Spectral Decomposition of Symmetric Matrices

This lecture covers the spectral decomposition theorem and the process of diagonalizing symmetric matrices. It explains how to find eigenspaces, orthogonal basis change matrices, and the properties of symmetric matrices. The instructor demonstrates the spectral decomposition of symmetric matrices and the conditions for a matrix to be orthogonally diagonalizable. The lecture also includes examples and exercises related to orthogonal matrices and eigenvalues. Additionally, it discusses the uniqueness of spectral decomposition and the properties of symmetric matrices. The theoretical part involves proofs of various theorems and lemmas, while the practical part consists of exercises and questions to test understanding.