Re: [R] eigenvalues of a circulant matrix

From: Rolf Turner <rolf_at_math.unb.ca>
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 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.)

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. 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.

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

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