【英文原版书】Circulant Matrices.pdf

Toeplitz and Circulant Matrices: A review

t0 t−1 t−2 ··· t−(n−1)


 t1 t0 t−1 
. 
. 
 t2 t1 t0 . 
. . 
. .. 
tn−1 ··· t0
Robert M. Gray
Information Systems Laboratory
Department of Electrical Engineering
Stanford University
Stanford, California 94305
Revised March 2000
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c 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.
ii
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 err