# RE: [R] eigenvalues of a circulant matrix

From: Huntsinger, Reid <reid_huntsinger_at_merck.com>
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

```>> 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.

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 ------------------------------

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