[R] eigenvalues of a circulant matrix
(Ted Harding)
Ted.Harding at nessie.mcc.ac.uk
Mon May 2 23:27:43 CEST 2005
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
> 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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