Translating Matrix Mathematics into Cellular Automata and Vice Versa

Introduction

Matrix mathematics and cellular automata (CA) represent two distinct but interrelated fields in the realm of computational science and mathematics. While matrix mathematics deals with the algebraic methods for representing and manipulating matrices, cellular automata model systems that are discretized into a grid of cells, each evolving over time based on simple rules.

From Matrix Mathematics to Cellular Automata

Defining Cellular Automata

Cellular automata consist of a grid of cells, each in a finite state (e.g., 0 or 1, 'on' or 'off'). The state of each cell at the next time step depends on its current state and the states of its neighbors according to predefined rules. These rules can be mathematically represented and manipulated using matrix operations.

Matrix Representation

You can represent the state of a cellular automaton as a matrix. For example, a 3x3 grid of cells can be represented as:

Transition Rules as Matrices
The rules for transitioning between states can also be expressed using matrices. For example, if you define a simple rule as a convolution operation, you could use a kernel matrix K such as:
This kernel can be applied to the state matrix A to determine the next state by counting neighbors. 
Example of Transition
If we apply the kernel to A we would calculate the sum of products for each cell in A using the kernel resulting in a new state matrix B. For instance, if we apply the kernel to the given matrix A, the new matrix B would be:
From Cellular Automata to Matrix Mathematics
Encoding CA States
Conversely, a cellular automaton can be described mathematically using matrices. The current state of the cellular automaton can be represented as a vector or matrix, and the rules can be expressed as matrix operations.
State Transition as Matrix Multiplication
Consider a simple CA that evolves according to rules that can be defined by a matrix T. The state vector s at time t can be transformed into the state at time t1 by:
[s_{t1}  T cdot s_t]

where T is a transition matrix that encodes the rules of the CA.
Example
For a one-dimensional CA like elementary cellular automata, the transition rules can be expressed in matrix form. For instance, if you have a rule that changes a cell based on its left neighbor and itself, you could express this as a matrix multiplication. If your CA has a rule set that transforms states from st to st1, and if T is a transformation matrix:
then applying T to your current state vector, for example, st  [1 0 1], gives you the next state:
Summary
By using matrices to represent states and transition rules, you can bridge the gap between matrix mathematics and cellular automata. This allows you to analyze and simulate the behavior of cellular automata using linear algebra techniques and vice versa. The application of convolution-like operations for neighbor interactions or the use of transformation matrices to represent state transitions are key concepts in this relation.
References:
Wikipedia - Cellular Automaton
Wikipedia - Matrix Mathematics
ResearchGate - Translating Cell Automata Rules into Matrices