A Singularly Valuable Decomposition: The SVD of a Matrix

Every teacher of linear algebra should be familiar with the matrix singular value deco~??positiolz (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 theo1-j~ of diagonalization for sylnmetric 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. Gilbert Strang was aware of these facts when he introduced the SVD in his now classical text [22, p. 1421, obselving that "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, p, xivl, stating that "perhaps the most recurring theme in the book is the practical and theoretical value" of the SVD. Additional evidence of the SVD's significance is its central role in a number of recent papers in :Matlgenzatics ivlagazine and the Atnericalz Mathematical ilironthly; for example, [2, 3, 17, 231. Although it is probably not feasible to include the SVD in the first linear algebra course, it definitely deselves 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.