Singular Value Decomposition

This lecture covers the Singular Value Decomposition (SVD) theorem, which states that for a matrix A of rank r, there exists a matrix U and V forming an orthogonal base of Rn, and a diagonal matrix Σ with singular values. The lecture explains how to factorize A as A = UΣV^T, where Σ is a diagonal matrix with singular values. It also discusses the properties of singular values and their relation to eigenvectors, emphasizing the orthogonality of the base formed by the eigenvectors of ATA. The lecture concludes with a proof of the SVD theorem and its application in decomposing matrices.