Gray R. M. - Toeplitz and Circulant Matrices.pdf - Algebra - Mathematics2014

Toeplitz and Circulant Matrices: A review

t 0 t − 1 t − 2

···

t − ( n− 1)

t 1 t 0 t − 1

t 2 t 1 t 0

. . .

t n− 1

···

t 0

Robert M. Gray

Information Systems Laboratory

Department of Electrical Engineering

Stanford University

Stanford, California 94305

Revised March 2000

This document available as an

Adobe portable document format (pdf) ﬁle at

http://www-isl.stanford.edu/~gray/toeplitz.pdf

Robert M. Gray, 1971, 1977, 1993, 1997, 1998, 2000.

The preparation of the original report was ﬁnanced in part by the National

Science Foundation and by the Joint Services Program at Stanford. Since then it

has been done as a hobby.

Abstract

In this tutorial report the fundamental theorems on the asymptotic be-

havior of eigenvalues, inverses, and products of “ﬁnite section”Toeplitz ma-

trices and Toeplitz matrices with absolutely summable elements are derived.

Mathematical elegance and generality are sacriﬁced for conceptual simplic-

ity and insight in the hopes of making these results available to engineers

lacking either the background or endurance to attack the mathematical lit-

erature 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.

Acknowledgements

The author gratefully acknowledges the assistance of Ronald M. Aarts of

the Philips Research Labs in correcting many typos and errors in the 1993

revision, Liu Mingyu in pointing out errors corrected in the 1998 revision,

Paolo Tilli of the Scuola Normale Superiore of Pisa for pointing out an in-

correct corollary and providing the correction, and to David Neuhoﬀ of the

University of Michigan for pointing out several typographical errors and some

confusing notation.

Contents

1 Introduction

2 The Asymptotic Behavior of Matrices

3 Circulant Matrices

4 Toeplitz Matrices 19

4.1 Finite Order Toeplitz Matrices . . ................ 23

4.2 Toeplitz Matrices ......................... 28

4.3 Toeplitz Determinants . ..................... 45

5 Applications to Stochastic Time Series 47

5.1 Moving Average Sources ..................... 48

5.2 Autoregressive Processes ..................... 51

5.3 Factorization . . ......................... 54

5.4 Diﬀerential Entropy Rate of Gaussian Processes ........ 57

Bibliography

CONTENTS

Chapter 1

Introduction

A toeplitz matrix is an n

n matrix T n = t k,j where t k,j = t k−j , i.e., a matrix

of the form

t 0 t − 1 t − 2

···

t − ( n− 1)

t 1 t 0 t − 1

t 2 t 1 t 0

T n =

(1.1)

. . .

t n− 1

···

t 0

Examples of such matrices are covariance matrices of weakly stationary

stochastic time series and matrix representations of linear time-invariant dis-

crete time ﬁlters. There are numerous other applications in mathematics,

physics, information theory, estimation theory, etc. A great deal is known

about the behavior of such matrices — the most common and complete ref-

erences being Grenander and Szego [1] and Widom [2]. A more recent text

devoted to the subject is Bottcher and Silbermann [15]. Unfortunately, how-

ever, the necessary level of mathematical sophistication for understanding

reference [1] is frequently beyond that of one species of applied mathemati-

cian for whom the theory can be quite useful but is relatively little under-

stood. This caste consists of engineers doing relatively mathematical (for an

engineering background) work in any of the areas mentioned. This apparent

dilemma provides the motivation for attempting a tutorial introduction on

Toeplitz matrices that proves the essential theorems using the simplest possi-

ble and most intuitive mathematics. Some simple and fundamental methods

that are deeply buried (at least to the untrained mathematician) in [1] are

here made explicit.

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