Diagonalization of Symmetric Matrices

This lecture covers the diagonalization of symmetric matrices, focusing on the theorem stating that for symmetric matrices, distinct eigenspaces associated with distinct eigenvalues are orthogonal. The instructor explains the process of finding eigenvalues, eigenvectors, and ensuring orthogonality between different eigenspaces. The lecture also delves into the Spectral Theorem, which asserts that a symmetric matrix is always diagonalizable. Additionally, the concept of Singular Value Decomposition (SVD) is introduced, emphasizing the importance of orthogonal matrices in reducing computational errors. Through examples, the lecture illustrates scenarios where matrices may not be diagonalizable, highlighting the significance of SVD in such cases.