From: Ted Harding <Ted.Harding_at_nessie.mcc.ac.uk>

Date: Tue 03 May 2005 - 07:27:43 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
*

*> 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 Received on Tue May 03 07:40:13 2005

Date: Tue 03 May 2005 - 07:27:43 EST

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! http://www.R-project.org/posting-guide.html Received on Tue May 03 07:40:13 2005

*
This archive was generated by hypermail 2.1.8
: Fri 03 Mar 2006 - 03:31:32 EST
*