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

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

© 2006 R. M. Gray

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**In this article:**

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.

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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: 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.

**Erratum for 3.1 Eigenvalues and Eigenvectors**

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 $e^{-2\pi i m(n - 1)/n}$

**Companion**

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