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### symmetric toeplitz and circulant question

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```Hi all,

Does anyone know of a good reference describing the properties of
symmetric Toeplitz matrices?  I have a single example from a paper
which describes that a symmetric Toeplitz matrix (m=5) is

a b c d e
b a b c d
T = c b a b c
d c b a b
e d c b a

and can be related to a circulant matrix

a b c c b
b a b c c
C = c b a b c
c c b a b
b c c b a.

I'm not sure I quite understand what happend to 'd' and 'e' in C?
Also, can anyone give an example when m is odd?  My best guess for m=4
is:

a b c d
T=  b a b c
c b a b
d c b a

and

a b b a
a a b b
C = b a a b
b b a a

Thx.  Dave
```
 0

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```Dave <Confused.Scientist@gmail.com> wrote in message <5c7e3a71-dacf-4893-96aa-9339634c100d@t10g2000vbg.googlegroups.com>...
> Hi all,
>
> Does anyone know of a good reference describing the properties of
> symmetric Toeplitz matrices?  I have a single example from a paper
> which describes that a symmetric Toeplitz matrix (m=5) is
>
>       a b c d e
>       b a b c d
> T = c b a b c
>       d c b a b
>       e d c b a
>
> and can be related to a circulant matrix
>
>       a b c c b
>       b a b c c
> C = c b a b c
>       c c b a b
>       b c c b a.
>
> I'm not sure I quite understand what happend to 'd' and 'e' in C?

What are you not sure about? T is a symmetric Teoplitz, but is
it a circulant matrix? No.

However, C is both symmetric Toeplitz and circulant.
I'll use my circulant function, as found on the file
exchange.

http://www.mathworks.com/matlabcentral/fileexchange/22858

circulant([1 2 3 3 2])
ans =
1     2     3     3     2
2     1     2     3     3
3     2     1     2     3
3     3     2     1     2
2     3     3     2     1

toeplitz([1 2 3 3 2])
ans =
1     2     3     3     2
2     1     2     3     3
3     2     1     2     3
3     3     2     1     2
2     3     3     2     1

> Also, can anyone give an example when m is odd?  My best guess for m=4
> is:
>
>       a b c d
> T=  b a b c
>       c b a b
>       d c b a
>
> and
>
>       a b b a
>       a a b b
> C = b a a b
>       b b a a
>

Here your C is circulant Toeplitz, but not symmetric.
T is symmetric Toeplitz, but not circulant.

Look at T. This is the general case of a symmetric
Toeplitz matrix of order 4. Can it be also circulant?

a b c d
T=  b a b c
c b a b
d c b a

To be also circulant, it would require that

b == d

So try this

toeplitz([1 2 3 2])
ans =
1     2     3     2
2     1     2     3
3     2     1     2
2     3     2     1

Indeed, this is all of symmetric, Toeplitz, and circulant.

John
```
 0
Reply woodchips (7942) 6/10/2009 3:44:03 PM

```Perfect.  Thank you John.
```
 0

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