gray - toeplitz and circulant matrices

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
/>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 errors and some
confusing notation.
Contents
1 Introduction 3
2 The Asymptotic Behavior of Matrices 5
3 Circulant Matrices 15
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 58
1
2 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
n
=










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









. (1.1)
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 Szeg¨o [1] and Widom [2]. A more recent text
devoted to the subject is B¨ottcher 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.
3
4 CHAPTER 1. INTRODUCTION
In addition to the fundamental theorems, several related results that nat-
urally follow but do not appear to be collected together anywhere are pre-
sented.
The essential prerequisites for this report are a knowledge of matrix the-
ory, an engineer’s knowledge of Fourier series and random processes, calculus
(Riemann integration), and hopefully a first course in analysis. Several of the
occasional results required of analysis are usually contained in one or more
courses in the usual engineering curriculum, e.g., the Cauchy-Schwarz and
triangle inequalities. Hopefully the only unfamiliar results are a corollary to
the Courant-Fischer Theorem and the Weierstrass Approximation Theorem.
The latter is an intuitive result which is easily believed even if not formally
proved. More advanced results from Lebesgue integration, functional analy-
sis, and Fourier series are not used.
The main approach of this report is to relate the properties of Toeplitz
matrices to those of their simpler, more structured cousin — the circulant or
cyclic matrix. These two matrices are shown to be asymptotically equivalent
in a certain sense and this is shown to imply that eigenvalues, inverses, prod-
ucts, and determinants behave similarly. This approach provides a simplified
and direct path (to the author’s point of view) to the basic eigenvalue distri-
bution and related theorems. This method is implicit but not immediately
apparent in the more complicated and more general results of Grenander
in Chapter 7 of [1]. The basic results for the special case of a finite order
Toeplitz matrix appeared in [16], a tutorial treatment of the simplest case

which was in turn based on the first draft of this work. The results were sub-
sequently generalized using essentially the same simple methods, but they
remain less general than those of [1].
As an application several of the results are applied to study certain models
of discrete time random processes. Two common linear models are studied
and some intuitively satisfying results on covariance matrices and their fac-
tors are given. As an example from Shannon information theory, the Toeplitz
results regarding the limiting behavior of determinants is applied to find the
differential entropy rate of a stationary Gaussian random process.
We sacrifices mathematical elegance and generality for conceptual sim-
plicity in the hope that this will bring an understanding of the interesting
and useful properties of Toeplitz matrices to a wider audience, specifically
to those who have lacked either the background or the patience to tackle the
mathematical literature on the subject.
Chapter 2
The Asymptotic Behavior of
Matrices
In this chapter we begin with relevant definitions and a prerequisite theo-
rem and proceed to a discussion of the asymptotic eigenvalue, product, and
inverse behavior of sequences of matrices. The remaining chapters of this
report will largely be applications of the tools and results of this chapter to
the special cases of Toeplitz and circulant matrices.
The eigenvalues λ
k
and the eigenvectors (n-tuples) x
k
of an n × n matrix
M are the solutions to the equation
Mx = λx (2.1)
and hence the eigenvalues are the roots of the characteristic equation of M:

det(M − λI)=0 . (2.2)
If M is Hermitian, i.e., if M = M
∗
, where the asterisk denotes conjugate
transpose, then a more useful description of the eigenvalues is the variational
description given by the Courant-Fischer Theorem [3, p. 116]. While we will
not have direct need of this theorem, we will use the following important
corollary which is stated below without proof.
Corollary 2.1 Define the Rayleigh quotient of an Hermitian matrix H and
a vector (complex n−tuple) x by
R
H
(x)=(x
∗
Hx)/(x
∗
x). (2.3)
5
6 CHAPTER 2. THE ASYMPTOTIC BEHAVIOR OF MATRICES
Let η
M
and η
m
be the maximum and minimum eigenvalues of H, respectively.
Then
η
m
= min
x
R

H
(x) = min
x
∗
x=1
x
∗
Hx (2.4)
η
M
= max
x
R
H
(x) = max
x
∗
x=1
x
∗
Hx (2.5)
This corollary will be useful in specifying the interval containing the eigen-
values of an Hermitian matrix.
The following lemma is useful when studying non-Hermitian matrices
and products of Hermitian matrices. Its proof is given since it introduces
and manipulates some important concepts.
Lemma 2.1 Let A be a matrix with eigenvalues α
k
. Define the eigenvalues
of the Hermitian matrix A

