Thanks to David Winsemius for his help about the use of La.svd. Two more
questions.
QUESTION 1
Your example works fine, but the one that I propose below does not.
Your example:
> Asymm
[,1] [,2] [,3]
[1,] 66 101 95
[2,] 101 76 104
[3,] 95 104 26
> La.svd(Asymm)
$d
[1] 257.72248 59.79550 29.92698
$u
[,1] [,2] [,3]
[1,] -0.5853410 -0.2966601 -0.75456525
[2,] -0.6225788 -0.4317295 0.65269081
[3,] -0.5193954 0.8518230 0.06801466
$vt
[,1] [,2] [,3]
[1,] -0.5853410 -0.6225788 -0.51939540
[2,] 0.2966601 0.4317295 -0.85182300
[3,] 0.7545652 -0.6526908 -0.06801466
> La.svd(Asymm)$vt*c(1,-1,-1)
[,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(Asymm)$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
My example:
> Asymmdefpos
[,1] [,2] [,3]
[1,] 2 -1 0
[2,] -1 2 -1
[3,] 0 -1 2
> La.svd(Asymmdefpos)
$d
[1] 3.4142136 2.0000000 0.5857864
$u
[,1] [,2] [,3]
[1,] -0.5000000 -7.071068e-01 0.5000000
[2,] 0.7071068 -1.288303e-16 0.7071068
[3,] -0.5000000 7.071068e-01 0.5000000
$vt
[,1] [,2] [,3]
[1,] -0.5000000 7.071068e-01 -0.5000000
[2,] -0.7071068 1.511107e-17 0.7071068
[3,] 0.5000000 7.071068e-01 0.5000000
> La.svd(Asymmdefpos)$vt*c(1,-1,-1)
[,1] [,2] [,3]
[1,] -0.5000000 7.071068e-01 -0.5000000
[2,] 0.7071068 -1.511107e-17 -0.7071068
[3,] -0.5000000 -7.071068e-01 -0.5000000
> t(La.svd(Asymmdefpos)$u)
[,1] [,2] [,3]
[1,] -0.5000000 7.071068e-01 -0.5000000
[2,] -0.7071068 -1.288303e-16 0.7071068
[3,] 0.5000000 7.071068e-01 0.5000000
Are these two matrices equal up to an orthonormal transformation? If yes, how
can I find out the right transformation?
QUESTION 2
One option of La.svd is nu=0.
I read carefully the manual, but I am not able to understand when this option
is impostant.
Is it important to set mu=0 (only) when I deal with singular matrices?
Thank you for your attention and your help
Stefano Sofia
> Dear R list users,
> the singluar value decomposition of a symmetric matrix M is UDV^(T),
> where U = V.
> La.svd(M) gives as output three elements: the diagonal of D and the
> two orthogonal matrices u and vt (which is already the transpose of
> v).
>
> I noticed that the transpose of vt is not exactly u. Why is that?
Well, it would have been u and t(vt) that you might have been thinking
should be equal. And they are equal up to an orthonormal transformation.
> samp <-sample(1:100, 9)
> M <- matrix(samp,3)+matrix(samp,3,byrow=T)
> all.equal( La.svd(M)$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.
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