From: Rolf Turner <rolf_at_math.unb.ca>

Date: Tue 03 May 2005 - 00:07:26 EST

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 00:12:48 2005

Date: Tue 03 May 2005 - 00:07:26 EST

I just Googled around a bit and found definitions of Toeplitz and
circulant matrices as follows:

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.)

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. 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.

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.

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 ______________________________________________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 00:12:48 2005

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