[R] La.svd of a symmetric matrix

[Stefano Sofia]

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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