Singular Value Decomposition: Fundamentals

This lecture introduces the Singular Value Decomposition (SVD) theorem, which states that any real matrix A can be decomposed into the product of three matrices: U, Σ, and Vᵀ. The lecture covers the properties of the SVD, the concept of orthogonal matrices, the calculation of singular values, and the application of SVD in low-rank approximation. It also explores the relationship between SVD and fundamental subspaces, providing insights into the orthobases for row space, column space, null space, and left and right singular vectors. Additionally, the lecture delves into the use of SVD in solving least squares problems and the concept of low-rank approximation error measurement.