Eigenvectors and 1D Wave Equation

This lecture discusses the discretization of the 1D wave equation on a bounded system, focusing on eigenvectors and eigenvalues. It covers the approximation of spatial derivatives, the solution to the wave equation, and the orthogonal basis of eigenvectors. The lecture also explores the eigenvalue problem, the positive definiteness of the symmetric matrix, and the solution to the discretized system using harmonic oscillators. Emphasis is placed on the systematic solution of the partial differential equation system and the reduction to independent harmonic oscillators. The importance of initial conditions and the concept of orthogonality are highlighted.