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Markov Chains and Transition Matrices

We use transition matrices to model processes and solve problems involving Markov chains. We start by recapping stochastic matrices, which are matrices where the sum of each column equals one. We use an example matrix representing three states—A, B, and C—to illustrate the concept of transitioning probabilities between these states. We discuss the significance of a special vector, denoted as u, whose entries sum to one and remains unchanged under the matrix's action. We then delve into positive stochastic matrices, emphasizing that when such a matrix is raised to increasingly high powers, the columns converge to resemble the vector u. This convergence often happens in practice when matrices are exponentiated to a large power, say in MATLAB. Further, we explore a practical application involving political parties, A, B, and C, creating a positive stochastic matrix to model voting behavior and predicting outcomes for future elections. This example helps illustrate how Markov chains can be applied to real-world scenarios. Throughout the lecture, we emphasize key properties of stochastic matrices, including how they converge over time and the interpretation of long-term probabilities. We conclude by highlighting that even non-positive stochastic matrices can display useful long-term behavior when raised to high powers. (Lecture 4.4 from Mathematical Modeling) #mathematics #mathematicalmodeling #markovchains #stochastic #linearalgebra #probabilitytheory