Explore the mathematical foundations and computational methods of Markov chains in this comprehensive lecture covering probability transition matrices, ergodic properties, and practical algorithms. Learn how transition matrices relate to column-L1-normalized adjacency matrices of graphs and understand key concepts including non-transient, non-absorbing, acyclic, and connected states. Discover the importance of limiting states and delicate balance principles in Markov chain analysis. Master four essential algorithms for working with Markov chains: eigenvalue methods, power method iterations, random walk simulations, and gain exposure to Metropolis algorithms for advanced applications in data science and probability theory.