Hi, The G matrix can be constructed from the SVD because GGt is square and symmetric, so the matrices of the left and right singular values (i.e. U and V) are the same.
Martyn -----Original Message----- From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On Behalf Of Peter Brady Sent: 14 August 2014 07:58 To: r-help@r-project.org Subject: [R] How Can SVD Reconstruct a Matrix 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: pdbr...@ans.com.au Skype: pbrady77 ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. ________________________________________________________________________ This e-mail has been scanned for all viruses by Star.\ _...{{dropped:3}} ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
