From: Huntsinger, Reid <reid_huntsinger_at_merck.com>

Date: Tue 03 May 2005 - 08:24:15 EST

*>
*

> 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
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*> 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).
*

[...]

Examples:

[4,] 4 3 2 1 2

[5,] 5 4 3 2 1

E-Mail: (Ted Harding) <Ted.Harding@nessie.mcc.ac.uk> Fax-to-email: +44 (0)870 094 0861

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide!

http://www.R-project.org/posting-guide.html

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 Tue May 03 08:31:47 2005

Date: Tue 03 May 2005 - 08:24:15 EST

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@stat.math.ethz.ch

[mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of
Ted.Harding@nessie.mcc.ac.uk

Sent: Monday, May 02, 2005 5:28 PM

To: r-help@stat.math.ethz.ch

Subject: Re: [R] eigenvalues of a circulant matrix

On 02-May-05 Ted Harding wrote:

> 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

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.

E-Mail: (Ted Harding) <Ted.Harding@nessie.mcc.ac.uk> Fax-to-email: +44 (0)870 094 0861

Date: 02-May-05 Time: 22:27:32 ------------------------------ XFMail ------------------------------ ______________________________________________R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide!

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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 Tue May 03 08:31:47 2005

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