linear algebra and its applications positive matrices

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1、CHAPTER16PositiveMatricesDefinition.ArealIxImatrixPiscalledentrywisepositiveifallitsentriespijarepositiverealnumbers.Caution:Thisnotionofpositivity,usedonlyinthischapter,isnottobeconfusedwithself-adjointmatricesthatarepositiveinthesenseofChapter10.Theorem1(Perron).Everypositi

2、vematrixPhasadominanteigenvalue,denotedbyX(P)whichhasthefollowingproperties:(i)I(P)ispositiveandtheassociatedeigenvectorhhaspositiveentries:Ph=A(P)h,h>0.(1)(ii)X(P)isasimpleeigenvalue.(iii)EveryothereigenvaluexofPislessthanx(P)inabsolutevalue:JJ

3、ctorfwithnonegativeentries.Proof.WerecallfromChapter13thatinequalitybetweenvectorsin1$"meansthattheinequalityholdsforallcorrespondingcomponents.Wedenotebyp(P)thesetofallnonnegativenumbersAforwhichthereisanonnegativevectorx#0suchthatPx>Ax,x>0.(3)LinearAlgebraandItsApplications

4、,SecondEdition,byPeterD.LaxCopyrightQ2007JohnWiley&Sons,Inc.237238LINEARALGEBRAANDITSAPPLICATIONSLemma2.ForPpositive,(i)p(P)isnonempty,andcontainsapositivenumber,(ii)p(P)isbounded,(iii)p(P)isclosed.Proof.Takeanypositivevectorx;sincePispositive,Pxisapositivevector.Clearly,(3)w

5、illholdforAsmallenoughpositive;thisproves(i)ofthelemma.Sincebothsidesof(3)arelinearinx,wecannormalizexsothatix=Exi=1,4_(1,...,1).(4)Multiply(3)byontheleft:Px>Aix=A.(5)DenotethelargestcomponentofiPbyb;thenb>iP.Settingthisinto(5)givesb>A;thisprovespart(ii)ofthelemma.Toprove(iii

6、),considerasequenceofA"inp(P);bydefinitionthereisacorrespondingx#0suchthat(3)holds:Pxn>An,xn.(6)Wemightaswellassumethatthexarenormalizedby(4):Thesetofnonnegativexnormalizedby(4)isaclosedboundedsetin01"andthereforecompact.Thusasubsequenceofxtendstoanonnegativexalsonormalizedby

7、(4),whileA"tendstoA.Passingtothelimitof(6)showsthatx,Asatisfy(3);thereforep(P)isclosed.Thisprovespart(iii)ofthelemma.OHavingshownthatp(P)isclosedandbounded,itfollowsthatithasamaximumAmax;by(i),Amax>0.WeshallshownowthatAmaxisthedominanteigenvalue.ThefirstthingtoshowisthatAmaxi

8、saneigenvalue.Since(3)issatisfiedbyAmax,thereisanonnegativevectorhfo