∗
A to be λ
k
. Then
n−1

k=0
λ
k
≥
n−1

k=0
|α
k
|
2
, (2.6)
with equality iff (if and only if) A is normal, that is, iff A
∗
A = AA
∗
. (If A
is Hermitian, it is also normal.)
Proof.
The trace of a matrix is the sum of the diagonal elements of a matrix.
The trace is invariant to unitary operations so that it also is equal to the
sum of the eigenvalues of a matrix, i.e.,
Tr{A
∗

A} =
n−1

k=0
(A
∗
A)
k,k
=
n−1

k=0
λ
k
. (2.7)
Any complex matrix A can be written as
A = WRW
∗
. (2.8)
where W is unitary and R = {r
k,j
} is an upper triangular matrix [3, p. 79].
The eigenvalues of A are the principal diagonal elements of R.Wehave
Tr{A
∗
A} =Tr{R
∗
R} =
n−1


k=0
n−1

j=0
|r
j,k
|
2
=
n−1

k=0
|α
k
|
2
+

k=j
|r
j,k
|
2
≥
n−1

k=0
|α
k
|

2
. (2.9)
7
Equation (2.9) will hold with equality iff R is diagonal and hence iff A is
normal.
Lemma 2.1 is a direct consequence of Shur’s Theorem [3, pp. 229-231]
and is also proved in [1, p. 106].
To study the asymptotic equivalence of matrices we require a metric or
equivalently a norm of the appropriate kind. Two norms — the operator or
strong norm and the Hilbert-Schmidt or weak norm — will be used here [1,
pp. 102-103].
Let A be a matrix with eigenvalues α
k
and let λ
k
be the eigenvalues of
the Hermitian matrix A
∗
A. The strong norm  A  is defined by
 A = max
x
R
A
∗
A
(x)
1/2
= max
x
∗

x=1
[x
∗
A
∗
Ax]
1/2
. (2.10)
From Corollary 2.1
 A 
2
= max
k
λ
k
∆
= λ
M
. (2.11)
The strong norm of A can be bounded below by letting e
M
be the eigenvector
of A corresponding to α
M
, the eigenvalue of A having largest absolute value:
 A 
2
= max
x
∗

x=1
x
∗
A
∗
Ax ≥ (e
∗
M
A
∗
)(Ae
M
)=|α
M
|
2
. (2.12)
If A is itself Hermitian, then its eigenvalues α
k
are real and the eigenvalues
λ
k
of A
∗
A are simply λ
k
= α
2
k
. This follows since if e

(k)
is an eigenvector of A
with eigenvalue α
k
, then A
∗
Ae
(k)
= α
k
A
∗
e
(k)
= α
2
k
e
(k)
. Thus, in particular,
if A is Hermitian then
 A = max
k
|α
k
| = |α
M
|. (2.13)
The weak norm of an n × n matrix A = {a
k,j

} is defined by
|A| =


n
−1
n−1

k=0
n−1

j=0
|a
k,j
|
2


1/2
=(n
−1
Tr[A
∗
A])
1/2
=

n
−1
n−1


k=0
λ
k

1/2
. (2.14)
From Lemma 2.1 we have
|A|
2
≥ n
−1
n−1

k=0
|α
k
|
2
, (2.15)
with equality iff A is normal.
8 CHAPTER 2. THE ASYMPTOTIC BEHAVIOR OF MATRICES
The Hilbert-Schmidt norm is the “weaker” of the two norms since
 A 
2
= max
k
λ
k
≥ n

−1
n−1

k=0
λ
k
= |A|
2
. (2.16)
A matrix is said to be bounded if it is bounded in both norms.
Note that both the strong and the weak norms are in fact norms in the
linear space of matrices, i.e., both satisfy the following three axioms:
1.  A ≥ 0 , with equality iff A =0 , the all zero matrix.
2.  A + B ≤ A  +  B 
3.  cA = |c|·  A 
.
(2.17)
The triangle inequality in (2.17) will be used often as is the following direct
consequence:
 A − B ≥ |  A −B  . (2.18)
The weak norm is usually the most useful and easiest to handle of the
two but the strong norm is handy in providing a bound for the product of
two matrices as shown in the next lemma.
Lemma 2.2 Given two n × n matrices G = {g
k,j
} and H = {h
k,j
}, then
|GH|≤G ·|H|. (2.19)
Proof.

