Foundations and Trends® in Communications and Information Theory > Vol 2 > Issue 3

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 TrendsĀ® in Communications and Information Theory: Vol. 2: No. 3, pp 155-239. http://dx.doi.org/10.1561/0100000006

Published: 31 Jan 2006
© 2006 R. M. Gray
 
Subjects
Signal processing for communications
 
 
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In this article:
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
ISBN: 978-1-933019-23-9
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Table of contents:
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

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.

 
CIT-006

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

Companion

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