Marchuk Institute of Numerical Mathematics
of the Russian Academy of Sciences

Marchuk Institute
of Numerical Mathematics RAS


Gubkin str., 8, Moscow, 119333, Russia.
tel.: (495) 984‑81‑20, (495) 989‑80‑24, fax: (495) 989‑80‑23, E‑mail:

  • Русский

Seminar by International Mathematical Center

On Tuesday, October 26th, at 11:00 am in the 727 auditorium, we will have a presentation by Professor Sergei Grudsky from the Department of Mathematics at CINVESTAV-IPN in Mexico. The topic of the presentation is “Eigenvalue Clusters of Large Banded Toeplitz Matrices“. The presentation will be broadcast via Zoom.

Abstract.Toeplitz matrices are typically non-Hermitian and hence they evade the well-elaborated machinery one can employ in the Hermitian case. In a
pioneering paper of 1960, Palle Schmidt and Frank Spitzer showed that the eigenvalues of large banded Toeplitz matrices cluster along a certain
limiting set which is the union of finitely many closed analytic arcs. Finding this limiting set nevertheless remains a challenge. We discuss the problems
arising here. Using numerical examples, we show that the limit set is completely different from the spectrum of the infinite Toeplitz matrices.
For tetradiagonal Toeplitz matrices, we describe all types of the limiting sets, classify their exceptional points, and establish asymptotic formulas
for the analytic arcs near their endpoints. At the end of the lecture, we discuss unsolved problems that naturally arise here.