## 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.

### MSC:

 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

### Citations:

Zbl 0080.09501; Zbl 0916.15012