[R] eigenvalues of a circulant matrix

[Rolf Turner]

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 at math.unb.ca