MIT18_06SCF11_Ses2.11Markov Matrices; Fourier Series

《MIT18_06SCF11_Ses2.11Markov Matrices; Fourier Series》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、Markovmatrices;FourierseriesInthislecturewelookatMarkovmatricesandFourierseries–twoapplicationsofeigenvaluesandprojections.EigenvaluesofATTheeigenvaluesofAandtheeigenvaluesofATarethesame:(A−λI)T=AT−λI,soproperty10ofdeterminantstellsusthatdet(A−λI)=det(AT−λI).Ifλisaneigenv

2、alueofAthendet(AT−λI)=0andλisalsoaneigenvalueofAT.MarkovmatricesAmatrixlike:⎡⎤.1.01.3A=⎣.2.99.3⎦.70.4inwhichallentriesarenon-negativeandeachcolumnaddsto1iscalledaMarkovmatrix.TheserequirementscomefromMarkovmatrices’useinprobability.SquaringorraisingaMarkovmatrixtoapowergi

3、vesusanotherMarkovmatrix.Whendealingwithsystemsofdifferentialequations,eigenvectorswiththeeigenvalue0representedsteadystates.Herewe’redealingwithpowersofmatricesandgetasteadystatewhenλ=1isaneigenvalue.Theconstraintthatthecolumnsaddto1guaranteesthat1isaneigenvalue.Allother

4、eigenvalueswillbelessthan1.Rememberthat(ifAhasnindependenteigenvectors)thesolutiontouk=Aku0isuk=c1λkx1+c2λkx2+···+12cnλknxn.Ifλ1=1andallotherseigenvaluesarelessthanonethesystemapproachesthesteadystatec1x1.Thisisthex1componentofu0.Whydoesthefactthatthecolumnssumto1guarante

5、ethat1isaneigenvalue?If1isaneigenvalueofA,then:⎡⎤−.9.01.3A−1I=⎣.2−.01.3⎦.70−.6shouldbesingular.Sincewe’vesubtracted1fromeachdiagonalentry,thesumoftheentriesineachcolumnofA−Iiszero.ButthenthesumoftherowsofA−Imustbethezerorow,andsoA−Iissingular.Theeigenvectorx1isinthe��.6nu

6、llspaceofA−Iandhaseigenvalue1.It’snotveryhardtofindx1=33..71We’restudyingtheequationuk+1=AukwhereAisaMarkovmatrix.Forexampleu1mightbethepopulationof(numberofpeoplein)Massachusettsandu2mightbethepopulationofCalifornia.Amightdescribewhatfractionofthepopulationmovesfromstatet

7、ostate,ortheprobabilityofasinglepersonmoving.Wecan’thavenegativenumbersofpeople,sotheentriesofAwillalwaysbepositive.Wewanttoaccountforallthepeopleinourmodel,sothecolumnsofAaddto1=100%.Forexample:������uCal.9.2uCal=uMasst=k+1.1.8uMasst=kassumesthatthere’sa90%chancethataper

8、soninCaliforniawillstayinCaliforniaandonlya10%chancethatsheorhewillmove,whilethere’sa20%percentc