This report investigates the possible spectra of large, finite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upper-left m x m block. The main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations, provided they have sufficiently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical non-Hermitian Toeplitz matrices there exist certain rank-one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large-dimensional limit

Boettcher, A.

Embree, Mark

Sokolov, V. I.

Albanian

ORA - Oxford University Research Archive

On large Toeplitz band matrices with an uncertain block

Mathematical Institute Eprints Archive

On large Toeplitz band matries
with an unertain blok
A. Botther
a;1
, M. Embree
b;2
and V. I. Sokolov
a;3
a
Fakultat fur Mathematik, TU Chemnitz, 09107 Chemnitz, Germany
b
Oxford University Computing Laboratory, Wolfson Building, Parks Road,
Oxford OX1 3QD, UK
Abstrat
This report investigates the possible spetra of large, nite dimensional Toeplitz
band matries with perturbations (impurities, unertainties) in the upper-leftmm
blok. The main result shows that the asymptoti spetrum of suh a matrix is not
aeted by these perturbations, provided they have suÆiently small norm. This
follows from analysis of strutured pseudospetra (strutured spetral value sets).
In ontrast, for typial non-Hermitian Toeplitz matries there exist ertain rank-
one perturbations of arbitrarily small norm that move an eigenvalue away from the
asymptoti spetrum in the large-dimensional limit.
1 Introdution
Let A be a bounded linear operator on `
2
(N) or on C
n
with the `
2
norm. We
denote the spetrum of A by spA. For a natural number m, let P
m
be the
projetion on `
2
(N) or C
n
that sends a sequene x = fx
k
g to the sequene
dened by (P
m
x)
k
= x
k
for 1  k  m and (P
m
x)
k
= 0 for k  m+1. We set
sp
m
"
A =
[
kKk"
sp(A + P
m
KP
m
): (1)
Thus, sp
m
"
A measures the extent to whih spA an inrease by a perturbation
(impurity, unertainty) of norm at most " loalized in the upper-left m m
blok of A.
1
Email: aboetthmathematik.tu-hemnitz.de.
2
Email: mark.embreeomlab.ox.a.uk. Supported by UK Engineering and Phys-
ial Sienes Researh Counil Grant GR/M12414.
3
Email: sokolovmathematik.tu-hemnitz.de.
Preprint submitted to Elsevier Preprint 27 April 2001
Sets like sp
m
"
A are referred to as strutured pseudospetra or strutured spe-
tral value sets. These sets have been introdued and applied to problems of lin-
ear systems theory by Hinrihsen, Kelb, Prithard, Gallestey and others (see,
e.g., [9,15,16℄). Consider, for example, the system _x = Ax+P
m
v; y = P
m
x, in
whih A is a large or innite matrix and only the rst m omponents of the
input and the output are available. The problem of the existene of a feedbak
v = Ky with kKk  " for whih the resulting system operator has  in its
spetrum is equivalent to the question of whether  2 sp
m
"
A.
Here, we study the ase where A is a Toeplitz band matrix. Random per-
turbations to suh matries (typially on the entire main diagonal, or in a
single entry) arise in non-Hermitian quantum mehanis [6{8,11,13℄, popula-
tion dynamis [18℄, and small world networks [21℄, for instane. A deterministi
version arises in the study of initial-boundary value problems, where a nite-
dierene disretization yields a Toeplitz matrix plus a orner perturbation
orresponding to boundary onditions [1,5,17℄. Pseudospetral analysis was
applied to a spei bidiagonal example in [23℄, while strutured pseudospe-
tra were rst used for perturbed Toeplitz matries in our papers [2,3℄.
In [2℄, we showed that if A is an innite non-diagonal Toeplitz band ma-
trix, then there exists an "
0
= "
0
(A;m) > 0 suh that sp
m
"
A = spA for
all " 2 (0; "
0
). This reveals that suÆiently small unertainties, loalized in
a xed nite set of sites, annot hange the spetrum of A. Here we prove
an analogous result for large nite Toeplitz band matries. This is partiu-
larly interesting beause the spetra of typial non-Hermitian nite Toeplitz
band matries are very sensitive to perturbations; the resolvent norm grows
exponentially in the matrix dimension, n, at points well separated from the
asymptoti spetrum [19℄. In the large-n limit of these nite matries, ertain
rank-one perturbations of arbitrarily small norm move an eigenvalue to any
point where the resolvent norm grows without bound in n [22℄. The present
work demonstrates a lass of rank-m perturbations of nite norm that fail to
move the eigenvalues anywhere beyond the asymptoti spetrum.
Single-entry perturbations of large nite Toeplitz matries are the fous of [3℄.
Let e
j
e

k
be the matrix that is zero everywhere exept in the (j; k) entry, whih
equals 1. The single-entry analogue of (1) is
sp
(j;k)
"
A =
[
!2"D
sp (A+ !e
j
e

