In this chapter we learn about one of the most fundamental and important matrix decompositions of linear algebra: the SVD. It bears some similarity with the eigendecomposition (ED), but is more general. In the signal processing context, the ED is typically applied only to symmetric square matrices, but the SVD may be applied to any arbitrary matrix. We study the relationship between the SVD and the ED. The SVD gives us important information about the rank, the column, and row spaces of the matrix and leads to very useful solutions and interpretations of least squares problems. We also discuss the concept of matrix projectors and their relationship with the SVD. We close the chapter with a discussion on the approximation of a matrix with another of lower rank. This approximation is a key element behind the latent variable methods for solving least squares problems, as discussed in Chap. 8 .