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Linear Algebra Application~ Markov Chains

LinearAlgebraApplication~ wayofdealingwitha sequenceofeventsbasedontheprobabilitiesd ictatingthemotionofapopulationamongvario usstates(Fraleigh105).Considera situationwherea a scciesofdisccctetimeinte,vaisove,whicha populationdistributionata giventime(t=n;n=0,1,2,..) canbecalculatedbasedonthethedistribution atanearliertime(t=n-l) ,a (Fraleigh105) givenstatecanneverbecomenegativeIfitiskn ownhowapopulationwillredistributeitselfa ftera giventimeinterval, ,calleda tcansit~atrix,descdbesthepwbabilistiemot ionofa populationmovestoa (thatis,thetotalpopulationisunchanging)a ndtherearenonegativeentries(logically,po pulationsarepositivequantities). ,denotedtij, (1)populationmovesfromstate3 ,the3 x3 matrixaboverepresentstransitionprobabili tiesbetween3 ,t23forexample,describesthelikelihoodtha ta~oftheIYl1m~Havinga meanstodescribethechangesina populationdistribution, columnvectorp=PIP2Pn/(2)AnelementPiofsuc ha vector,knownasapopulationdistributionvec tor,providesthee transitionmatrix,thesumoftheentriesinpmu staddto1 transitionmatrixtoa populationvectorprovidesthepopulationdis -tributionata , ,whentheoriginalmatrixTisraisedtosomepow erm, matrixiscalledaregularchain(Fraleigh107) .

2. The population of a given state can never become negative If it is known how a population will redistribute itself after a given time interval, the initial and final populations can be related using the tools of linear algebra. A matrix T, called a tcansit~atrix, descdbes the pwbabilistie motion of a popnlation between vac­ ious states.

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