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! ______________________________________________ R-help at 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