k
);
where "D = fz 2 C : jzj  "g. In [3℄, we used the well-known equality
sp
(j;k)
"
A = spA [ f 2 C n spA : j[(A  I)
 1
℄
kj
j  1="g (2)
2
to determine sp
(j;k)
"
A when A is a large Toeplitz band matrix. The analogue
of (2) for the strutured pseudospetra (1) is the formula
sp
m
"
A = spA [ f 2 C n spA : kP
m
(A  I)
 1
P
m
k  1="g: (3)
This formula, established only reently by Gallestey, Hinrihsen, and Prithard
in [9℄, is the key to our present analysis.
2 Main result
For a omplex-valued ontinuous funtion a on the omplex unit irle T, the
innite Toeplitz matrix T (a) and the nite Toeplitz matries T
n
(a) are dened
by
T (a) = (a
j `
)
1
j;`=1
and T
n
(a) = (a
j `
)
n
j;`=1
;
where a
k
is the kth Fourier oeÆient of a,
a
k
=
1
2
2
Z
0
a(e
i
)e
 ik
d; k 2 Z:
Here, we require a to be a trigonometri polynomial, a 2 P, whih means that
at most a nite number of the Fourier oeÆients are nonzero, or, equivalently,
that T (a) is a banded matrix. The matrix T (a) indues a bounded operator
on `
2
(N), and we think of T
n
(a) as a bounded operator on C
n
with the `
2
norm.
The spetrum spA of a bounded linear operator is the set of all  2 C for
whih A I is not invertible. The spetrum of T (a) was identied by Gohberg
in 1952 [10℄:
sp T (a) = a(T) [ f 2 C n a(T) : wind(a; ) 6= 0g; (4)
where wind(a; ) is the winding number of a (on the ounter-lokwise oriented
unit irle) about . Eah point in spT (a) n a(T) is an eigenvalue of nite
multipliity, while a point on a(T) may be an eigenvalue or not.
Suppose that a 2 P and that T (a) is not triangular. Thus we an write
a(t) =
q
X
k= p
a
k
t
k
; p  1; q  1; a
 p
a
q
6= 0: (5)
3
Theorem 2.1 If a is of the form (5), then
sp
m
"
T (a) = spT (a) for all " 2 (0; "
1
)
where
"
1
= max(ja
 p
j; ja
q
j)
0

m 1
X
j=0
(m  j)p
2j
1
A
 1=2
0

m 1
X
j=0
(m  j)q
2j
1
A
 1=2
: (6)
A slightly dierent (and more general) version of this theorem was established
in our paper [2℄. For the reader's onveniene, we give a proof of Theorem 2.1
in Setion 4.
The spetra of T
n
(a) for large n were haraterized by Shmidt and Spitzer
in 1960 [20℄. They showed that the sets sp T
n
(a) onverge in the Hausdor
metri to some \very thin" set (a),
lim
n!1
spT
n
(a) = (a); (7)
and they gave the following desription of (a). It is lear that (a) = fa
0
g if
T (a) is triangular. So suppose that T (a) is not triangular and a is given by (5).
For % > 0, dene a
%
2 P by a
%
(t) =
P
q
k= p
a
k
%
k
t
k
. In [20℄, it was shown that
(a) =
\
%>0
sp T (a
%
): (8)
If T (a) is triangular, then, obviously,
sp
m
"
T
n
(a)  sp
1
"
T
n
(a) = a
0
+ "D:
This implies that the limit of sp
m
"
T
n
(a) as n ! 1 is stritly larger than
(a) = fa
0
g for every " > 0. As the following theorem shows, this annot
happen for non-triangular Toeplitz band matries.
Theorem 2.2 If a is of the form (5), then there exists an "
2
= "
2
(a;m) > 0
suh that
lim
n!1
sp
m
"
T
n
(a) = (a) for all " 2 (0; "
2
);
the limit being taken in the Hausdor metri.
4
Theorems 2.1 and 2.2 show that sp
m
"
T (a) and limsp
m
"
T
n
(a) stabilize at on-
stant sets before " reahes zero and that, moreover, these two onstant sets
are in general dierent. Thus, exept for some trivial ases, we always have
lim
n!1
sp
m
"
T
n
(a) $ sp
m
"
T (a)
for suÆiently small ", implying that the passage from the \nite volume ase"
to the \innite volume ase" is disontinuous.
3 An example
We provide a onrete demonstration of these results for the symbol a(t) =
t+t
 2
. In this ase, the set lim
n!1
spT
n
(a) = (a) onsists of three line segments
joined at the origin,
(a) = fre
i
: 0  r  3  2
 2=3
;  2 f0;
2
3
;
4
3
gg;
see [20, Setion 7℄. Figure 1 illustrates this set, together with sp T (a).
Now onsider perturbations to the upper-left 3 3 blok of T
n
(a), i.e., m = 3.
We abbreviate P
3
KP
3
to E. Figure 2 shows the eigenvalues of T
50
(a) + E for
1000 values of E, where E has omplex normally distributed entries saled so
that kEk = " for " =
1
2
, " =
3
4
, and " = 1. These sets suggest sp
3
"
T
50
(a). Note
that sp
3
"
T
50
(a) losely resembles (a) for " =
1
2
, while sp
3
"
T
50
(a) deviates from
(a) as " inreases. Figure 3 oers more onvining evidene, illustrating the
maximum distane that some eigenvalue of T
n
(a) + E varies from (a),
max
2sp(T
n
(a)+E)
dist (;(a)) := max
2sp(T
n
(a)+E)
min
z2(a)
j  zj;
as a funtion of " = kEk for n = 25 and n = 100. (For eah xed ", we take
the maximum over 1000 randomly-generated perturbations E.) For values
of " roughly less than
4
10
, the perturbed eigenvalues don't dier muh from
the asymptoti spetrum, while for larger values of " this dierene appears
to grow linearly with ". As the matrix dimension inreases, the dierene
between sp (T
n
(a) + E) and (a) redues for small ". (Note that T
3
(a) is a
normal matrix, so that sp (T
3
(a) + E) an vary from (a) by " for all "  0.)
The omputations presented here were exeuted with MATLAB.
5
Fig. 1. The sets spT (a) (left) and lim
n!1
spT
n
(a) = (a) (right) for a(t) = t+ t
 2
.
Fig. 2. Superimposed eigenvalues of T
50
(a)+E (with a(t) = t+t
 2
) for 1000 omplex
random perturbations E of norm
1
2
,
3
4
, and 1. The solid urves on eah plot reveal
the boundary of sp
50
"
T
50
(a), omputed using the software [24℄.
4 Proofs
Proof of Theorem 2.1. Pik  2 C n spT (a). We have
a(t)   = t
 p
a
q
(t  z
1
()) : : : (t  z
p+q
()); (9)
and sine a   has no zeros on T, we an write
a(t)   = t
 p
a
q
(t  Æ
1
) : : : (t  Æ
r
)(t  
1
) : : : (t  
s
)
where jÆ
j
j = jÆ
j
()j < 1 and j
j
j = j
j
()j > 1. As wind (a; ) =  p + r and
wind (a; ) = 0 due to (4), we see that neessarily p = r and hene q = s.
Thus, a(t)   = a
q
'
 
(t)'
+
(t) with
'
 
(t) :=
 
1 
Æ
1
t
!
: : :
 
1 
Æ
p
t
!
; '
+
(t) = (t  
1
) : : : (t  
q
):
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
"
m
a
x

2
s
p
(
T
n
(
a
)
+
E
)
d
i
s
t
(

;

(
a
)
)
n = 25
n = 100
Fig. 3. Deviation of sp(T
n
(a) + E) from (a) for the symbol a(t) = t + t
 2
. Eah
data point represents the maximum distane over 1000 randomly generated pertur-
bations E of norm ".
Standard omputations with Toeplitz matries (see, e.g., [4℄) now give
T (a  ) = a
q
T ('
 
'
+
) = a
q
T ('
 
)T ('
+
);
T
 1
(a  ) = a
 1
q
T ('
 1
+
)T ('
 1
 
);
P
m
T
 1
(a  )P
m
= a
 1
q
P
m
T ('
 1
+
)T ('
 1
 
)P
m
= a
 1
q
T
m
('
 1
+
)T
m
('
 1
 
):
We have
'
 1
 
(t) =
 
1 +
Æ
1
t
+
Æ
2
1
t
2
+ : : :
!
: : :
 
1 +
Æ
p
t
+
Æ
2
p
t
2
+ : : :
!
=:
1
X
n=0
b
n
t
 n
with
jb
n
j =






X

1
+:::+
p
=n
Æ

1
1
: : : Æ

p
p






 (jÆ
1
j+ : : :+ jÆ
p
j)
n
< p
n
;
and beause
T
m
('
 1
 
) =
0
B
B
B
B
B
B
B
B

b
0
b
1
: : : b
m 1
b
0
: : : b
m 2
.
.
.
.
.
.
b
0
1
C
C
C
C
C
C
C
C
A
;
7
it follows that
kT
m
('
 1
 
)k
2
mjb
0
j
2
+ (m  1)jb
1
j
2
+ : : :+ jb
m 1
j
2
m+ (m  1)p
2
+ : : :+ p
2m 2
:
Analogously,
kT
m
('
 1
+
)k
2
 m + (m  1)q
2
+ : : :+ q
2m 2
:
In summary,
kP
m
T
 1
(a  )P
m
k
2

1
ja
q
j
2
0

m 1
X
j=0
(m  j)p
2j
1
A
0

m 1
X
j=0
(m  j)q
2j
1
A
: (10)
Denote the right-hand side of (10) by M=ja
q
j
2
. From (3) we now infer that
sp
m
"
T (a) is the union of sp T (a) and the set

 =2 sp T (a) : kP
m
T
 1
(a  )P
m
k
2

1
"
2


(
 =2 spT (a) :
M
ja
q
j
2

1
"
2
)
:
Clearly, the last set is empty whenever M=ja
q
j
2
< 1="
2
, whih gives the as-
sertion with "
1
= ja
q
j=
p
M . Considering the transpose of T (a), we get the
assertion with "
1
= ja
 p
j=
p
M . This implies Theorem 2.1 with "
1
given by
formula (6).
We now turn to the proof of Theorem 2.2.
Lemma 4.1 There exists a onstant Æ = Æ(a) > 1 suh that
(a) =
\
%2[1=Æ;Æ℄
spT (a
%
):
A proof of this lemma is in [3℄.
For a sequene fM
n
g
1
n=1
of nonempty ompat sets M
n
 C, we onsider the
two limiting sets
lim inf
n!1
M
n
:= f 2 C :  is the limit of some
sequene f
n
g
1
n=1
with 
n
2M
n
g;
lim sup
n!1
M
n
:= f 2 C :  is a partial limit of some
sequene f
n
g
1
n=1
with 
n
2M
n
g:
8
It is well known that the two equalities
lim inf
n!1
M
n
= lim sup
n!1
M
n
= M
are equivalent to the onvergene of M
n
to M in the Hausdor metri (see,
e.g., [12, Setions 3.1.1 and 3.1.2℄ or [14, Setion 28℄).
Lemma 4.2 For every " > 0,
lim sup
n!1
sp
m
"
T
n
(a)  sp
m
"
T (a):
Proof. Pik  2 C n sp
m
"
T (a). Then  =2 spT (a) and, by (3),
kP
m
T
 1
(a  )P
m
k < 1=":
It follows that there is an open neighborhood U  C of  suh that if  2 U ,
then  =2 spT (a) and
kP
m
T
 1
(a  )P
m
k < 1=":
A well-known result on the nite setion method for Toeplitz operators (see,
e.g., [4, p. 42℄) says that T
 1
n
(a   ) onverges strongly (i.e., pointwise) to
T
 1
(a  ). Consequently,
kP
m
T
 1
n
(a  )P
m
  P
m
T
 1
(a  )P
m
k ! 0 as n!1; (11)
and it is straightforward to hek that the onvergene in (11) is uniform with
respet to  in ompat subsets of U . Thus, there exist an open neighborhood
V  U of  and a natural number n
0
suh that
kP
m
T
 1
n
(a  )P
m
k < 1=" for all  2 V and all n  n
0
:
This, in onjuntion with (3), implies that V \ sp
m
"
T (a) = ; for all n  n
0
,
whene  =2 lim sup
n!1
sp
m
"
T (a).
Lemma 4.3 Let Æ be the onstant given by Lemma 4.1. If % 2 [1=Æ; Æ℄ and
" > 0, then
lim sup
n!1
sp
m
"
T
n
(a)  sp
m

T (a
%
); (12)
where  = "max(%
m+1
; %
 m 1
).
9
Proof. We have T
n
(a   ) = D
 1
n
(%)T
n
(a
%
  )D
n
(%), where D
n
(%) is the
diagonal matrix diag(%; %
2
; : : : ; %
n
). It results that
P
m
T
 1
n
(a  )P
m
= D
 1
m
(%)P
m
T
 1
n
(a
%
  )P
m
D
m
(%);
whene
kP
m
T
 1
n
(a  )P
m
k  
m
(%)kP
m
T
 1
n
(a
%
  )P
m
k;
where 
m
(%) := kD
 1
m
(%)k kD
m
(%)k. Sine also spT
n
(a) = sp T
n
(a
%
), we on-
lude from (3) that
sp
m
"
T
n
(a) = sp T
n
(a) [ f =2 spT
n
(a) : kP
m
T
 1
n
(a  )P
m
k  1="g
= sp T
n
(a
%
) [ f =2 sp T
n
(a
%
) : kP
m
T
 1
n
(a  )P
m
k  1="g
 sp T
n
(a
%
) [ f =2 sp T
n
(a
%
) : 
m
(%)kP
m
T
 1
n
(a
%
  )P
m
k  1="g
= sp
m
"
m
(%)
T
n
(a
%
);
and now Lemma 4.2 yields the inlusion
lim sup
n!1
sp
m
"
T
n
(a)  sp
m
"
m
(%)
T (a
%
):
Beause 
m
(%) = %
m 1
if %  1 and 
m
(%) = %
 m+1
if % < 1, we arrive at the
assertion.
Proof of Theorem 2.2. Denote the right-hand side of (6) by
C(p; q;m)max(ja
 p
j; ja
q
j);
let Æ be the onstant from Lemma 4.1, and put
"
2
:= C(p; q;m)ja
q
j=Æ
q+m 1
:
We laim that Theorem 2.2 is true with this hoie of "
2
.
Let " 2 (0; "
2
). Lemma 4.2 shows that if % 2 [1=Æ; Æ℄, then (12) holds with
 = "max(%
m 1
; %
 m+1
)
< C(p; q;m)ja
q
jmax(%
m 1
; %
 m+1
)=Æ
q+m 1
 C(p; q;m)ja
q
j%
q
Æ
q
max(%
m 1
; %
 m+1
)=Æ
q+m 1
(sine 1  %Æ)
= C(p; q;m)ja
q
j%
q
max(%
m 1
; %
 m+1
)=Æ
m 1
 C(p; q;m)ja
q
j%
q
: (13)
10
Applying Theorem 2.1 to T (a
%
) and taking into aount that ja
q
j does not
exeed max(ja
 p
j; ja
q
j), we see that
sp
m

T (a
%
) = sp T (a
%
) for  < C(p; q;m)ja
q
j%
q
:
Thus, (12) and (13) give lim sup
n!1
sp
m
"
T
n
(a)  spT (a
%
) for every % 2 [1=Æ; Æ℄.
By Lemma 4.1, this implies that
lim sup
n!1
sp
m
"
T
n
(a)  (a):
To establish inlusion in the other diretion, note that (7) implies
(a)  lim inf
n!1
sp T
n
(a)  lim inf
n!1
sp
m
"
T
n
(a);
thus ompleting the proof of Theorem 2.2.
Referenes
[1℄ R. M. Beam and R. F. Warming, The asymptoti spetra of banded Toeplitz
and quasi-Toeplitz matries, SIAM J. Si. Comput. 14 (1993) 971{1006.
[2℄ A. Botther, M. Embree, and V. I. Sokolov, Innite Toeplitz and Laurent
matries with loalized impurities, Linear Algebra Appl., to appear.
[3℄ A. Botther, M. Embree, and V. I. Sokolov, The spetra of large Toeplitz band
matries with a randomly perturbed entry, in preparation.
[4℄ A. Botther and B. Silbermann, Introdution to Large Trunated Toeplitz
Matries (Universitext, Springer-Verlag, New York, 1998).
[5℄ B. Cahlon, D. M. Kulkarni, and P. Shi, Stepwise stability for the heat equation
with a nonloal onstraint, SIAM J. Numer. Anal. 32 (1995) 571{593.
[6℄ E. B. Davies, Spetral properties of random non-self-adjoint matries and
operators, Pro. Roy. So. London Ser. A 457 (2001) 191{206.
[7℄ J. Feinberg and A. Zee, Non-Hermitian loalization and deloalization, Phys.
Rev. E 59 (1999) 6433{6443.
[8℄ J. Feinberg and A. Zee, Spetral urves of non-hermitian hamiltonians, Nu.
Phys. B 522 (1999) 599{623.
[9℄ E. Gallestey, D. Hinrihsen, and A. J. Prithard, Spetral value sets of losed
linear operators, Pro. Roy. So. London Ser. A 456 (2000) 1397{1418.
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[10℄ I. Gohberg, On an appliation of the theory of normed rings to singular integral
equations, Uspekhi Matem. Nauk SSSR 112 (1952) 149{156 [Russian℄.
[11℄ I. Ya. Goldsheid and B. A. Khoruzhenko, Distribution of eigenvalues in non-
Hermitian Anderson models, Phys. Rev. Lett. 80 (1998) 2897{2900.
[12℄ R. Hagen, S. Roh, and B. Silbermann, C

-Algebras and Numerial Analysis
(Dekker, New York, 2001).
[13℄ N. Hatano and D. R. Nelson, Vortex pinning and non-Hermitian quantum
mehanis, Phys. Rev. B 77 (1996) 8651{8673.
[14℄ F. Hausdor, Set Theory (Chelsea, New York, 1957).
[15℄ D. Hinrihsen and B. Kelb, Spetral value sets: a graphial tool for robustness
analysis, Systems Control Lett. 21 (1993) 127{136.
[16℄ D. Hinrihsen and A. J. Prithard, Real and omplex stability radii: a survey,
in: D. Hinrihsen and B. Martensson, eds., Control of Unertain Systems (Prog.
Systems Control Theory, Vol. 6, Birkhauser Verlag, Basel, 1990) 119{162.
[17℄ D. Kulkarni, D. Shmidt, and S.-K. Tsui, Eigenvalues of tridiagonal pseudo-
Toeplitz matries, Linear Algebra Appl. 297 (1999) 63{80.
[18℄ D. R. Nelson and N. M. Shnerb, Non-Hermitian loalization and population
biology, Phys. Rev. E 58 (1998) 1383{1403.
[19℄ L. Reihel and L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz
matries, Linear Algebra Appl. 162{164 (1992) 153{185.
[20℄ P. Shmidt and F. Spitzer, The Toeplitz matries of an arbitrary Laurent
polynomial, Math. Sand. 8 (1960) 15{38.
[21℄ G. Strang, X. Liu, S. Ott, and L. He, Loalized eigenvetors from loalized
matrix modiations, in preparation; see also G. Strang, \From the SIAM
President", SIAM News, April 2000, May 2000.
[22℄ L. N. Trefethen, Spetra and pseudospetra: the behavior of non-normal
matries and operators, in: M. Ainsworth, J. Levesley, and M. Marletta, eds.,
The Graduate Student's Guide to Numerial Analysis '98, (Springer-Verlag,
Berlin, 1999) 217{250.
[23℄ L. N. Trefethen, M. Contedini, and M. Embree, Spetra, pseudospetra, and
loalization for random bidiagonal matries, Comm. Pure Appl. Math. 54 (2001)
594{623.
[24℄ T. G. Wright, MATLAB Pseudospetra GUI, 2001. Available online at:
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12


Oxford University Research Archive

On large Toeplitz band matrieswith an unertain blokA. Botther a;1, M. Embree b;2 and V. I. Sokolov a;3aFakultat fur Mathematik, TU Chemnitz, 09107 Chemnitz, GermanybOxford University Computing Laboratory, Wolfson Building, Parks Road,Oxford OX1 3QD, UKAbstratThis report investigates the possible spetra of large, nite dimensional Toeplitzband matries with perturbations (impurities, unertainties) in the upper-leftmmblok. The main result shows that the asymptoti spetrum of suh a matrix is notaeted by these perturbations, provided they have suÆiently small norm. Thisfollows from analysis of strutured pseudospetra (strutured spetral value sets).In ontrast, for typial non-Hermitian Toeplitz matries there exist ertain rank-one perturbations of arbitrarily small norm that move an eigenvalue away from theasymptoti spetrum in the large-dimensional limit.1 IntrodutionLet A be a bounded linear operator on `2(N) or on Cn with the `2 norm. Wedenote the spetrum of A by spA. For a natural number m, let Pm be theprojetion on `2(N) or Cn that sends a sequene x = fxkg to the sequenedened by (Pmx)k = xk for 1  k  m and (Pmx)k = 0 for k  m+1. We setspm" A = [kKk" sp(A + PmKPm): (1)Thus, spm" A measures the extent to whih spA an inrease by a perturbation(impurity, unertainty) of norm at most " loalized in the upper-left m mblok of A.1 Email: aboetthmathematik.tu-hemnitz.de.2 Email: mark.embreeomlab.ox.a.uk. Supported by UK Engineering and Phys-ial Sienes Researh Counil Grant GR/M12414.3 Email: sokolovmathematik.tu-hemnitz.de.Preprint submitted to Elsevier Preprint 27 April 2001Sets like spm" A are referred to as strutured pseudospetra or strutured spe-tral value sets. These sets have been introdued and applied to problems of lin-ear systems theory by Hinrihsen, Kelb, Prithard, Gallestey and others (see,e.g., [9,15,16℄). Consider, for example, the system _x = Ax+Pmv; y = Pmx, inwhih A is a large or innite matrix and only the rst m omponents of theinput and the output are available. The problem of the existene of a feedbakv = Ky with kKk  " for whih the resulting system operator has  in itsspetrum is equivalent to the question of whether  2 spm" A.Here, we study the ase where A is a Toeplitz band matrix. Random per-turbations to suh matries (typially on the entire main diagonal, or in asingle entry) arise in non-Hermitian quantum mehanis [6{8,11,13℄, popula-tion dynamis [18℄, and small world networks [21℄, for instane. A deterministiversion arises in the study of initial-boundary value problems, where a nite-dierene disretization yields a Toeplitz matrix plus a orner perturbationorresponding to boundary onditions [1,5,17℄. Pseudospetral analysis wasapplied to a spei bidiagonal example in [23℄, while strutured pseudospe-tra were rst used for perturbed Toeplitz matries in our papers [2,3℄.In [2℄, we showed that if A is an innite non-diagonal Toeplitz band ma-trix, then there exists an "0 = "0(A;m) > 0 suh that spm" A = spA forall " 2 (0; "0). This reveals that suÆiently small unertainties, loalized ina xed nite set of sites, annot hange the spetrum of A. Here we provean analogous result for large nite Toeplitz band matries. This is partiu-larly interesting beause the spetra of typial non-Hermitian nite Toeplitzband matries are very sensitive to perturbations; the resolvent norm growsexponentially in the matrix dimension, n, at points well separated from theasymptoti spetrum [19℄. In the large-n limit of these nite matries, ertainrank-one perturbations of arbitrarily small norm move an eigenvalue to anypoint where the resolvent norm grows without bound in n [22℄. The presentwork demonstrates a lass of rank-m perturbations of nite norm that fail tomove the eigenvalues anywhere beyond the asymptoti spetrum.Single-entry perturbations of large nite Toeplitz matries are the fous of [3℄.Let ejek be the matrix that is zero everywhere exept in the (j; k) entry, whihequals 1. The single-entry analogue of (1) issp(j;k)" A = [!2"D sp (A+ !ejek);where "D = fz 2 C : jzj  "g. In [3℄, we used the well-known equalitysp(j;k)" A = spA [ f 2 C n spA : j[(A  I) 1℄kjj  1="g (2)2to determine sp(j;k)" A when A is a large Toeplitz band matrix. The analogueof (2) for the strutured pseudospetra (1) is the formulaspm" A = spA [ f 2 C n spA : kPm(A  I) 1Pmk  1="g: (3)This formula, established only reently by Gallestey, Hinrihsen, and Prithardin [9℄, is the key to our present analysis.2 Main resultFor a omplex-valued ontinuous funtion a on the omplex unit irle T, theinnite Toeplitz matrix T (a) and the nite Toeplitz matries Tn(a) are denedby T (a) = (aj `)1j;`=1 and Tn(a) = (aj `)nj;`=1;where ak is the kth Fourier oeÆient of a,ak = 12 2Z0 a(ei)e ikd; k 2 Z:Here, we require a to be a trigonometri polynomial, a 2 P, whih means thatat most a nite number of the Fourier oeÆients are nonzero, or, equivalently,that T (a) is a banded matrix. The matrix T (a) indues a bounded operatoron `2(N), and we think of Tn(a) as a bounded operator on Cn with the `2norm.The spetrum spA of a bounded linear operator is the set of all  2 C forwhih A I is not invertible. The spetrum of T (a) was identied by Gohbergin 1952 [10℄: sp T (a) = a(T) [ f 2 C n a(T) : wind(a; ) 6= 0g; (4)where wind(a; ) is the winding number of a (on the ounter-lokwise orientedunit irle) about . Eah point in spT (a) n a(T) is an eigenvalue of nitemultipliity, while a point on a(T) may be an eigenvalue or not.Suppose that a 2 P and that T (a) is not triangular. Thus we an writea(t) = qXk= p aktk; p  1; q  1; a paq 6= 0: (5)3Theorem 2.1 If a is of the form (5), thenspm" T (a) = spT (a) for all " 2 (0; "1)where "1 = max(ja pj; jaqj)0m 1Xj=0 (m  j)p2j1A 1=20m 1Xj=0 (m  j)q2j1A 1=2 : (6)A slightly dierent (and more general) version of this theorem was establishedin our paper [2℄. For the reader's onveniene, we give a proof of Theorem 2.1in Setion 4.The spetra of Tn(a) for large n were haraterized by Shmidt and Spitzerin 1960 [20℄. They showed that the sets sp Tn(a) onverge in the Hausdormetri to some \very thin" set (a),limn!1 spTn(a) = (a); (7)and they gave the following desription of (a). It is lear that (a) = fa0g ifT (a) is triangular. So suppose that T (a) is not triangular and a is given by (5).For % > 0, dene a% 2 P by a%(t) = Pqk= p ak%ktk. In [20℄, it was shown that(a) = \%>0 sp T (a%): (8)If T (a) is triangular, then, obviously,spm" Tn(a)  sp1" Tn(a) = a0 + "D:This implies that the limit of spm" Tn(a) as n ! 1 is stritly larger than(a) = fa0g for every " > 0. As the following theorem shows, this annothappen for non-triangular Toeplitz band matries.Theorem 2.2 If a is of the form (5), then there exists an "2 = "2(a;m) > 0suh that limn!1 spm" Tn(a) = (a) for all " 2 (0; "2);the limit being taken in the Hausdor metri.4Theorems 2.1 and 2.2 show that spm" T (a) and limspm" Tn(a) stabilize at on-stant sets before " reahes zero and that, moreover, these two onstant setsare in general dierent. Thus, exept for some trivial ases, we always havelimn!1 spm" Tn(a) $ spm" T (a)for suÆiently small ", implying that the passage from the \nite volume ase"to the \innite volume ase" is disontinuous.3 An exampleWe provide a onrete demonstration of these results for the symbol a(t) =t+t 2. In this ase, the set limn!1 spTn(a) = (a) onsists of three line segmentsjoined at the origin,(a) = frei : 0  r  3  2 2=3;  2 f0; 23 ; 43 gg;see [20, Setion 7℄. Figure 1 illustrates this set, together with sp T (a).Now onsider perturbations to the upper-left 3 3 blok of Tn(a), i.e., m = 3.We abbreviate P3KP3 to E. Figure 2 shows the eigenvalues of T50(a) + E for1000 values of E, where E has omplex normally distributed entries saled sothat kEk = " for " = 12 , " = 34 , and " = 1. These sets suggest sp3" T50(a). Notethat sp3" T50(a) losely resembles (a) for " = 12 , while sp3" T50(a) deviates from(a) as " inreases. Figure 3 oers more onvining evidene, illustrating themaximum distane that some eigenvalue of Tn(a) + E varies from (a),max2sp(Tn(a)+E) dist (;(a)) := max2sp(Tn(a)+E) minz2(a) j  zj;as a funtion of " = kEk for n = 25 and n = 100. (For eah xed ", we takethe maximum over 1000 randomly-generated perturbations E.) For valuesof " roughly less than 410 , the perturbed eigenvalues don't dier muh fromthe asymptoti spetrum, while for larger values of " this dierene appearsto grow linearly with ". As the matrix dimension inreases, the dierenebetween sp (Tn(a) + E) and (a) redues for small ". (Note that T3(a) is anormal matrix, so that sp (T3(a) + E) an vary from (a) by " for all "  0.)The omputations presented here were exeuted with MATLAB.5Fig. 1. The sets spT (a) (left) and limn!1 spTn(a) = (a) (right) for a(t) = t+ t 2.Fig. 2. Superimposed eigenvalues of T50(a)+E (with a(t) = t+t 2) for 1000 omplexrandom perturbations E of norm 12 , 34 , and 1. The solid urves on eah plot revealthe boundary of sp50" T50(a), omputed using the software [24℄.4 ProofsProof of Theorem 2.1. Pik  2 C n spT (a). We havea(t)   = t paq(t  z1()) : : : (t  zp+q()); (9)and sine a   has no zeros on T, we an writea(t)   = t paq(t  Æ1) : : : (t  Ær)(t  1) : : : (t  s)where jÆjj = jÆj()j < 1 and jjj = jj()j > 1. As wind (a; ) =  p + r andwind (a; ) = 0 due to (4), we see that neessarily p = r and hene q = s.Thus, a(t)   = aq' (t)'+(t) with' (t) :=  1  Æ1t ! : : : 1  Æpt ! ; '+(t) = (t  1) : : : (t  q):60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.6"max 2sp(T n(a)+E)dist(;(a)) n = 25n = 100Fig. 3. Deviation of sp(Tn(a) + E) from (a) for the symbol a(t) = t + t 2. Eahdata point represents the maximum distane over 1000 randomly generated pertur-bations E of norm ".Standard omputations with Toeplitz matries (see, e.g., [4℄) now giveT (a  ) = aqT (' '+) = aqT (' )T ('+);T 1(a  ) = a 1q T (' 1+ )T (' 1  );PmT 1(a  )Pm = a 1q PmT (' 1+ )T (' 1  )Pm = a 1q Tm(' 1+ )Tm(' 1  ):We have' 1  (t) =  1 + Æ1t + Æ21t2 + : : :! : : : 1 + Æpt + Æ2pt2 + : : :! =: 1Xn=0 bnt nwith jbnj =  X1+:::+p=n Æ11 : : : Æpp   (jÆ1j+ : : :+ jÆpj)n < pn;and beause Tm(' 1  ) = 0BBBBBBBB b0 b1 : : : bm 1b0 : : : bm 2. . . ...b01CCCCCCCCA ;7it follows thatkTm(' 1  )k2mjb0j2 + (m  1)jb1j2 + : : :+ jbm 1j2m+ (m  1)p2 + : : :+ p2m 2:Analogously, kTm(' 1+ )k2  m + (m  1)q2 + : : :+ q2m 2:In summary,kPmT 1(a  )Pmk2  1jaqj2 0m 1Xj=0 (m  j)p2j1A0m 1Xj=0 (m  j)q2j1A : (10)Denote the right-hand side of (10) by M=jaqj2. From (3) we now infer thatspm" T (a) is the union of sp T (a) and the set =2 sp T (a) : kPmT 1(a  )Pmk2  1"2  ( =2 spT (a) : Mjaqj2  1"2) :Clearly, the last set is empty whenever M=jaqj2 < 1="2, whih gives the as-sertion with "1 = jaqj=pM . Considering the transpose of T (a), we get theassertion with "1 = ja pj=pM . This implies Theorem 2.1 with "1 given byformula (6).We now turn to the proof of Theorem 2.2.Lemma 4.1 There exists a onstant Æ = Æ(a) > 1 suh that(a) = \%2[1=Æ;Æ℄ spT (a%):A proof of this lemma is in [3℄.For a sequene fMng1n=1 of nonempty ompat sets Mn  C, we onsider thetwo limiting setslim infn!1 Mn := f 2 C :  is the limit of somesequene fng1n=1 with n 2Mng;lim supn!1 Mn := f 2 C :  is a partial limit of somesequene fng1n=1 with n 2Mng:8It is well known that the two equalitieslim infn!1 Mn = lim supn!1 Mn = Mare equivalent to the onvergene of Mn to M in the Hausdor metri (see,e.g., [12, Setions 3.1.1 and 3.1.2℄ or [14, Setion 28℄).Lemma 4.2 For every " > 0,lim supn!1 spm" Tn(a)  spm" T (a):Proof. Pik  2 C n spm" T (a). Then  =2 spT (a) and, by (3),kPmT 1(a  )Pmk < 1=":It follows that there is an open neighborhood U  C of  suh that if  2 U ,then  =2 spT (a) and kPmT 1(a  )Pmk < 1=":A well-known result on the nite setion method for Toeplitz operators (see,e.g., [4, p. 42℄) says that T 1n (a   ) onverges strongly (i.e., pointwise) toT 1(a  ). Consequently,kPmT 1n (a  )Pm   PmT 1(a  )Pmk ! 0 as n!1; (11)and it is straightforward to hek that the onvergene in (11) is uniform withrespet to  in ompat subsets of U . Thus, there exist an open neighborhoodV  U of  and a natural number n0 suh thatkPmT 1n (a  )Pmk < 1=" for all  2 V and all n  n0:This, in onjuntion with (3), implies that V \ spm" T (a) = ; for all n  n0,whene  =2 lim supn!1 spm" T (a).Lemma 4.3 Let Æ be the onstant given by Lemma 4.1. If % 2 [1=Æ; Æ℄ and" > 0, then lim supn!1 spm" Tn(a)  spm T (a%); (12)where  = "max(%m+1; % m 1). 9Proof. We have Tn(a   ) = D 1n (%)Tn(a%   )Dn(%), where Dn(%) is thediagonal matrix diag(%; %2; : : : ; %n). It results thatPmT 1n (a  )Pm = D 1m (%)PmT 1n (a%   )PmDm(%);whene kPmT 1n (a  )Pmk  m(%)kPmT 1n (a%   )Pmk;where m(%) := kD 1m (%)k kDm(%)k. Sine also spTn(a) = sp Tn(a%), we on-lude from (3) thatspm" Tn(a) = sp Tn(a) [ f =2 spTn(a) : kPmT 1n (a  )Pmk  1="g= sp Tn(a%) [ f =2 sp Tn(a%) : kPmT 1n (a  )Pmk  1="g sp Tn(a%) [ f =2 sp Tn(a%) : m(%)kPmT 1n (a%   )Pmk  1="g= spm"m(%) Tn(a%);and now Lemma 4.2 yields the inlusionlim supn!1 spm" Tn(a)  spm"m(%) T (a%):Beause m(%) = %m 1 if %  1 and m(%) = % m+1 if % < 1, we arrive at theassertion.Proof of Theorem 2.2. Denote the right-hand side of (6) byC(p; q;m)max(ja pj; jaqj);let Æ be the onstant from Lemma 4.1, and put"2 := C(p; q;m)jaqj=Æq+m 1:We laim that Theorem 2.2 is true with this hoie of "2.Let " 2 (0; "2). Lemma 4.2 shows that if % 2 [1=Æ; Æ℄, then (12) holds with = "max(%m 1; % m+1)< C(p; q;m)jaqjmax(%m 1; % m+1)=Æq+m 1 C(p; q;m)jaqj%qÆq max(%m 1; % m+1)=Æq+m 1 (sine 1  %Æ)= C(p; q;m)jaqj%q max(%m 1; % m+1)=Æm 1 C(p; q;m)jaqj%q: (13)10Applying Theorem 2.1 to T (a%) and taking into aount that jaqj does notexeed max(ja pj; jaqj), we see thatspm T (a%) = sp T (a%) for  < C(p; q;m)jaqj%q:Thus, (12) and (13) give lim supn!1 spm" Tn(a)  spT (a%) for every % 2 [1=Æ; Æ℄.By Lemma 4.1, this implies thatlim supn!1 spm" Tn(a)  (a):To establish inlusion in the other diretion, note that (7) implies(a)  lim infn!1 sp Tn(a)  lim infn!1 spm" Tn(a);thus ompleting the proof of Theorem 2.2.Referenes[1℄ R. M. Beam and R. F. Warming, The asymptoti spetra of banded Toeplitzand quasi-Toeplitz matries, SIAM J. Si. Comput. 14 (1993) 971{1006.[2℄ A. Botther, M. Embree, and V. I. Sokolov, Innite Toeplitz and Laurentmatries with loalized impurities, Linear Algebra Appl., to appear.[3℄ A. Botther, M. 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Available online at:http://web.omlab.ox.a.uk/oul/work/tom.wright/psgui/index.html.12

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On large Toeplitz band matrices with an uncertain block

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