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Re: Eigenvalues of a random Hermitian Toeplitz matrix
Posted:
Jul 7, 2002 12:04 PM
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Hello Weerakhan:
I found some sources on Hermitian/Toeplitz matrices which might be helpful
for you. One is at:
http://ee-www.stanford.edu/~gray/toeplitz.pdf.
This is a paper called: "Toeplitz and circulant matrices: a review", by
Robert Gray. This source is also an excellent source for other references on
Toeplitz matrices.
This paper discusses some properties of the eigenvalues of a Hermitian
matrix and it also has an intriguing lemma in "Chapter 4", pages 26-29
(below, f(lambda) denotes a Fourier series defined in the article, and {t_k}
denotes a sequence also defined in the article):
Let M_f and m_f denote the l.u.b and the g.l.b of f(lambda). Let {t_k}
denote {a certain infinite sequence. Read the section.}. Let f(lambda) = Sum
from -infinity to + infinity of t_k*e^iklambda.
Lemma 4.1: Let tau_n, k be the eigenvalues of a Toeplitz matrix T_n(f). If
T_n(f) is Hermitian, then m_f <= tau_n, k <= M_f.
Whether or not T_n(f) is Hermitian, norm(T_n(f)) <= 2M_abs(f), so that the
matrix is uniformly bounded over n if f is bounded.
Even though that Lemma does not specifically specify that the first row is
made of gaussian random variables, I think it will be very helpful for you
in your investigations.
Section 5.4 in R. Gray's paper, however, gives a gorgeous application of the
Toeplitz eigenvalue distribution Theorem to information theory and Gaussian
processes. If this is not exactly the research source you want (but I doubt
you will find nothing useful here!) I have no doubt you will find a wealth
of material on Toeplitz/Hermitian matrices, their eigenvalue distributions,
and on Gaussian processes (again, see section 5.4) along with an excellent
bibliography at the end of the paper, which can take you to other written
sources to answer your questions.
Take Care,
R. B.
PS: Some other good references are:
Hirshman, Jr. (1967): "The Spectra of Certain Toeplitz Matrices", pp. 27-29,
Illinois Journal Math. 11, 145-159. Math
Review 34. 4905.
"Eigenvalues and pseudoeigenvalues of Toeplitz matrices. Lin. Alg. Appl.
162-164, 153-185. Math Review 92k: 15028.
Also: http://www.math.washington.edu/~ejpecp/Ejpvol5/paper1b.bibl.html.
--
[Response to]
"Weerakhan Tantiphaiboontana" <wtantiph@ee.tamu.edu> wrote in message
news://fL2M8.12477$U7.144956@vixen.cso.uiuc.edu...
> I am trying to find the eigenvalues distribution of an NxN random
Hermitian
> Toeplitz matrix, whose elements in the first row are complex Gaussian
random
> variables. So far, with my very limited ability, I just found the
eigenvalues
> distribution of a random Hermitian matrix according to Wigner's theorem.
But
> the matrix is not Toeplitz. Can anyone point me to a reference or any
> solutions to this problem?
>
> Thank you very much.
> Sincerely,
> Weerakhan
>
>
>
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