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.

Michael W. Berry (berry@cs.utk.edu)

Sun May 19 11:34:27 EDT 1996