From: Janusz Kawczak <jkawczak_at_uncc.edu>

Date: Mon 02 May 2005 - 23:46:26 EST

R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Mon May 02 23:52:07 2005

Date: Mon 02 May 2005 - 23:46:26 EST

Again, what's your "kinv"?

On Mon, 2 May 2005, Globe Trotter wrote:

> OK, lets redo this again, and ensure that we start with a row that will indeed

*> lead to a symmetric matrix for the circulant matrix:
**>
**> x<-scan("kinv")
**> y<-x[c(109:1,2:108)]
**>
**> X=toeplitz(y)
**> Z=y
**> for (i in 2:216) Z=rbind(Z,y[c((216-i+2):216,1:(216-i+1))])
**>
**> range(X-Z)
**> [1] 0 0
**>
**> eigen(X) is the same as eigen(Z), but we know that Z is a circulant matrix so
**> the eigenvectors are complex....
**>
**> Any thoughts/screams?
**>
**>
**> --- Prof Brian Ripley <ripley@stats.ox.ac.uk> wrote:
**> > On Sun, 1 May 2005, someone who didn't give his name wrote:
**> >
**> > > It is my understanding that the eigenvectors of a circulant matrix are
**> > > given as follows:
**> > >
**> > > 1,omega,omega^2,....,omega^{p-1}
**> > >
**> > > where the matrix has dimension given by p x p and omega is one of p complex
**> > > roots of unity. (See Bellman for an excellent discussion on this).
**> >
**> > What is the relevance of this? Also, your reference is useless to us,
**> > which is important as this all hinges on your definitions.
**> >
**> > > The matrix created by the attached row and obtained using the following
**> > > commands indicates no imaginary parts for the eigenvectors. It appears
**> > > that the real values are close, but not exactly so, and there is no
**> > > imaginary part whatsoever.
**> > >
**> > > x<-scan("kinv.dat") #length(x) = 216
**> > > y<-x[c(109:216,1:108)]
**> > > X<-toeplitz(y)
**> > > eigen(X)$vectors
**> >
**> > We don't have "kinv.dat", but X is not circulant as usually defined.
**> >
**> > > Note that the eigenvectors are correct, and they are indeed real,
**> > > because X is symmetric.
**> > >
**> > > Is this a bug in R? Any insight if not, please!
**> >
**> > Well, first R calls LAPACK or EISPACK, so it would be a bug in one of
**> > those. But in so far as I understand you, X is a real symmetric matrix,
**> > and those have real eigenvalues and eigenvectors.
**> >
**> > I think you are confused about the meaning of Toeplitz and circulant.
**> > Compare
**> >
**> > http://mathworld.wolfram.com/CirculantMatrix.html
**> > http://mathworld.wolfram.com/ToeplitzMatrix.html
**> >
**> > and note that ?toeplitz says it computes the *symmetric* Toeplitz matrix.
**> >
**> > There is a very regretable tendency here for people to assume their
**> > lack of understanding is `a bug in R'.
**> >
**> > --
**> > Brian D. Ripley, ripley@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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**>
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R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Mon May 02 23:52:07 2005

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