Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol101.Issue1-2.1999:
Journal of Computational and Applied Mathematics 101 (1999) 255–263
Invertibly convergent in nite products of matrices
William F. Trench
413 Lake Drive West, Divide, CO 80814, USA
Received 15 July 1998
Abstract
The standard de nition of convergence of an in nite product of scalars is extended to the in nite product P = ∞
n=1 Bn
of k × k matrices; that is, P is convergent according to the de nition given here if and only if there is an integer N such
that Bn is invertible for n¿N and P = limn→∞ nm=N Bm is invertible. A family of sucient conditions for this kind of
c 1999 Elsevier Science B.V.
convergence is given, along with examples showing that they have nontrivial applications.
All rights reserved.
Keywords: Convergence; In nite products; Invertible
1. Introduction
A scalar in nite product p = ∞
numbers is said to converge if bn is nonzero
n=1 bn of complex
Qn
,
and
q
=
lim
for n suciently large, say
n¿N
n→∞
m=N bn exists and is nonzero. If this is so then
QN −1
p is de ned to be p = q n=1
bn . With this de nition, a convergent in nite product vanishes if and
only if one of its factors vanishes.
If {Bn } are k × k complex matrices we de ne
Q
s
Y
n=r
Bj =
Bs Bs−1 · · · Br
I
if r6s;
if r¿s;
thus, successive
terms multiply on the left. The
standard de nition of convergence of an in nite
Q
Qn
B
requires
only
that
P
=
lim
product ∞
n→∞
n=1 n
m=1 Bm exist. With this de nition, P may be singular
even if Bn is nonsingular for all n¿1.
We gave the following de nition in [1].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 9 1 - 5
256
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
De nition 1. An in nite product ∞
n=1 Bn of k × k matrices converges invertibly if there is an integer
N such that Bn is invertible for n¿N and
Q = lim
n→∞
n
Y
Q
(1)
Bm
m=N
exists and is invertible. In this case we de ne
Q∞
n=1
Bn = Q
QN −1
n=1
Bn .
Trgo [3] de nes convergence of an in nite product of matrices as in De nition 1, without the
adverb “invertibly”.
De nition 1 has the following obvious consequence.
Theorem 2. An invertibly convergent in nite product is singular if and only if at least one of its
factors is singular.
As discussed in [1], the motivation for De nition 1 stems from a question about linear systems
∞
of di erence equations: Under what conditions on {Bn }∞
n=1 does the solution {xn }n=0 of the system xn = Bn xn−1 ; n = 1; 2; : : : ; approach a nite nonzero limit whenever x0 6= 0? A system with this
property is said to have linear aysmptotic equilibrium. It is easy to show that the
system has linQ
ear asymptotic equilibrium if and only if Bn is invertible for every n¿1 and ∞
n=1 Bn converges
invertibly.
The following theorem was proved in [1].
Theorem 3. If
Q∞
Bn converges invertibly then limn→∞ Bn = I .
Because of Theorem 3 we consider only in nite products of the form ∞ (I + An ) where limn→∞
An = 0. (Since invertible convergence of an in nite is independent of the rst nitely many factors,
we will usually not specify the lower limit of the product when discussing conditions for invertible
convergence.)
In [1] we gave some sucient conditions for invertible convergence of in nite products of k × k
matrices. In this paper, we present a succession of invertible convergence tests, along with examples
showing that they have nontrivial application. These results generalize results obtained in [2] for
conditional convergence of scalar in nite products.
Q
2. Sucient conditions for invertible convergence
Throughout this paper, if A is a k × k matrix then |A| is some p-norm of A. The following theorem
is proved in [1]; however, for convenience we include a shorter and more direct proof here.
Theorem 4. The product
Q∞
(I + An ) converges invertibly if
P∞
|An |¡∞.
Proof. Let N be an integer such that ∞
n=N |An |¡1, and let BN be the Banach space of all bounded
sequences X = {Xn }∞
of
k
×
k
matrices,
with norm kXk = supn¿N |Xn |. Then the transformation
n=N
P
257
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
Y = T X de ned by
Yn = I −
∞
X
Am Xm ;
n¿N;
m=n
is a contraction mapping of BN into itself. Let X̂ be the xed point of this mapping; thus,
X̂ n = I −
∞
X
Am X̂ m ;
(2)
n¿N;
m=n
so X̂ n+1 = (I + An )X̂ n ; n¿N , and therefore
X̂ n =
n−1
Y
!
