Using the Singular Value Decomposition
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Emmett J. Ientilucci
Chester F. Carlson Center for Imaging Science
Rochester Institute of Technology
emmett@cis.rit.edu
May 29, 2003
Abstract
This report introduces the concept of factorizing a matrix based on
the singular value decomposition. It discusses methods that operate on
square-symmetric matrices such as spectral decomposition. The singu-
lar value decomposition technique is explained and related to solving
linear systems of equations. Examples are presented based on over and
under determined systems.
1 Introduction
The Singular Value Decomposition (SVD) is a widely used technique to
decompose a matrix into several component matrices, exposing many of the
useful and interesting properties of the original matrix. The decomposition
of a matrix is often called a factorization. Ideally, the matrix is decomposed
into a set of factors (often orthogonal or independent) that are optimal based
on some criterion. For example, a criterion might be the reconstruction of
the decomposed matrix. The decomposition of a matrix is also useful when
the matrix is not of full rank. That is, the rows or columns of the matrix
are linearly dependent. Theoretically, one can use Gaussian elimination to
reduce the matrix to row echelon form and then count the number of nonzero
rows to determine the rank. However, this approach is not practical when
working in finite precision arithmetic. A similar case presents itself when
using LU decomposition where L is in lower triangular form with 1’s on the
diagonal and U is in upper triangular form. Ideally, a rank-deficient matrix
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This file has version number v1.0, last revised on 5/29/03
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