Singular Value Decomposition

Date: 11/04/2005 at 19:02:45
From: Volkan
Subject: Singular Value Decomposition Steps

Hi,

First of all I want to know if a matrix like  -1  1  0 , has a 
                                               0 -1  1  
unique singular value decomposition.

Can you explain what kind of difficulties may arise while trying to 
find SVD? For example, can we find eigenvectors for corresponding 
eigenvalues in every situation?  Can you make an SVD for the above matrix?

In SVD, I can easily find eigenvalues. But by iterative methods I 
could not find eigenvectors.

Thank you.

Date: 11/07/2005 at 08:40:57
From: Doctor George
Subject: Re: Singular Value Decomposition Steps

Hi Volkan,

Thanks for writing to Doctor Math.

Just to make sure we are using correct terminology, the SVD produces
singular values and singular vectors.  Eigen decomposition produces
eigenvalues and eigenvectors.

The SVD will always exist, but it is not unique. Permutations of the factor matrices can produce the same product.

Let the SVD of your matrix be A = UWV', where ' denotes transpose.  To
get V and W, look at

    A'A = (UWV')'(UWV')

        = VW'U'UWV'

        = V(W^2)V'

Notice that this is just the Eigen decomposition of A'A.  In a similar
way we get U from the Eigen decomposition of AA'.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/