Singular Value Decomposition

We first present an approach to singular value decomposition (SVD) for non-square matrices based only on standard methods of linear algebra. Using the Courant-Fischer formula, we then link SVD to the greedy algorithm already discussed in Chapter 6 . This is followed by several applications such as dimensionality reduction of datasets and lower-rank approximation of matrices. As a concrete example, we discuss image compression. Finally, we illustrate the technique of principal component analysis (PCA), as well as the method of collaborative filtering in the context of movie ratings.