(I + An ) X̂ N ;
m=N
(3)
n¿N:
From (2), limn→∞ X̂ n = I . Therefore (3) implies that I = (
Q∞
m=N (I
+ An ))X̂ N , so
Q∞
m=N (I
The following theorem is a weaker sucent condition for invertible convergence of
−1
+ An ) = X̂ N .
Q∞
(I + An ).
Theorem 5. If there is a sequence {Rn } of k × k matrices such that
lim Rn = I
(4)
n→∞
and
then
∞
X
|(I + An )Rn − Rn+1 |¡∞
Q∞
(5)
(I + An ) converges invertibly.
Proof. Let Gn = (I + An )Rn − Rn+1 . Then ∞ |Gn |¡∞ from (5), so limn→∞ Gn = 0 and therefore
Choose N so that Rn ; I + An and I + R−1
limn→∞ An = 0 by (4).
n+1 Gn are invertible if n¿N . Now de ne
Qn
−1
−1
PN −1 = I and Pn = m=N (I + Am ); n¿N . If n¿N then I + An = Pn Pn−1
, so Gn = Pn Pn−1
Rn − Rn+1 ,
−1
−1
and therefore Pn = Rn+1 (I + Rn+1 Gn )Rn Pn−1 , which implies that
Pn = Rn+1
P
"
n
Y
#
−1
(I + R−1
m+1 Gm ) RN :
m=N
(6)
Since
(4) and (5) imply that ∞ |R−1
m+1 Gm |¡∞, Theorem 4 implies that the in nite product Q =
Q∞
−1
(
I
+
R
G
)
converges
invertibly;
moreover Q is invertible because I + R−1
m+1 m
m+1 Gm is invertible
m=N
if m¿N . Now (4) and (6) imply that limn→∞ Pn = QR−1
is
nite
and
invertible.
N
P
To apply this theorem nontrivially we must exhibit sequences {Rn } that yield results even if
|An | = ∞. The following theorem provides ways to do this.
P∞
258
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
Theorem 6. Suppose that for some positive integer q the sequences
An(k) =
∞
X
Am Am(k−1) ;
k = 1; : : : ; q (with A(0)
m = I );
m=n
are all de ned, and
Then
∞
X
|An An(q) |¡∞:
Q∞
(7)
(I + An ) converges invertibly.
Proof. De ne
Rn(l) = I +
l
X
(−1) j A(nj) ;
16l6q:
j=1
We show by nite induction on l that
(l)
(I + An )Rn(l) − Rn+1
= (−1)l An An(l)
(8)
for 16l6q. Since limn→∞ Rn(q)
= I we can then set l = q in (8) and conclude from (7) and
Q
Theorem 5 with Rn = Rn(q) that ∞ (I + An ) converges invertibly.
(1)
Since R(1)
n = I − An the left-hand side of (8) with l = 1 is
(1)
(1)
(1)
(1)
(1)
(I + An )(I − A(1)
n ) − (I − An+1 ) = An − An − An An + An+1 = − An An ;
(1)
since A(1)
n+1 + An = An . This proves (8) for l = 1.
Now suppose that (8) holds if 16l¡q − 1. Since
Rn(l) = Rn(l+1) + (−1)l An(l+1) ;
(8) implies that
(l+1)
(l+1)
− (−1)l An+1
(I + An )(Rn(l+1) + (−1)l An(l+1) ) − Rn+1
= (−1)l An An(l) :
Therefore,
(l+1)
(l+1)
)
= (−1)l (An An(l) − An(l+1) − An An(l+1) + An+1
(I + An )Rn(l+1) − Rn+1
= (−1)(l+1) An An(l+1) ;
since
(l+1)
An+1
+ An An(l) = An(l+1) :
This completes the induction.
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
259
3. Preparation for applications of Theorem 6
We now prepare for speci c applications of Theorem 6. Henceforth let
E (t ) = diag (ei 1 t ; ei 2 t ; : : : ; ei k t );
where 1 ; 2 ; : : : ; k and t are real numbers. Let be the forward di erence operator;
thus, Gm = Gm+1 − Gm .
