Toeplitz and circulant matrices: a review. Print version of Foundations and Trends in Communications and Information Theory Vol. 2, No. 3 (2005). (English) Zbl 1115.15021

Foundations and Trends in Communications and Information Theory. Boston, MA: now (ISBN 978-1-933019-23-9). x, 93 p. (2006).
[Original version: Tech. Report 6502-1, Stanford Electronic Laboratory, June 1971.]
This paper gives a tutorial on some of the asymptotic properties of the sequence of finite segments of an infinite Toeplitz matrix. Consider a doubly infinite sequence \(\left\{ t_{k}\right\} \) and the associated function represented by the Fourier series \(f(\lambda)=\sum_{k=-\infty}^{\infty} t_{k}e^{ik\lambda}\) (the author restricts himself to the case where \(\sum\left| t_{k}\right| <\infty\) and so the Fourier series converges absolutely on \(\left[ 0,2\pi\right] \)). Let \(T_{n}(f)\) be the \(n\times n\) Toeplitz matrix whose \((i,j)\)th entry is \(t_{i-j}\); in particular, \(T_{n}(f)\) is Hermitian when \(t_{-k}\) and \(t_{k}\) are complex conjugate for all \(k\).
Asymptotic properties of the sequence \(\left\{ T_{n}(f)\right\} \), such as properties of the eigenvalues of the \(T_{n}(f)\), are of interest in applications to statistical signal processing and information theory and have been studied in detail by U. Grenander and G. Szegö [Toeplitz forms and their applications, Univ. Calif. Press (1958; Zbl 0080.09501)] and by A. Böttcher and B. Silbermann [Introduction to large truncated Toeplitz matrices. New York, NY: Springer (1999; Zbl 0916.15012)].
The author’s intention here is to give a more elementary approach to some of the key results. After defining what is meant by two asymptotically equivalent sequences \(\left\{ A_{n}\right\} \) and \(\left\{ B_{n}\right\} \) of matrices of increasing degrees, he shows that \(\left\{ T_{n}(f)\right\} \) is asymptotically equivalent to a sequence \(\left\{ C_{n}(f)\right\} \) of circulant matrices (under the assumption \(\sum\left| t_{k}\right| <\infty\).). He then derives asymptotic properties of \(\left\{ T_{n}(f)\right\} \) from simple, known properties of circulants including the fundamental eigenvalue distribution theorem of Szegö, and properties of inverses and products of Toeplitz matrices and the Cholesky decomposition.


15B05 Toeplitz, Cauchy, and related matrices
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators