MATLAB Implementation of Markov Chain Concepts with Code Descriptions

A Markov chain is a significant stochastic process characterized by its "memoryless property," meaning the next state depends solely on the current state. The core implementation of Markov chains in MATLAB involves constructing a probability transition matrix, which defines the probability distribution of system transitions from one state to other states.

For a 5×5 probability transition matrix implementation, three critical aspects require attention: First, each matrix row represents a current state; second, the five elements in each row correspond to transition probabilities to respective states; finally, the sum of elements in each row must strictly equal 1 (probability normalization). In MATLAB code, this is typically achieved by generating a random matrix using rand(5,5) and then performing row-wise normalization with diag(1./sum(P,2))*P to ensure valid probability distributions.

The standard implementation workflow begins by creating a random matrix, followed by row normalization to guarantee probability validity. Such matrices can effectively model transition patterns in scenarios like weather changes or market state transitions. For analysis, eigenvalue computation combined with the eig function is commonly employed to study steady-state distributions - one of the fundamental properties of Markov chains where [V,D] = eig(P') helps identify the eigenvector corresponding to the eigenvalue 1, representing the stationary distribution.