[R] SVD and spectral decompositions of a hermitian matrix
Prof Brian Ripley
ripley at stats.ox.ac.uk
Thu Jul 3 20:21:27 CEST 2003
On Thu, 3 Jul 2003, Ravi Varadhan wrote:
> I create a hermitian matrix
You didn't succeed, if you meant Hermitian.
> and then perform its singular value
> decomposition. But when I put it back, I don't get the original
> hermitian matrix. I am having the same problem with spectral value
> decomposition as well.
> I am using R 1.7.0 on Windows. Here is my code:
> X <- matrix(rnorm(16)+1i*rnorm(16),4)
> X <- X + t(X)
> X[upper.tri(X)] <- Conj(X[upper.tri(X)])
and I get
> X - Conj(t(X))
[,1] [,2] [,3] [,4]
[1,] 0-7.044789i 0+0.000000i 0+0.000000i 0+0.000000i
[2,] 0+0.000000i 0+4.255175i 0+0.000000i 0+0.000000i
[3,] 0+0.000000i 0+0.000000i 0+6.163605i 0+0.000000i
[4,] 0+0.000000i 0+0.000000i 0+0.000000i 0+3.021553i
so X is not Hermitian.
> Y <- La.svd(X)
> Y$u %*% diag(Y$d) %*% t(Y$v) # this doesn't give back X
The result has component vt, not v: you can't read the help page!
> Y$u %*% diag(Y$d) %*% Y$v # this works fine.
but is really matching Y$u %*% diag(Y$d) %*% Y$vt
> Z <- La.eigen(X) # the eigen values should be real, but are not.
The matrix is not Hermitian.
> Z$vec %*% diag(Z$val) %*% t(Z$vec) # this doesn't give back X
Nor should it: for a Hermitian matrix try
Z$vec %*% diag(Z$val) %*% Conj(t(Z$vec))
> The help for "La.svd" says that the function return U, D, and V such
> that X = U D V'
It doesn't: please work on improving your reading skills.
> Furthermore, the help for "La.eigen" says that if the
> argument "symmetric" is not specified, the matrix is inspected for
> symmetry, so I expect that I should get real eigen values to a
> hermitian matrix.
Yes, so check your matrix!
> Are there any problems with these 2 functions, or
> what is it that I am not understanding?
There is now no real point in using La.svd() and La.eigen() rather than
svd() and eigen().
Brian D. Ripley, ripley at stats.ox.ac.uk
Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
University of Oxford, Tel: +44 1865 272861 (self)
1 South Parks Road, +44 1865 272866 (PA)
Oxford OX1 3TG, UK Fax: +44 1865 272595
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