singular value decomposition
Any real matrix can be decomposed into
where is an orthogonal matrix, is an orthogonal matrix, and is a unique diagonal matrix with real, non-negative elements , , in descending order:
The are the singular values of and the first columns of and are the left and right (respectively) singular vectors of . has the form:
where is a diagonal matrix with the diagonal elements . We assume now . If , then
If and , then is the rank of . In this case, becomes an matrix, and and shrink accordingly. SVD can thus be used for rank determination.
The SVD provides a numerically robust solution to the least-squares problem. The matrix-algebraic phrasing of the least-squares solution is
Then utilizing the SVD by making the replacement we have
References
-
•
Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
Title | singular value decomposition |
Canonical name | SingularValueDecomposition |
Date of creation | 2013-03-22 12:05:17 |
Last modified on | 2013-03-22 12:05:17 |
Owner | akrowne (2) |
Last modified by | akrowne (2) |
Numerical id | 9 |
Author | akrowne (2) |
Entry type | Definition |
Classification | msc 15-00 |
Classification | msc 65-00 |
Synonym | SVD |
Synonym | singular value |
Synonym | singular vector |
Related topic | Eigenvector |
Related topic | Eigenvalue |