Discrete Dynamical Systems
If you are dealing with a system whose state can be described by a list of numbers, you can describe the state using a vector, and the different states of the system can be shown using a linear transformation. These types of systems are called discrete dynamical systems and each state is calculated using the formula:
$$ x_{n+1} = Ax_{n} = A^nx_0 $$
Where $A$ is the state matrix and $x_0$ is the initial state of the system. When solving discrete dynamical systems, Diagonalization proves to be a very useful technique.