Lemma 7. Suppose that t is not an integral multiple of any of the numbers 2= 1 ; : : : ; 2= k ; and
{Gm }∞
m=0 is a sequence of complex k × k matrices such that lim m→∞ Gm = 0 and
∞
X
| Gm |¡∞
(9)
for some positive integer . Then
∞
X
P∞
" −1
X
E (mt )Gm = (I − E (t ))−
m=0
E (mt )Gm converges and
Qs Gs + E (t )
s=0
∞
X
#
E (mt )( Gm ) ;
m=0
(10)
where
Qs =
−1
X
(−1)m−s
m=s
!
E (mt );
m−s
06s6 − 1:
(11)
Proof. Our assumptions imply that I − E (t ) is invertible. Suppose that M ¿2 and let
SM = (I − E (t ))
M
X
E (mt )Gm :
(12)
m=0
Since
(I − E (t )) E (mt ) =
X
(−1)r
r=0
!
E ((m + r )t );
r
we have
SM =
M
X
m=0
=
X
X
(−1)r
r=0
r
(−1)
r=0
=
−1
X
E (mt )
m=0
+
M
X
X
E ((m + r )t )Gm
E ((m + r )t ) Gm =
(−1)r
r m=0
r
r=0
!
+r
MX
E (mt )Gm−r
r m=r
!
m
X
(−1)
r
r=0
M
+
X
m=M +1
E (mt )
!
!
X
!
Gm−r
r
r=m−M
(−1)
r
!
+
!
M
X
E (mt )
m=
!
Gm−r :
r
X
r=0
(−1)
r
!
Gm−r
r
!
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W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
Since limm→∞ Gm = 0 the last sum on the right converges to 0 as M → ∞. The second sum on the
right is
M
X
E (mt ) ( Gm− ) = E (t )
m=
M
−
X
E (mt ) ( Gm ) ;
m=0
which converges as M → ∞ because of (9). Therefore,
lim SM = S =
M →∞
−1
X
E (mt )
m=0
m
X
(−1)
r=0
r
!
Gm−r
r
!
+ E (t )
∞
X
E (mt ) ( Gm ) ;
m=0
which can also be written as
−1
X
S=
Qs Gs + E (t )
s=0
∞
X
E (mt ) ( Gm ) ;
m=0
with Qs as in (11). This and (12) imply (10).
De nition 8. If ¿0 let F be the set of in nitely di erentiable k × k matrix functions F = (Frs )kr; s=1
on [1; ∞) such that
Frs() (x) = O(x− − );
16r; s6n;
= 0; 1; : : : :
The following lemma provides an in nite set of examples of functions belonging to F .
Lemma 9. For r; s =1; : : : ; n let Frs = urs
rs ; where
rs ¿0 and urs is a rational function with positive values on [1; ∞) and a zero of order prs ¿0 at ∞. Then F =(Frs )nr; s=1 is in F ; with = min{prs
rs }kr; s = 1 .
Proof. Applying the formula of Faa di Bruno [4] for the derivatives of a composite function to
f(u) = u
where u = u(x) yields
′
X
X
d
u()(x)
u (x) r1 u′′ (x) r2
r!
(r)
−r
·
·
·
f
(
u
(
x
))
=
(
)
u
(
x
)
d x
r1 ! · · · r !
1!
2!
!
r
r=1
where
P
r
!r
;
is over all partitions of r as a sum of nonnegative integers,
r 1 + r2 + · · · + r = r
(13)
such that
r1 + 2r2 + · · · + r =
and (
)(r) =
(
− 1) · · · (
− r + 1). Since u(l) (x) = O(x−p−l ), it follows that
u
−r (x))(u′ (x))r1 (u′′ (x))r2 · · · (u() (x))r = O(x−p
− );
(14)
261
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
since
p(
− r ) + (p + 1)r1 + (p + 2)r2 + · · · + (p + )r = p
+
because of (13) and (14). Applying this argument with f = Frs , r; s = 1; : : : ; k yields the conclusion.
Note that F is a vector space over the complex numbers. Moreover, if Fi ∈ F i ; i = 1; 2, then
F1 F2 ∈ F 1 + 2 .
Henceforth, if F is a k × k matrix function then F (x) = F (x +1) −F (x). We write F (x) = O(x− )
to indicate that |F (x)| = O(x− ).
Lemma 10. If F ∈ F then
F (x) = O(x− − );
= 0; 1; 2; : : : :
Proof. Taylor’s theorem shows that
| F (x)| 6 K max |F () ()|;
x66x+
where K is a constant independent of F . Since F () (x) = O(x− − ), this implies the conclusion.
