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 <[EMAIL PROTECTED]> 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, [EMAIL PROTECTED] > 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 > ______________________________________________ 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