Eigenvalues and Eigenvectors of Markov Chains

This lecture covers the main theorem related to the rate of convergence in Markov chains, focusing on eigenvalues and eigenvectors of the transition matrix. The instructor explains the concept of ergodic Markov chains, initial and limiting distributions, and the balance condition. The lecture introduces a new matrix Q derived from the transition matrix P and discusses its properties. It delves into the symmetric nature of matrix Q, the existence of real eigenvalues, and eigenvectors. The lecture concludes with facts about the eigenvalues of P, to be proven in the next session, and presents a theorem under various assumptions.