Lemma 11. Suppose that F ∈ F . Let be a xed positive integer and t be a real number, not
an integral multiple of any of the numbers 2= 1 ; : : : ; 2= k . Then
∞
X
E (mt )F (m) = E (nt )T (n) + O(n− −+1 );
m=n
where T ∈ F (and T depends upon ).
Proof. We write
∞
X
E (mt )F (m) = E (nt )
m=n
∞
X
E (mt )F (n + m):
m=0
From Lemma 10, F (n + m) = O((n + m)− − ); therefore,
∞
X
| F (n + m)| = O(n− −+1 ):
m=0
Applying Lemma 7 (speci cally, (10)) with Gm = F (n + m) and n xed shows that
∞
X
E (mt )F (n + m) = T (n) + O(n− −+1 );
m=0
with
T (x) = (I − E (t ))−
−1
X
Qs (t )F (x + s);
s=0
so T ∈ F . Now (15) implies the conclusion.
(15)
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W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
4. Applications of Theorem 6
The following theorem shows that Theorem 6 has nontrivial applications for every positive
integer q.
Theorem 12. Suppose that
An = E (n)F (n);
n = 1; 2; 3; : : : ;
(16)
where is real and F ∈ F
for some
∈ (0; 1]. Let q be the smallest integer such that
(q + 1)
¿1;
(17)
and de ne
Nq = {(2)= (p i ) | = integer;
Then the in nite product
Q∞
p = 1; : : : ; q; i = 1; : : : ; k}:
(I + An ) converges invertibly if ∈= Nq .
Proof. We show by nite induction on p that if p = 1; : : : ; q then
−(p+1)
−q+p
);
An A(p)
n = E ((p + 1)n)Fp (n) + O(n
(18)
where Fp ∈ F(p+1)
. In particular, (18) with p = q implies that An An(q) = O(n−(q+1)
), so (17) implies
(7) and P converges invertibly, by Theorem 6.
From (16) and Lemma 11 with t = , =
, and = q,
A(1)
n =
∞
X
E (m)F (m) = E ((n))T1 (n) + O(n−
−q+1 );
m=n
with T1 ∈ F
. Therefore
−
−q+1
An A(1)
)):
n = E (n)F (n)(E (n)T1 (n) + O(n
(19)
However, F (n)E (n)= E (n)F̂ (n), where F̂rs (n) = ei( s − r )n Frs (n) for 16r; s6k , so F̂ ∈ F
. Therefore
−2
−q+1
(19) can be rewritten as An A(1)
), with F1 = F̂T1 ∈ F2
. This establishes
n = E (2n)F1 (n) + O(n
(18) with p = 1, so we are nished if q = 1.
Now suppose that q¿1 and (18) holds if 16p6q. Since ∈= Nq , Lemma 11 with t = (p + 1),
F = Fp , = (p + 1)
, and = q − p implies that
∞
X
E ((p + 1)m)Fp (m) = E ((p + 1)n)Tp (n) + O(n−(p+1)
−q+p+1 );
m=n
where Tp ∈ F(p+1)
. This and (18) imply that
=
A(p+1)
n
∞
X
−(p+1)
−q+p+1
);
Am A(p)
m = E ((p + 1)n)Tp (n) + O(n
m=n
so
= E (n)F (n)(E ((p + 1)n)Tp (n) + O(n−(p+1)
−q+p+1 )):
An A(p+1)
n
(20)
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W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
However, F (n)E ((p + 1)n) = E ((p + 1)n)F̃ (n), where F̃rs (n) = ei( s − r )(p+1)n Frs (n) for 16r; s6k ,
so F̂ ∈ F
. Therefore (20) can be rewritten as
= E ((p + 2)n)Fp+1 (n) + O(n−(p+2)
−q+p+1 );
An A(p+1)
n
with Fp+1 = F̃Tp ∈ F(p+2)
. This completes the induction.
In the following corollaries N∞ = {(2)= i | rational; i = 1; : : : ; k} .
Corollary 13. If {An }∞ is as de ned in Theorem 12 then
∈= N∞ .
Q∞
(I + An ) converges invertibly if
Corollary 14. Let F = (urs
rs )kr; s=1 where, for 16r; s6k ,
rs ¿0 and urs is a rational function
with
Q
positive values on [1; ∞) and a zero of positive order at ∞. Then the in nite product ∞ (I +
E (n)F (n)) converges invertibly if ∈= N∞ .
