eigenvalues of a circulant matrix

When the matrix is symmetric and omega is not real, omega and its conjugate
(= inverse) give the same eigenvalue, so you have a 2-dimensional
eigenspace. R chooses a real basis of this, which is perfectly fine since
it's not looking for circulant structure.

For example,

m

     [,1] [,2] [,3] [,4] [,5]
[1,]    1    2    3    3    2
[2,]    2    1    2    3    3
[3,]    3    2    1    2    3
[4,]    3    3    2    1    2
[5,]    2    3    3    2    1

eigen(m)

$values
[1] 11.000000 -0.381966 -0.381966 -2.618034 -2.618034

$vectors
          [,1]      [,2]       [,3]       [,4]      [,5]
[1,] 0.4472136  0.000000 -0.6324555  0.6324555  0.000000
[2,] 0.4472136  0.371748  0.5116673  0.1954395  0.601501
[3,] 0.4472136 -0.601501 -0.1954395 -0.5116673  0.371748
[4,] 0.4472136  0.601501 -0.1954395 -0.5116673 -0.371748
[5,] 0.4472136 -0.371748  0.5116673  0.1954395 -0.601501

and you can match these columns up with the "canonical" eigenvectors
exp(2*pi*1i*(0:4)*j/5) for j = 0,1,2,3,4. E.g.,

Im(exp(2*pi*1i*(0:4)*3/5))

[1]  0.0000000 -0.5877853  0.9510565 -0.9510565  0.5877853

which can be seen to be a scalar multiple of column 2. 

Reid Huntsinger

Reid Huntsinger

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Huntsinger, Reid
Sent: Monday, May 02, 2005 10:43 AM
To: 'Globe Trotter'; Rolf Turner
Cc: r-help at stat.math.ethz.ch
Subject: RE: [R] eigenvalues of a circulant matrix


It's hard to argue against the fact that a real symmetric matrix has real
eigenvalues. The eigenvalues of the circulant matrix with first row v are
*polynomials* (not the roots of 1 themselves, unless as Rolf suggested you
start with a vector with all zeros except one 1) in the roots of 1, with
coefficients equal to the entries in v. This is the finite Fourier transform
of v, by the way, and takes real values when the coefficients are real and
symmetric, ie when the matrix is symmetric.

Reid Huntsinger

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Globe Trotter
Sent: Monday, May 02, 2005 10:23 AM
To: Rolf Turner
Cc: r-help at stat.math.ethz.ch
Subject: Re: [R] eigenvalues of a circulant matrix



--- Rolf Turner <rolf at 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 at math.unb.ca

Many thanks and best wishes!

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