Discrete dynamical systems - Eigenvalue and Eigenvectors

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Discrete dynamical systems


Assume that AA is diagonalizable, with nn linearly independent eigenvectors v1,v2,,vnv_1, v_2, \cdots , v_n, and corresponding eigenvalues λ1,λ2,λn\lambda _1, \lambda _2, \cdots \lambda _n. Then we can write an initial vector x0x_0 to be:
x0=c1v1+c2v2++cnvnx_0=c_1 v_1+c_2 v_2+ \cdots +c_n v_n

Let's say we want to transform x0x_0 with matrix AA. Let's call the transformed vector to be x1x_1. Then,
x1=Ax0=c1Av1+c2Av2++cnAvnx_1=Ax_0=c_1 Av_1+c_2 Av_2+\cdots+c_n Av_n
=c1λ1v1+c2λ2v2++cnλnvn=c_1 \lambda_1 v_1+c_2 \lambda_2 v_2+\cdots+c_n \lambda_n v_n

Let's say we want to keep transforming it with matrix A  kA\; k times. Then we can generalize this to be:
xk=c1(λ1)kv1+c2(λ2)kv2++cn(λn)kvnx_k=c_1 (\lambda_1 )^k v_1+c_2 (\lambda_2 )^k v_2+\cdots+c_n (\lambda_n )^k v_n

This is useful because we get to know the behaviour of this equation when kk \infty.
  • Intro Lesson
    Discrete Dynamical Systems Overview:
  • 4.
    Predator and Prey Model
    Let the eagle and rabbit population at time kk be denoted as initial vector k, where kk is the time in years, EkE_k is the number of eagles at time kk, and RkR_k is the number of rabbits at time kk (all measured in thousands). Suppose there are two equations describing the relationship between these two species:

    Ek+1=(.4)Ek+(.5)Rk E_{k+1}=(.4) E_k+(.5)R_k
    Rk+1=(.207)Ek+(1.2)Rk R_{k+1}=(-.207) E_k+(1.2) R_k
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Discrete dynamical systems

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