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Numerical Linear Algebra with Applications by William Ford

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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 ...

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