|GH|
2
= n
−1

i

j
|

k
g
i,k
h
k,j
|
2
= n
−1

i

j

k

m
g
i,k
¯g

i,m
h
k,j
¯
h
m,j
= n
−1

j
h
∗
j
G
∗
Gh
j
,
(2.20)
9
where ∗ denotes conjugate transpose and h
j
is the j
th
column of H. From
(2.10)
(h
∗
j
G

∗
Gh
j
)/(h
∗
j
h
j
) ≤ G 
2
and therefore
|GH|
2
≤ n
−1
 G 
2

j
h
∗
j
h
j
= G 
2
·|H|
2
.
Lemma 2.2 is the matrix equivalent of 7.3a of [1, p. 103]. Note that the

lemma does not require that G or H be Hermitian.
We will be considering n × n matrices that approximate each other when
n is large. As might be expected, we will use the weak norm of the difference
of two matrices as a measure of the “distance” between them. Two sequences
of n × n matrices A
n
and B
n
are said to be asymptotically equivalent if
1. A
n
and B
n
are uniformly bounded in strong (and hence in weak) norm:
 A
n
 ,  B
n
≤ M<∞ (2.21)
and
2. A
n
− B
n
= D
n
goes to zero in weak norm as n →∞:
lim
n→∞
|A

n
− B
n
| = lim
n→∞
|D
n
| =0.
Asymptotic equivalence of A
n
and B
n
will be abbreviated A
n
∼ B
n
. If one
of the two matrices is Toeplitz, then the other is said to be asymptotically
Toeplitz. We can immediately prove several properties of asymptotic equiv-
alence which are collected in the following theorem.
Theorem 2.1
1. If A
n
∼ B
n
, then
lim
n→∞
|A
n

| = lim
n→∞
|B
n
|. (2.22)
2. If A
n
∼ B
n
and B
n
∼ C
n
, then A
n
∼ C
n
.
3. If A
n
∼ B
n
and C
n
∼ D
n
, then A
n
C
n

∼ B
n
D
n
.
10 CHAPTER 2. THE ASYMPTOTIC BEHAVIOR OF MATRICES
4. If A
n
∼ B
n
and  A
−1
n
,  B
−1
n
≤ K<∞, i.e., A
−1
n
and B
−1
n
exist
and are uniformly bounded by some constant independent of n, then
A
−1
n
∼ B
−1
n

.
5. If A
n
B
n
∼ C
n
and  A
−1
n
≤ K<∞, then B
n
∼ A
−1
n
C
n
.
Proof.
1. Eqs. (2.22) follows directly from (2.17).
2. |A
n
− C
n
| = |A
n
− B
n
+ B
n

− C
n
|≤|A
n
− B
n
| + |B
n
− C
n
|
−→
n→∞
0
3. Applying Lemma 2.2 yields
|A
n
C
n
− B
n
D
n
| = |A
n
C
n
− A
n
D

n
+ A
n
D
n
− B
n
D
n
|
≤ A
n
·|C
n
− D
n
|+  D
n
·|A
n
− B
n
|
−→
n→∞
0.
4. |A
−1
n
− B

−1
n
| = |B
−1
n
B
n
A
n
− B
−1
n
A
n
A
−1
n
≤ B
−1
n
·A
−1
n
·|B
n
− A
n
|
−→
n→∞

0.
5. B
n
− A
−1
n
C
n
= A
−1
n
A
n
B
n
− A
−1
n
C
n
≤ A
−1
n
·|A
n
B
n
− C
n
|

−→
n→∞
0.
The above results will be useful in several of the later proofs.
Asymptotic equality of matrices will be shown to imply that eigenvalues,
products, and inverses behave similarly. The following lemma provides a
prelude of the type of result obtainable for eigenvalues and will itself serve
as the essential part of the more general theorem to follow.
Lemma 2.3 Given two sequences of asymptotically equivalent matrices A
n
and B
n
with eigenvalues α
n,k
and β
n,k
, respectively, then
lim
n→∞
n
−1
n−1

k=0
α
n,k
= lim
n→∞
n
−1

n−1

k=0
β
n,k
. (2.23)
11
Proof.
Let D
n
= {d
k,j
} = A
n
− B
n
. Eq. (2.23) is equivalent to
lim
n→∞
n
−1
Tr(D
n
)=0. (2.24)
Applying the Cauchy-Schwartz inequality [4, p. 17] to Tr(D
n
) yields
|Tr(D
n
)|

