From: Huntsinger, Reid <reid_huntsinger_at_merck.com>

Date: Tue 03 May 2005 - 00:43:27 EST

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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 00:50:32 2005

Date: Tue 03 May 2005 - 00:43:27 EST

- Rolf Turner <rolf@math.unb.ca> wrote:

> I just Googled around a bit and found definitions of Toeplitz and

*> circulant matrices as follows:**>**> A Toeplitz matrix is any n x n matrix with values constant along each**> (top-left to lower-right) diagonal. matrix has the form**>**> a_0 a_1 . . . . ... a_{n-1}**> a_{-1} a_0 a_1 ... a_{n-2}**> a_{-2} a_{-1} a_0 a_1 ... .**> . . . . . .**> . . . . . .**> . . . . . .**> a_{-(n-1)} a_{-(n-2)} ... a_1 a_0**>**> (A Toeplitz matrix ***may*** be symmetric.)*

Agreed. As may a circulant matrix if a_i = a_{p-i+2}

*>
*

> 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.
**>
**> The eigenvalues of the forgoing circulant matrix are 10, 2 + 2i,
**> 2 - 2i, and 2 --- certainly not roots of unity.
*

The eigenvalues are 4+1*omega+2*omega^2+3*omega^3. omega=cos(2*pi*k/4)+isin(2*pi*k/4) as k ranges over 1, 2, 3, 4, so the above holds.

Bellman may have

> been talking about the particular (important) case of a circulant

*> matrix where the vector from which it is constructed is a canonical
**> basis vector e_i with a 1 in the i-th slot and zeroes elsewhere.
*

No, that is not true: his result can be verified for any circulant matrix, directly.

> Such a matrix is in fact a unitary matrix (operator), whence its

*> spectrum is contained in the unit circle; its eigenvalues are indeed
**> n-th roots of unity.
**>
**> Such matrices are related to the unilateral shift operator on
**> Hilbert space (which is the ``primordial'' Toeplitz operator).
**> It arises as multiplication by z on H^2 --- the ``analytic''
**> elements of L^2 of the unit circle.
**>
**> On (infinite dimensional) Hilbert space the unilateral shift
**> looks like
**>
**> 0 0 0 0 0 ...
**> 1 0 0 0 0 ...
**> 0 1 0 0 0 ...
**> 0 0 1 0 0 ...
**> . . . . . ...
**> . . . . . ...
**>
**> which maps e_0 to e_1, e_1 to e_2, e_2 to e_3, ... on and on
**> forever. On (say) 4 dimensional space we can have a unilateral
**> shift operator/matrix
**>
**> 0 0 0 0
**> 1 0 0 0
**> 0 1 0 0
**> 0 0 1 0
**>
**> but its range is a 3 dimensional subspace (e_4 gets ``killed'').
**>
**> The ``corresponding'' circulant matrix is
**>
**> 0 0 0 1
**> 1 0 0 0
**> 0 1 0 0
**> 0 0 1 0
**>
**> which is an onto mapping --- e_4 gets sent back to e_1.
**>
**> I hope this clears up some of the confusion.
**>
**> cheers,
**>
**> Rolf Turner
**> rolf@math.unb.ca
*

Many thanks and best wishes!

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 00:50:32 2005

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