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Toeplitz and Circulant Matrices: A Review

• Robert M. Gray 1

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

Short description

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.

Keywords

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

Foundations and Trends® in Communications and Information Theory

(Vol 2, Issue 3, 2006, pp 155-239)

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.

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

100 pages

DOI: 10.1561/9781933019680

E-ISBN: 978-1-933019-68-0

ISBN: 978-1-933019-23-9

Description

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.

Update

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

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