On Jan 16, 2010, at 8:10 AM, Stefano Sofia wrote:
Dear R list users,
the singluar value decomposition of a symmetric matrix M is UDV^(T),
where U = V.
La.svd(M) gives as output three elements: the diagonal of D and the
two orthogonal matrices u and vt (which is already the transpose of
v).
I noticed that the transpose of vt is not exactly u. Why is that?
Well, it would have been u and t(vt) that you might have been thinking
should be equal. And they are equal up to an orthonormal transformation.
> samp <-sample(1:100, 9)
> M <- matrix(samp,3)+matrix(samp,3,byrow=T)
> all.equal( La.svd(M)$vt *c(1,-1,1) , t(La.svd(M)$u) )
[1] "Mean relative difference: 0.6169069"
> M
[,1] [,2] [,3]
[1,] 66 101 95
[2,] 101 76 104
[3,] 95 104 26
> La.svd(M)$vt
[,1] [,2] [,3]
[1,] -0.5853410 -0.6225788 -0.51939540
[2,] 0.2966601 0.4317295 -0.85182300
[3,] 0.7545652 -0.6526908 -0.06801466
> t(La.svd(M)$u)
[,1] [,2] [,3]
[1,] -0.5853410 -0.6225788 -0.51939540
[2,] -0.2966601 -0.4317295 0.85182300
[3,] -0.7545652 0.6526908 0.06801466
> all.equal( La.svd(M)$vt *c(1,-1,-1) , t(La.svd(M)$u) )
[1] TRUE
And, of course, numerical accuracy gets in the way of exact equality:
> La.svd(M)$vt *c(1,-1,-1) == t(La.svd(M)$u)
[,1] [,2] [,3]
[1,] FALSE FALSE FALSE
[2,] FALSE FALSE FALSE
[3,] TRUE FALSE TRUE
--
David.
thank you for your attention and your help
Stefano
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