Symmetric Matrices, Eigenvalues, Eigenvectors, Spectral Theorem

This lecture covers the properties of symmetric matrices, focusing on eigenvalues and eigenvectors. It explains that for a symmetric matrix, orthogonal eigenvectors correspond to distinct eigenvalues. The spectral theorem states that a symmetric matrix is orthogonally diagonalizable. Additionally, it discusses the factorization of symmetric matrices and the implications for eigenvalues. The lecture concludes with remarks on the orthogonality and diagonalizability of symmetric matrices, emphasizing the real roots and properties of eigenspaces.