Papers in Physics, vol. 2, art. 020004 (2010)

Received: 29 September 2010, Accepted: 4 October 2010
Edited by: A. G. Green
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.020004

www.papersinphysics.org

ISSN 1852-4249

Commentary on “Expansions for eigenfunctions and eigenvalues of
large-n Toeplitz matrices”

Torsten Ehrhardt1∗

The paper by L. P. Kadanoff [1] is concerned with
the problem of describing the asymptotics of the
eigenvalues and eigenvectors of Toeplitz matrices,

Tn(φ) = (φj−k)
n−1
j,k=0,

as the matrix size n goes to infinity. Here, φk are
the Fourier coefficients of the generating function
φ, i.e.,

φ(z) =

∞∑
k=−∞

φkz
k, |z| = 1.

The concrete focus of the paper are the specific
symbols

a(z) = (2 −z − 1/z)α(−z)β, |z| = 1,

with 0 < α < |β| < 1 being real parameters. The
function a(z) is smooth (in fact, analytic) except at
the point z = 1, where it has a “mild” zero. Its im-
age describes a simple closed curve in the complex
plane as z passes along the unit circle.

The problem of the asymptotics of the eigenval-
ues of Toeplitz matrices has a long history and is a
multi-faceted and difficult topic. It is closedly con-
nected with the asymptotics of the determinants of
Toeplitz matrices and thus with the Szegö Limit

∗E-mail: tehrhard@ucsc.edu

1 Department of Mathematics, University of California,
Santa Cruz, CA-95064, USA.

Theorem and its generalizations. The reader is ad-
vised to consult, e.g. [3] for many more details and
references.

For various classes of symbols φ, descriptions
have been given for the limiting set (in the Haus-
dorff metric) of the spectrum of Tn(φ) as n goes
to infinity. For instance, a result of Widom [5]
states that for certain symbols, the eigenvalues of
Tn(φ) accumulate asymptotically along the curve
described by φ(z), |z| = 1. The result applies to the
symbols considered here, where it is of importance
that a(z) is non-smooth at precisely one point.

Moreover, under certain conditions on φ, one
variant of the Szegö Limit Theorem states that

lim
n→∞

1

n

n∑
k=1

f(λ
(n)
k ) =

1

2π

∫ 2π
0

f(φ(eix)) dx,

where f is a smooth test function and λ
(n)
k are the

eigenvalues of Tn(φ).
One should point out that there are other classes

of symbols which show a completely different
asymptotics of the Toeplitz eigenvalues. For in-
stance, if φ is a rational function, then it is proved
that the eigenvalues do (in general) accumulate
along arcs which lie inside the curve described by
φ. This case is best understood because there is an
explicit formula for the characteristic polynomial of
Tn(φ).

Furthermore, if φ is a piecewise continuous func-
tion with at least two jump discontinuities, then it
is conjectured and numerically substantiated that

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Papers in Physics, vol. 2, art. 020004 (2010) / T. Ehrhardt

“most”, but not all eigenvalues, accumulate along
the image. If there is precisely one jump disconti-
nuity, then one expects that all eigenvalues accu-
mulate along the image.

For continuous real-valued symbols, i.e. for Her-
mitian Toeplitz matrices, the asymptotics of the
eigenvalues is again “canonical”, i.e. the eigenval-
ues accumulate along the image and the above for-
mula holds for continuous test functions f.

The two afore-mentioned results give some, but
limited information about the eigenvalues of the
Toeplitz matrices. The paper under consideration
(together with a preceding paper [4]) makes a sig-
nificant first attempt to determine the asymptotics
of the individual eigenvalues of Tn(a). The asymp-
totics are obtained up to third order and can be
described by

λ
(n)
k = a

(
e−ip

(n)
k

)
,

p
(n)
k = 2π

k

n
− i(1 + 2α)

ln n

n
+

1

n
d

(
k

n

)
+ o

(
1

n

)
,

as n →∞, with an explicit expression for d = d(x),
which is continuous in 0 < x < 1. (We made some
slight changes regarding notation and formulation
in comparison with the main formula (25) in [1].)
The asymptotics holds uniformly in k under the
assumption 0 < ε ≤ k/n ≤ 1−ε. The latter means
that the description does not catch the eigenvalues
accumulating near the point 0 = a(1) where the
curve a(z) is not smooth. At this point, perhaps a
different, more complicated asymptotics holds.

The derivation of the results in the paper is not
completely rigorous, despite the arguments being
quite convincing. The methods are appropriate for
dealing with Toeplitz systems. For instance, it is
made use of the fact that the finite matrices Tn(φ)
are naturally related to two semi-infinite Toeplitz
systems T(φ) = (φj−k) and T(φ̃) = (φk−j), j,k ≥
0. In the paper, this is reflected by the use of the
auxiliary functions φ− and φ+. The symbol φ is
equal to K(z) = a(z)−λ, where λ is an eigenvalue
(which is to be determined). The two semi-infinite
systems are analyzed by Wiener-Hopf factorization,
the factors of which serve as approximations for
the auxiliary functions φ− and φ+. Both auxiliary
functions allow to reconstruct the eigenfunction for
Tn(φ).

In view of the argumentation, it seems plausi-
ble that the results can be generalized without too
much effort to slightly more general symbols,

a(z) = (2 −z − 1/z)α(−z)βb(z), |z| = 1,

where b(z) is a smooth (or analytic) function for
|z| = 1 such that a(z) describes a simple closed
curve in the complex plane. On the other hand,
notice that symbols with two or more singularities
could produce a more complicated eigenvalue be-
havior [5].

After a preprint version of the paper appeared,
Bogoya, Böttcher and Grudsky [2] gave a rigorous
proof of the eigenvalue asymptotics in the special
case of symbols a(z) with β = α− 1. The general
case is (as of now) still open.

Acknowledgements - Supported in part by NSF
grant DMS-0901434.

[1] L P Kadanoff, Expansions for eigenfunctions
and eigenvalues of large-n Toeplitz matrices,
Pap. Phys. 2, 020003 (2010).

[2] J M Bogoya, A Böttcher, S M Grudsky,
Asymptotics of individual eigenvalues of large
Hessenberg Toeplitz matrices, Preprint 2010-8,
Fakultät für Mathematik, Technische Univer-
sität Chemnitz, ISSN 1614-8835.

[3] A Böttcher, B Silbermann, Introduction to
large truncated Toeplitz matrices, Universitext,
Springer, New York (1999).

[4] H Dai, Z Geary, L P Kadanoff, Asymptotics of
eigenvalues and eigenvectors of Toeplitz matri-
ces, J. Stat. Mech. P05012 (2009).

[5] H Widom, Eigenvalue distribution of non-
selfadjoint Toeplitz matrices and the asymp-
totics of Toeplitz determinants in the case
of nonvanishing index, Oper. Theory: Adv.
Appl. 48, 387 (1990).

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