Given an 
matrix A, where 
and rank(A) = r,
the singular value decomposition of A, denoted by SVD(A), is defined as
where 
, 
, and
. The first r columns of the orthogonal
matrices U and V
define the orthonormal eigenvectors associated with the r nonzero eigenvalues
of 
and 
, respectively. U and V are referred to as the
left and right singular vectors, respectively. The singular values of A are
defined as the diagonal elements of 
which are the non-negative square roots of
the n eigenvalues of 
.
A discussion of the properties and applications of the SVD
can be found elsewhere [11,21].
The SVD can reveal important information about the structure of a matrix as illustrated by the following well-known theorem [3].
The rank property illustrates how the singular values of A can be used as quantitative measures of the qualitative notion of rank. The dyadic decomposition, which is the rationale for data reduction or compression in many scientific applications, provides a canonical description of a matrix as a sum of r rank-one matrices of decreasing importance, as measured by the singular values.