2
=





n−1

k=0
d
k,k





2
≤ n
n−1

k=0
|d
k,k
|
2
≤ n
n−1

k=0

n−1

j=0
|d
k,j
|
2
= n
2
|D
n
|
2
.
.
Dividing by n
2
, and taking the limit, results in
0 ≤|n
−1
Tr(D
n
)|
2
≤|D
n
|
2
−→
n→∞

0. (2.25)
which implies (2.24) and hence (2.23).
Similarly to (2.23), if A
n
and B
n
are Hermitian then (2.22) and (2.15)
imply that
lim
n→∞
n
−1
n−1

k=0
α
2
n,k
= lim
n→∞
n
−1
n−1

k=0
β
2
n,k
. (2.26)
Note that (2.23) and (2.26) relate limiting sample (arithmetic) averages of

eigenvalues or moments of an eigenvalue distribution rather than individual
eigenvalues. Equations (2.23) and (2.26) are special cases of the following
fundamental theorem of asymptotic eigenvalue distribution.
Theorem 2.2 Let A
n
and B
n
be asymptotically equivalent sequences of ma-
trices with eigenvalues α
n,k
and β
n,k
, respectively. Assume that the eigenvalue
moments of either matrix converge, e.g., lim
n→∞
n
−1
n−1

k=0
α
s
n,k
exists and is finite
for any positive integer s. Then
lim
n→∞
n
−1
n−1


k=0
α
s
n,k
= lim
n→∞
n
−1
n−1

k=0
β
s
n,k
. (2.27)
12 CHAPTER 2. THE ASYMPTOTIC BEHAVIOR OF MATRICES
Proof.
Let A
n
= B
n
+ D
n
as in Lemma 2.3 and consider A
s
n
− B
s
n

∆
=∆
n
. Since
the eigenvalues of A
s
n
are α
s
n,k
, (2.27) can be written in terms of ∆
n
as
lim
n→∞
n
−1
Tr∆
n
=0. (2.28)
The matrix ∆
n
is a sum of several terms each being a product of ∆

n
s and
B

n
s but containing at least one D

n
. Repeated application of Lemma 2.2 thus
gives
|∆
n
|≤K

|D
n
|
−→
n→∞
0. (2.29)
where K

does not depend on n. Equation (2.29) allows us to apply Lemma
2.3 to the matrices A
s
n
and D
s
n
to obtain (2.28) and hence (2.27).
Theorem 2.2 is the fundamental theorem concerning asymptotic eigen-
value behavior. Most of the succeeding results on eigenvalues will be appli-
cations or specializations of (2.27).
Since (2.26) holds for any positive integer s we can add sums correspond-
ing to different values of s to each side of (2.26). This observation immedi-
ately yields the following corollary.
Corollary 2.2 Let A

n
and B
n
be asymptotically equivalent sequences of ma-
trices with eigenvalues α
n,k
and β
n,k
, respectively, and let f(x) be any poly-
nomial. Then
lim
n→∞
n
−1
n−1

k=0
f (α
n,k
) = lim
n→∞
n
−1
n−1

k=0
f (β
n,k
) . (2.30)
Whether or not A

n
and B
n
are Hermitian, Corollary 2.2 implies that
(2.30) can hold for any analytic function f(x) since such functions can be
expanded into complex Taylor series, i.e., into polynomials. If A
n
and B
n
are Hermitian, however, then a much stronger result is possible. In this
case the eigenvalues of both matrices are real and we can invoke the Stone-
Weierstrass approximation Theorem [4, p. 146] to immediately generalize
Corollary 2.3. This theorem, our one real excursion into analysis, is stated
below for reference.
Theorem 2.3 (Stone-Weierstrass) If F (x) is a continuous complex function
on [a, b], there exists a sequence of polynomials p
n
(x) such that
lim
n→∞
p
n
(x)=F (x)
13
uniformly on [a, b].
Stated simply, any continuous function defined on a real interval can be
approximated arbitrarily closely by a polynomial. Applying Theorem 2.3 to
Corollary 2.2 immediately yields the following theorem:
Theorem 2.4 Let A
n

and B
n
be asymptotically equivalent sequences of Her-
mitian matrices with eigenvalues α
n,k
and β
n,k
, respectively. Since A
n
and
B
n
are bounded there exist finite numbers m and M such that
m ≤ α
n,k
,β
n,k
≤ M, n=1, 2, k =0, 1, ,n− 1. (2.31)
Let F (x) be an arbitrary function continuous on [m, M]. Then
lim
n→∞
n
−1
n−1

k=0
F [α
n,k
] = lim
n→∞

n
−1
n−1

k=0
F [β
n,k
] (2.32)
if either of the limits exists.
Theorem 2.4 is the matrix equivalent of Theorem (7.4a) of [1]. When two
real sequences {α
n,k
; k =0, 1, ,n−1} and {β
n,k
; k =0, 1, ,n−1} satisfy
(2.31)-(2.32), they are said to be asymptotically equally distributed [1, p. 62].
As an example of the use of Theorem 2.4 we prove the following corollary
on the determinants of asymptotically equivalent matrices.
Corollary 2.3 Let A
n
and B
n
be asymptotically equivalent Hermitian matri-
ces with eigenvalues α
n,k
and β
n,k
, respectively, such that α
n,k
,β

n,k
≥ m>0.
Then
lim
n→∞
(det A
n
)
1/n
= lim
n→∞
(det B
n
)
1/n
. (2.33)
Proof.
From Theorem 2.4 we have for F (x)=lnx
lim
n→∞
n
−1
n−1

k=0
ln α
n,k
= lim
n→∞
n

−1
n−1

k=0
ln β
n,k
14 CHAPTER 2. THE ASYMPTOTIC BEHAVIOR OF MATRICES
and hence
lim
n→∞
exp

n
−1
ln
n−1

k=0
α
n,k

= lim
n→∞
exp

n
−1
ln
n−1


k=0
β
n,k

or equivalently
lim
n→∞
exp[n
−1
ln det A
n
] = lim
n→∞
exp[n
−1
ln det B
n
],
from which (2.33) follows.
With suitable mathematical care the above corollary can be extended to
the case where α
n,k
,β
n,k
> 0, but there is no m satisfying the hypothesis of
the corollary, i.e., where the eigenvalues can get arbitrarily small but are still
strictly positive.
In the preceding chapter the concept of asymptotic equivalence of matri-
ces was defined and its implications studied. The main consequences have
been the behavior of inverses and products (Theorem 2.1) and eigenvalues

(Theorems 2.2 and 2.4). These theorems do not concern individual entries
in the matrices or individual eigenvalues, rather they describe an “average”
behavior. Thus saying A
−1
n
∼ B
−1
n
means that that |A
−1
n
− B
−1
n
|
−→
n→∞
0 and
says nothing about convergence of individual entries in the matrix. In certain
cases stronger results on a type of elementwise convergence are possible using
the stronger norm of Baxter [7, 8]. Baxter’s results are beyond the scope of
this report.
The major use of the theorems of this chapter is that we can often study
the asymptotic behavior of complicated matrices by studying a more struc-
tured and simpler asymptotically equivalent matrix.
Chapter 3
Circulant Matrices
The properties of circulant matrices are well known and easily derived [3, p.
267],[19]. Since these matrices are used both to approximate and explain the
behavior of Toeplitz matrices, it is instructive to present one version of the

relevant derivations here.
A circulant matrix C is one having the form
C =













c
0
c
1
c
2
··· c
n−1
c
n−1
c
0
c
1

c
2
.
.
.
c
n−1
c
0
c
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
c
2
c
1

c
1
··· c
n−1
c
0













, (3.1)
where each row is a cyclic shift of the row above it. The matrix C is itself
a special type of Toeplitz matrix. The eigenvalues ψ
k
and the eigenvectors
y
(k)
of C are the solutions of
Cy = ψy (3.2)
or, equivalently, of the n difference equations
m−1


k=0
c
n−m+k
y
k
+
n−1

k=m
c
k−m
y
k
= ψy
m
; m =0, 1, ,n− 1. (3.3)
Changing the summation dummy variable results in
n−1−m

k=0
c
k
y
k+m
+
n−1

k=n−m
c

k
y
k−(n−m)
= ψy
m
; m =0, 1, ,n− 1. (3.4)
15
16 CHAPTER 3. CIRCULANT MATRICES
One can solve difference equations as one solves differential equations — by
guessing an (hopefully) intuitive solution and then proving that it works.
Since the equation is linear with constant coefficients a reasonable guess is
y
k
= ρ
k
(analogous to y(t)=e
sτ
in linear time invariant differential equa-
tions). Substitution into (3.4) and cancellation of ρ
m
yields
n−1−m

k=0
c
k
ρ
k
+ ρ
−n

n−1

k=n−m
c
k
ρ
k
= ψ.
Thus if we choose ρ
−n
=1, i.e., ρ is one of the n distinct complex n
th
roots
of unity, then we have an eigenvalue
ψ =
n−1

k=0
c
k
ρ
k
(3.5)
with corresponding eigenvector
y = n
−1/2

1,ρ,ρ
2
, ,ρ

n−1

, (3.6)
where the normalization is chosen to give the eigenvector unit energy. Choos-
ing ρ
j
as the complex n
th
root of unity, ρ
j
= e
−2πij/n
, we have eigenvalue
ψ
m
=
n−1

k=0
c
k
e
−2πimk/n
(3.7)
and eigenvector
y
(m)
= n
−1/2


1,e
−2πim/n
, ···,e
−2πi(n−1)/n

.
From (3.7) we can write
C = U
∗
ΨU, (3.8)
where
U =

y
(0)
|y
(1)
|···|y
(n−1)

= n
−1

e
−2πimk/n
; m, k =0, 1, ,n− 1

Ψ={ψ
k
δ

k,j
}
17
To verify (3.8) we note that the (k, j)
th
element of C,saya
k,j
,is
a
k,j
= n
−1
n−1

m=0
e
2πim(k− j)/n
ψ
m
= n
−1
n−1

m=0
e
2πim(k− j)/n
n−1

r=0
c

r
e
2πimr/n
= n
−1
n−1

r=0
c
r
n−1

m=0
e
2πim(k− j+r)/n
.
(3.9)
But we have
n−1

m=0
e
2πim(k− j+r)/n
=

nk− j = −r mod n
0 otherwise
so that a
k,j
= c

−(k−j)mod n
. Thus (3.8) and (3.1) are equivalent. Furthermore
(3.9) shows that any matrix expressible in the form (3.8) is circulant.
Since C is unitarily similar to a diagonal matrix it is normal. Note that
all circulant matrices have the same set of eigenvectors. This leads to the
following properties.
Theorem 3.1 Let C = {c
k−j
} and B = {b
k−j
} be circulant n × n matrices
with eigenvalues
ψ
m
=
n−1

k=0
c
k
e
−2πimk/n
β
m
=
n−1

k=0
b
k

e
−2πimk/n
,
respectively.
1. C and B commute and
CB = BC = U
∗
γU ,
where γ = {ψ
m
β
m
δ
k,m
}, and CB is also a circulant matrix.
18 CHAPTER 3. CIRCULANT MATRICES
2. C + B is a circulant matrix and
C + B = U
∗
ΩU,
where Ω={(ψ
m
+ β
m
)δ
k,m
}
3. If ψ
m
=0; m =0, 1, ,n− 1, then C is nonsingular and

C
−1
= U
∗
Ψ
−1
U
so that the inverse of C can be straightforwardly constructed.
Proof.
We have C = U
∗
ΨU and B = U
∗
ΦU where Ψ and Φ are diagonal matrices
with elements ψ
m
δ
k,m
and β
m
φ
k,m
, respectively.
1. CB = U
∗
ΨUU
∗
ΦU
= U
∗

ΨΦU
= U
∗
ΦΨU = BC
Since ΨΦ is diagonal, (3.9) implies that CB is circulant.
2. C + B = U
∗
(Ψ + Φ)U
3. C
−1
=(U
∗
ΨU)
−1
= U
∗
Ψ
−1
U
if Ψ is nonsingular.
Circulant matrices are an especially tractable class of matrices since in-
verses, products, and sums are also circulants and hence both straightforward
to construct and normal. In addition the eigenvalues of such matrices can
easily be found exactly.
In the next chapter we shall see that certain circulant matrices asymp-
totically approximate Toeplitz matrices and hence from Chapter 2 results
similar to those in Theorem 3 will hold asymptotically for Toeplitz matrices.
Chapter 4
Toeplitz Matrices
In this chapter the asymptotic behavior of inverses, products, eigenvalues,

and determinants of finite Toeplitz matrices is derived by constructing an
asymptotically equivalent circulant matrix and applying the results of the
previous chapters. Consider the infinite sequence {t
k
; k =0, ±1, ±2, ···}and
define the finite (n × n) Toeplitz matrix T
n
= {t
k−j
} as in (1.1). Toeplitz
matrices can be classified by the restrictions placed on the sequence t
k
.If
there exists a finite m such that t
k
=0, |k| >m,then T
n
is said to be a
finite order Toeplitz matrix. If t
k
is an infinite sequence, then there are two
common constraints. The most general is to assume that the t
k
are square
summable, i.e., that
∞

k=−∞
|t
k

|
2
< ∞ . (4.1)
Unfortunately this case requires mathematical machinery beyond that as-
sumed in this paper; i.e., Lebesgue integration and a relatively advanced
knowledge of Fourier series. We will make the stronger assumption that the
t
k
are absolutely summable, i.e.,
∞

k=−∞
|t
k
| < ∞. (4.2)
This assumption greatly simplifies the mathematics but does not alter the
fundamental concepts involved. As the main purpose here is tutorial and we
wish chiefly to relay the flavor and an intuitive feel for the results, this paper
will be confined to the absolutely summable case. The main advantage of
(4.2) over (4.1) is that it ensures the existence and continuity of the Fourier
19
20 CHAPTER 4. TOEPLITZ MATRICES
series f(λ) defined by
f(λ)=
∞

k=−∞
t
k
e

ikλ
= lim
n→∞
n

k=−n
t
k
e
ikλ
. (4.3)
Not only does the limit in (4.3) converge if (4.2) holds, it converges uniformly
for all λ, that is, we have that






f(λ) −
n

k=−n
t
k
e
ikλ







=






−n−1

k=−∞
t
k
e
ikλ
+
∞

k=n+1
t
k
e
ikλ







≤






−n−1

k=−∞
t
k
e
ikλ






+






∞


k=n+1
t
k
e
ikλ






≤
−n−1

k=−∞
|t
k
| +
∞

k=n+1
|t
k
|
,
where the righthand side does not depend on λ and it goes to zero as n →∞
from (4.2), thus given  there is a single N, not depending on λ, such that







f(λ) −
n

k=−n
t
k
e
ikλ






≤ , all λ ∈ [0, 2π] , if n ≥ N. (4.4)
Note that (4.2) is indeed a stronger constraint than (4.1) since
∞

k=−∞
|t
k
|
2
≤




∞

k=−∞
|t
k
|



2
.
Note also that (4.2) implies that f (λ) is bounded since
|f(λ)|≤
∞

k=−∞
|t
k
e
ikλ
|
≤
∞

k=−∞
|t
k
|
∆
= M

|f|
< ∞ .
The matrix T
n
will be Hermitian if and only if f is real, in which case we
denote the least upper bound and greatest lower bound of f(λ)byM
f
and
m
f
, respectively. Observe that max(|m
f
|, |M
f
|) ≤ M
|f|
.
21
Since f(λ) is the Fourier series of the sequence t
k
, we could alternatively
begin with a bounded and hence Riemann integrable function f(λ)on[0, 2π]
(|f(λ)|≤M
|f|
< ∞ for all λ) and define the sequence of n × n Toeplitz
matrices
T
n
(f)=


(2π)
−1

2π
0
f(λ)e
−i(k−j)
dλ ; k, j =0, 1, ···,n− 1

. (4.5)
As before, the Toeplitz matrices will be Hermitian iff f is real. The as-
sumption that f(λ) is Riemann integrable implies that f(λ) is continuous
except possibly at a countable number of points. Which assumption is made
depends on whether one begins with a sequence t
k
or a function f(λ)—
either assumption will be equivalent for our purposes since it is the Riemann
integrability of f(λ) that simplifies the bookkeeping in either case. Before
finding a simple asymptotic equivalent matrix to T
n
, we use Corollary 2.1
to find a bound on the eigenvalues of T
n
when it is Hermitian and an upper
bound to the strong norm in the general case.
Lemma 4.1 Let τ
n,k
be the eigenvalues of a Toeplitz matrix T
n
(f).IfT

n
(f)
is Hermitian, then
m
f
≤ τ
n,k
≤ M
f
. (4.6)
Whether or not T
n
(f) is Hermitian,
 T
n
(f) ≤ 2M
|f|
(4.7)
so that the matrix is uniformly bounded over n if f is bounded.
Proof.
Property (4.6) follows from Corollary 2.1:
max
k
τ
n,k
= max
x
(x
∗
T

n
x)/(x
∗
x) (4.8)
min
k
τ
n,k
= min
x
(x
∗
T
n
x)/(x
∗
x)
22 CHAPTER 4. TOEPLITZ MATRICES
so that
x
∗
T
n
x =
n−1

k=0
n−1

j=0

t
k−j
x
k
¯x
j
=
n−1

k=0
n−1

j=0

(2π)
−1

2π
0
f(λ)e
i(k−j)λ
dλ

x
k
¯x
j
=(2π)
−1


2π
0





n−1

k=0
x
k
e
ikλ





2
f(λ) dλ
(4.9)
and likewise
x
∗
x =
n−1

k=0
|x

k
|
2
=(2π)
−1

2π
0
dλ|
n−1

k=0
x
k
e
ikλ
|
2
. (4.10)
Combining (4.9)-(4.10) results in
m
f
≤

2π
0
dλf(λ)






n−1

k=0
x
k
e
ikλ





2

2π
0
dλ





n−1

k=0
x
k
e

ikλ





2
=
x
∗
T
n
x
x
∗
x
≤ M
f
, (4.11)
which with (4.8) yields (4.6). Alternatively, observe in (4.11) that if e
(k)
is
the eigenvector associated with τ
n,k
, then the quadratic form with x = e
(k)
yields x
∗
T
n

x = τ
n,k

n−1
k=0
|x
k
|
2
. Thus (4.11) implies (4.6) directly.
We have already seen in (2.13) that if T
n
(f) is Hermitian, then  T
n
(f) =
max
k
|τ
n,k
|
∆
= |τ
n,M
|, which we have just shown satisfies |τ
n,M
|≤max(|M
f
|, |m
f
|)

which in turn must be less than M
|f|
, which proves (4.7) for Hermitian ma-
trices Suppose that T
n
(f) is not Hermitian or, equivalently, that f is not
real. Any function f can be written in terms of its real and imaginary parts,
f = f
r
+ if
i
, where both f
r
and f
i
are real. In particular, f
r
=(f + f
∗
)/2
and f
i
=(f − f
∗
)/2i. Since the strong norm is a norm,
 T
n
(f)  =  T
n
(f

r
+ if
i
) 
=  T
n
(f
r
)+iT
n
(f
i
) 
≤T
n
(f
r
)  +  T
n
(f
i
) 
≤ M
|f
r
|
+ M
|f
i
|

.
4.1. FINITE ORDER TOEPLITZ MATRICES 23
Since |(f ±f
∗
)/2 ≤ (|f|+|f
∗
|)/2 ≤ M
|f|
, M
|f
r
|
+M
|f
i
|
≤ 2M
|f|
, proving (4.7).
Note for later use that the weak norm between Toeplitz matrices has a
simpler form than (2.14). Let T
n
= {t
k−j
} and T

n
= {t

k−j

} be Toeplitz, then
by collecting equal terms we have
|T
n
− T

n
|
2
= n
−1
n−1

k=0
n−1

j=0
|t
k−j
− t

k−j
|
2
= n
−1
n−1

k=−(n−1)
(n −|k|)|t

k
− t

k
|
2
=
n−1

k=−(n−1)
(1 −|k|/n)|t
k
− t

k
|
2
. (4.12)
We are now ready to put all the pieces together to study the asymptotic
behavior of T
n
. If we can find an asymptotically equivalent circulant matrix
then all of the results of Chapters 2 and 3 can be instantly applied. The
main difference between the derivations for the finite and infinite order case
is the circulant matrix chosen. Hence to gain some feel for the matrix chosen
we first consider the simpler finite order case where the answer is obvious,
and then generalize in a natural way to the infinite order case.
4.1 Finite Order Toeplitz Matrices
Let T
n

be a sequence of finite order Toeplitz matrices of order m + 1, that is,
t
i
= 0 unless |i|≤m. Since we are interested in the behavior or T
n
for large n
we choose n>>m. A typical Toeplitz matrix will then have the appearance
of the following matrix, possessing a band of nonzero entries down the central
diagonal and zeros everywhere else. With the exception of the upper left and
lower right hand corners that T
n
looks like a circulant matrix, i.e. each row