[R] How Can SVD Reconstruct a Matrix

[Richard M. Heiberger]

This looks like a variant of the Woodbury formula
http://en.wikipedia.org/wiki/Woodbury_matrix_identity

On Thu, Aug 14, 2014 at 2:57 AM, Peter Brady
<subscriptions at simonplace.net> wrote:
> Hi All,
>
> I've inherited some R code that I can't work out what they've done.  It
> appears to work and give sort of reasonable answers, I'm just trying to
> work out why they've done what they have.  I suspect that this is a
> simple vector identity that I've just been staring at too long and have
> forgotten...
>
> The code:
>
> GGt <- M0 - M1 %*% M0inv %*% t(M1)
> svdGG <- svd(GGt)
> Gmat <- svdGG$u %*% diag(sqrt(svdGG$d))
>
> It is supposed to solve:
>
> G*G^T = M0 - M1*M0^-1*M1^T
>
> for G, where G^T is the transpose of G.  It is designed to reproduce a
> numerical method described in two papers:
>
> Srikanthan and Pegram, Journal of Hydrology, 371 (2009) 142-153,
> Equation A13, who suggest the SVD method but don't describe the
> specifics, eg: "...G is found by singular value decomposition..."
>
> Alternatively, Matalas (1967) Water Resources Research 3 (4) 937-945,
> Equation 17, say that the above can be solved using Principle Component
> Analysis (PCA).
>
> I use PCA (specifically POD) and SVD to look at the components after
> decomposition, so I'm a bit lost as to how the original matrix G can be
> constructed in this case from only the singular values and the left
> singular vectors.  Like I said earlier, I suspect that this is a simple
> array identity that I've forgotten.  My Google Fu is letting me down at
> this point.
>
> My questions:
> 1) What is the proof, or where can I better find it to satisfy myself,
> that the above works?
>
> 2) Alternatively, can anyone suggest how I could apply PCA in R to
> compute the same?
>
> Thanks in advance,
> -pete
>
> --
> Peter Brady
> Email: pdbrady at ans.com.au
> Skype: pbrady77
>
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