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**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.

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.

Reviewer: John D. Dixon (Ottawa)

### 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 |