eigenvalues of a circulant matrix

The construction was

y<-x[c(109:1,2:108)]

so y is symmetric in the sense of the usual way of writing a function on
integers mod n as a vector with 1-based indexing. I.e., y[i+1] = y[n-(i+1)]
for i=0,1,...,n-1. So the assignment

Z <- toeplitz(y)

*does* create a symmetric circulant matrix. It is diagonalizable but does
not have distinct eigenvalues, hence the eigenspaces may be more than
one-dimensional, so you can't just pick a unit vector and call it "the"
eigenvector for that eigenvalue. You choose a basis for each eigenspace. R
detects the symmetry:

...
symmetric: if `TRUE', the matrix is assumed to be symmetric (or
          Hermitian if complex) and only its lower triangle is used. If
          `symmetric' is not specified, the matrix is inspected for
          symmetry.

(from help(eigen))
and knows that computations can be done with real arithmetic.

As for why you get NaN, you should submit --along with your example--
details of your platform (machine, R version, how R was built and installed,
etc).

Reid Huntsinger

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of
Ted.Harding at nessie.mcc.ac.uk
Sent: Monday, May 02, 2005 5:28 PM
To: r-help at stat.math.ethz.ch
Subject: Re: [R] eigenvalues of a circulant matrix

On 02-May-05 Rolf Turner wrote:

I just Googled around a bit and found definitions of Toeplitz and
circulant matrices as follows:
[...]
A circulant matrix is an n x n matrix whose rows are composed of
cyclically shifted versions of a length-n vector. For example, the
circulant matrix on the vector (1, 2, 3, 4)  is

      4 1 2 3
      3 4 1 2
      2 3 4 1
      1 2 3 4

So circulant matrices are a special case of Toeplitz matrices.
However a circulant matrix cannot be symmetric.

I suspect the confusion may lie in what's meant by "cyclically
shifted". In Rolf's example above, each row is shifted right by 1
and the one that falls off the end is put at the beginning. This
cannot be symmetric for general values in the fist row.

However, if you shift left instead, then you get

        4 1 2 3
        1 2 3 4
        2 3 4 1
        3 4 1 2

and this *is* symmetric (and indeed will always be so, for
general values in the first row).

I just had a look at ?toeplitz

(We should have done that earlier!)

toeplitz                package:stats                R Documentation
Form Symmetric Toeplitz Matrix
     *********

Description:
     Forms a symmetric Toeplitz matrix given its first row.
             *********
[...]
Examples:

     x <- 1:5
     toeplitz (x)

x <- 1:5
     toeplitz (x)

[,1] [,2] [,3] [,4] [,5]
[1,]    1    2    3    4    5
[2,]    2    1    2    3    4
[3,]    3    2    1    2    3
[4,]    4    3    2    1    2
[5,]    5    4    3    2    1

Since "Globe Trotter's" construction was

  Y<-toeplitz(x)

it's not surprising what he got (and it *certainly* wasn't
a circulant!!!).

Everybody barking up the wring tree here!

Best wishes to all,
Ted.


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Date: 02-May-05                                       Time: 22:27:32
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