Singular Value DecompositionDate: 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/ |
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