Singular Value Decomposition A Primer

The singular value decomposition SVD is a powerful technique in many matrix computa tions and analyses Using the SVD of a matrix in computations rather than the original matrix has the advantage of being more robust to numerical error Additionally the SVD exposes the geometric structure of a matrix an important aspect of many matrix calcula tions A matrix can be described as a tranformation from one vector space to another The components of the SVD quantify the resulting change between the underlying geometry of those vector spaces The SVD is employed in a variety of applications from least squares problems to solving systems of linear equations Each of these applications exploit key properties of the SVD its relation to the rank of a matrix and its ability to approximate matrices of a given rank Many fundamental aspects of linear algebra rely on determining the rank of a matrix making the SVD an important and widely used technique This primer serves as a short introduction to the SVD and its applications More com prehensive coverage can be found in numerous references such as GVL Dep Vac Organization of the paper is as follows Section introduces the de nition of the SVD followed by a discussion of the properties of the components of the SVD Section explores further properties of the SVD and provides a geometric interpretation of the singular values Section lists a number of interesting applications and Section concludes the paper with a discussion of the advantages and disadvantages of using the SVD