A singularly valuable decomposition The SVD of a matrix.pdf

A Singularly Valuable Decomposition: The SVD of a Matrix Dan Kalman The American University Washington, DC 20016 February 13, 2002 Every teacher of linear algebra should be familiar with the matrix singular value decomposition (or SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and theoretical insights about linear transformations. The close connection between the SVD and the well known theory of diagonalization for symmetric matrices makes the topic immediately accessible to linear algebra teachers, and indeed, a natural extension of what these teachers already know. At the same time, the SVD has fundamental importance in several different applications of linear algebra. Strang was aware of these facts when he introduced the SVD in his now classical text [22, page 142], observing . . . it is not nearly as famous as it should be. Golub and Van Loan ascribe a central significance to the SVD in their definitive explication of numerical matrix methods [8, page xiv] stating . . . perhaps the most recurring theme in the book is the practical and theoretical value of [the SVD]. Additional evidence of the significance of the SVD is its central role in a number of papers in recent years in Mathematics Magazine and The American Mathematical Monthly (for example [2, 3, 17, 23]). Although it is probably not feasible to include the SVD in the first linear algebra course, it definitely deserves a place in more advanced undergraduate courses, particularly those with a numerical or applied emphasis. My primary goals in this article are to bring the topic to the attention of a broad audience, and to reveal some of the facets that give it both practical and theoretical significance. 1 Theory The SVD is intimately related to the familiar theory of diagonalizing a symmetric matrix. Recall that if A is a symmetric real n × n matrix, there is an orthogonal matrix V and a diagonal D such that A = V DV T . Here the columns of V ar