Eigenvalue Problems: Methods and Applications

This lecture covers the eigenvalue problem and its decomposition, basics of iterative methods, the power method for eigenvalue estimation, examples of eigenproblems, inverse power methods, Gershgorin circles theorem, Rayleigh quotient method, and numerical solutions for quantum particles in arbitrary potentials. It also discusses the limitations of direct methods compared to iterative methods, convergence properties of the power method, and applications in connectivity networks and Google PageRank(TM) matrix. The lecture concludes with the numerical solution of the Schrödinger equation for quantum particles in arbitrary potentials, emphasizing the importance of iterative methods for solving complex eigenvalue problems.