Discrete dynamical systems  Eigenvalue and Eigenvectors
Discrete dynamical systems
Lessons
Notes:
Assume that $A$ is diagonalizable, with $n$ linearly independent eigenvectors $v_1, v_2, \cdots , v_n$, and corresponding eigenvalues $\lambda _1, \lambda _2, \cdots \lambda _n$. Then we can write an initial vector $x_0$ to be:
$x_0=c_1 v_1+c_2 v_2+ \cdots +c_n v_n$
Let's say we want to transform $x_0$ with matrix $A$. Let's call the transformed vector to be $x_1$. Then,
$x_1=Ax_0=c_1 Av_1+c_2 Av_2+\cdots+c_n Av_n$
$=c_1 \lambda_1 v_1+c_2 \lambda_2 v_2+\cdots+c_n \lambda_n v_n$
Let's say we want to keep transforming it with matrix $A\; k$ times. Then we can generalize this to be:
$x_k=c_1 (\lambda_1 )^k v_1+c_2 (\lambda_2 )^k v_2+\cdots+c_n (\lambda_n )^k v_n$
This is useful because we get to know the behaviour of this equation when $k$→$\infty$.
$=c_1 \lambda_1 v_1+c_2 \lambda_2 v_2+\cdots+c_n \lambda_n v_n$

Intro Lesson
Discrete Dynamical Systems Overview:

4.
Predator and Prey Model
Let the eagle and rabbit population at time $k$ be denoted as , where $k$ is the time in years, $E_k$ is the number of eagles at time $k$, and $R_k$ is the number of rabbits at time $k$ (all measured in thousands). Suppose there are two equations describing the relationship between these two species:
$E_{k+1}=(.4) E_k+(.5)R_k$
$R_{k+1}=(.207) E_k+(1.2) R_k$