Matrix decomposition is a fundamental tool in linear algebra for understanding the action of a matrix, establishing its suitability to solve a problem, and for solving linear systems more efficiently and effectively. The topic of this chapter, the singular value decomposition (SVD), can be thought of as an extension of the eigendecomposition; it is a tool for more general, even non-square matrices. This chapter allows us to revisit several themes from past chapters: eigenvalues and eigenvectors, the condition number, the least squares solution to an overdetermined system, and more. Thus, it provides a good review of some important ideas in linear algebra. We will look at some applications of SVD, one of which is image compression. A section is devoted to principal component analysis and applications of this tool are described: axis alignment, regression, and data compression. A face authentication application applies some of the tools from this chapter. Sketches and figures illustrate the concepts.