53. Understanding Linear Algebra 4.5: Markov Chains, Eigenvalues, and Steady-State Vectors

In this video, we see one of the most important applications of eigenvalues and eigenvectors: Markov chains. Using a model of exercise habits in a population, we build a transition matrix, find its eigenvalues and eigenvectors, and use powers of the matrix to predict the long-term behavior of the system. Along the way, we introduce steady-state vectors and the Perron–Frobenius Theorem, which explains why positive Markov chains converge to a unique long-term equilibrium regardless of the initial state. Based on Section 4.5 of Understanding Linear Algebra by David Austin.