvectorsxandyoflengthone,whicharesimplyscalarsxandy,themostnaturalnotionofdistancebetweenxandyisobtainedfromtheabsolutevalue;wedenethedistan

3、cetobejxyj.Wethereforedeneadistancefunctionforvectorsthathassimilarproperties.Afunctionk
k:Rn!Riscalledavectornormifithasthefollowingproperties:1.kxk0foranyvectorx2Rn,andkxk=0ifandonlyifx=02.kxk=jjkxkforanyvectorx2Rnandanyscalar2R3.kx+ykkxk+kykforanyvectorsx,y2Rn.Thelastpropertyiscalledthetr

4、iangleinequality.Itshouldbenotedthatwhenn=1,theabsolutevaluefunctionisavectornorm.Themostcommonlyusedvectornormsbelongtothefamilyofp-norms,or`p-norms,whicharedenedbyn!1=pXkxk=jxjp:pii=1Itcanbeshownthatforanyp>0,k
kpdenesavectornorm.Thefollowingp-normsareofparticularinterest:p=1:The`1-normkxk1

5、=jx1j+jx2j+


+jxnjp=2:The`2-normorEuclideannormqpkxk2=x21+x22+


+x2n=xTx1p=1:The`1-normkxk1=maxjxij1inItcanbeshownthatthe`2-normsatisestheCauchy-Bunyakovsky-SchwarzinequalityjxTyjkxkkyk22foranyvectorsx,y2Rn.Toseewhythisistrue,wenotethat!2XnXnXnX(xTy)2=xy=xyxy=x2y2+2xyxyiiiijjiiiijji=1i;j

6、=1i=1i

7、fthesize,ormagnitude,ofavector,wecandiscussdistanceandconvergence.Givenavectornormk
k,andvectorsx;y2Rn,wedenethedistancebetweenxandy,withrespecttothisnorm,bykxyk.Then,wesaythatasequenceofn-vectorsfx(k)g1convergestoavectorx