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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’snotveryhardto?ndx1=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
