Chapter 15 - The Singular Value Decomposition

Abstract

This chapter covers the singular value decomposition (SVD), one of the greatest results in linear algebra. After proving the SVD theorem, the SVD is used to determine the four fundamental subspaces of a matrix and to develop formula for the Frobenius norm in terms of the singular values of a matrix. A geometric interpretation of the SVD is discussed, followed by a demonstration with a 2 × 2 matrix. The chapter shows how to use the MATLAB svd function, and provides examples. Although it should rarely be computed, the SVD can be used to compute the matrix inverse. One very interesting application is image compression using the SVD. It is shown that any matrix can be written as a sum of rank 1 matrices, each involving one singular value. Image compression is done by converting the image to a matrix, computing the SVD, and forming the rank k approximation to the matrix. This approximates the matrix using only the k largest singular values, and is capable of reasonably good compression. There are many other applications of the SVD including least-squares and principal components analysis.