## Toeplitz and Circulant Matrices: A Review

Robert M. Gray, Stanford University, USA, rmgray@stanford.edu

Suggested Citation
Robert M. Gray (2006), "Toeplitz and Circulant Matrices: A Review", Foundations and Trends® in Communications and Information Theory: Vol. 2: No. 3, pp 155-239. http://dx.doi.org/10.1561/0100000006

Published: 31 Jan 2006

Subjects
Signal processing for communications

#### Journal details

1. Introduction
2. The Asymptotic Behavior of Matrices
3. Circulant Matrices
4. Toeplitz Matrices
5. Matrix Operations on Toeplitz Matrices
6. Applications to Stochastic Time Series
Acknowledgements
References

#### Abstract

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

DOI:10.1561/0100000006

#### Book details

ISBN: 978-1-933019-23-9
100 pp. $40.00 ISBN: 978-1-933019-68-0 100 pp.$100.00
1. Introduction
2. The Asymptotic Behavior of Matrices
3. Circulant Matrices
4. Toeplitz Matrices
5. Matrix Operations on Toeplitz Matrices
6. Applications to Stochastic Time Series
Acknowledgements
References

#### Toeplitz and Circulant Matrices

Toeplitz and Circulant Matrices: A Review derives in a tutorial manner the fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

Toeplitz and Circulant Matrices: A Review is written for students and practicing engineers in an accessible manner bringing this important topic to a wider audience.

#### Supplementary information

Erratum for 3.1 Eigenvalues and Eigenvectors

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Commentary Submitted By: Robert M. Gray , Stanford University, rmgray@stanford.edu. Date Accepted: 3/5/2006

• Description: On p. 33, the first equation: y^{(m)} = \frac{1}{\sqrt{n}}$$1, e^{-2\pi i m/n}, \ldots, e^{-2\pi i (n - 1)/n}$$. The last exponent should be $e^{-2\pi i m(n - 1)/n}$

#### Related publications

Companion

Block Toeplitz Matrices: Asymptotic Results and Applications , Foundations and Trends® in Communications and Information Theory, Volume 8, Issue 3 10.1561/0100000066