[R] eigenvalues of a circulant matrix

[Janusz Kawczak]

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 at 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 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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