Iterated Linear Maps in the Plane

Iterated Linear Maps in the Plane.   That sounds complex, doesn’t it?   In fact, it’s something we are already very familiar with.  At the beginning of the semester we investigated repeated iteration of a single linear equation $f(x)= ax+b$.   This time we are investigating the repeated iteration of a matrix, which is a nice way to write a system of equations.

We can take the matrix $A$= $\left( \begin{array}{cc} a & b \\ c & d \\\end{array} \right)$

and initial values $x_0, y_0$ placed into a vector $x_n$ = $\left( \begin{array}{c} x_0 \\ y_0 \end{array} \right)$

We then investigate repeated iterations of $Ax_n = x_{n+1}$

Analysis of the convergence of these matrices  sees a connection to the direction of the matrix and a special number called an $eigenvalue (\lambda)$ and its corresponding $eigenvector$