Corollary 15. If F =(n−
rs )kr; s=1 where,
rs ¿0 for 1 6 r; s 6 n; then the in nite product
E (n)F (n)) converges invertibly if ∈= N∞ .
Q∞
(I +
References
[1] W.F. Trench, Invertibly convergent in nite products of matrices, with applications to di erence equations, Comput.
Math. Appl. 30 (1995) 39 – 46.
[2] W.F. Trench, Conditional convergence of in nite products, Amer. Math. Monthly, to appear.
[3] A. Trgo, Monodromy matrix for linear di erence operators with almost constant coecients, J. Math. Anal. Appl.
194 (1995) 697–719.
[4] Ch.-J. de La Vallee Poussin, Cours d’analyse in nitesimale, vol. 1, 12th ed., Libraire Universitaire Louvain, Gauthier–
Villars, Paris, 1959.
Invertibly convergent in nite products of matrices
William F. Trench
413 Lake Drive West, Divide, CO 80814, USA
Received 15 July 1998
Abstract
The standard de nition of convergence of an in nite product of scalars is extended to the in nite product P = ∞
n=1 Bn
of k × k matrices; that is, P is convergent according to the de nition given here if and only if there is an integer N such
that Bn is invertible for n¿N and P = limn→∞ nm=N Bm is invertible. A family of sucient conditions for this kind of
c 1999 Elsevier Science B.V.
convergence is given, along with examples showing that they have nontrivial applications.
All rights reserved.
Keywords: Convergence; In nite products; Invertible
1. Introduction
A scalar in nite product p = ∞
numbers is said to converge if bn is nonzero
n=1 bn of complex
Qn
,
and
q
=
lim
for n suciently large, say
n¿N
n→∞
m=N bn exists and is nonzero. If this is so then
QN −1
p is de ned to be p = q n=1
bn . With this de nition, a convergent in nite product vanishes if and
only if one of its factors vanishes.
If {Bn } are k × k complex matrices we de ne
Q
s
Y
n=r
Bj =
Bs Bs−1 · · · Br
I
if r6s;
if r¿s;
thus, successive
terms multiply on the left. The
standard de nition of convergence of an in nite
Q
Qn
B
requires
only
that
P
=
lim
product ∞
n→∞
n=1 n
m=1 Bm exist. With this de nition, P may be singular
even if Bn is nonsingular for all n¿1.
We gave the following de nition in [1].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 9 1 - 5
256
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
De nition 1. An in nite product ∞
n=1 Bn of k × k matrices converges invertibly if there is an integer
N such that Bn is invertible for n¿N and
Q = lim
n→∞
n
Y
Q
(1)
Bm
m=N
exists and is invertible. In this case we de ne
Q∞
n=1
Bn = Q
QN −1
n=1
Bn .
Trgo [3] de nes convergence of an in nite product of matrices as in De nition 1, without the
adverb “invertibly”.
De nition 1 has the following obvious consequence.
Theorem 2. An invertibly convergent in nite product is singular if and only if at least one of its
factors is singular.
As discussed in [1], the motivation for De nition 1 stems from a question about linear systems
∞
of di erence equations: Under what conditions on {Bn }∞
n=1 does the solution {xn }n=0 of the system xn = Bn xn−1 ; n = 1; 2; : : : ; approach a nite nonzero limit whenever x0 6= 0? A system with this
property is said to have linear aysmptotic equilibrium. It is easy to show that the
system has linQ
ear asymptotic equilibrium if and only if Bn is invertible for every n¿1 and ∞
n=1 Bn converges
invertibly.
The following theorem was proved in [1].
Theorem 3. If
Q∞
Bn converges invertibly then limn→∞ Bn = I .
Because of Theorem 3 we consider only in nite products of the form ∞ (I + An ) where limn→∞
An = 0. (Since invertible convergence of an in nite is independent of the rst nitely many factors,
we will usually not specify the lower limit of the product when discussing conditions for invertible
convergence.)
In [1] we gave some sucient conditions for invertible convergence of in nite products of k × k
matrices. In this paper, we present a succession of invertible convergence tests, along with examples
showing that they have nontrivial application. These results generalize results obtained in [2] for
conditional convergence of scalar in nite products.
Q
2. Sucient conditions for invertible convergence
Throughout this paper, if A is a k × k matrix then |A| is some p-norm of A. The following theorem
is proved in [1]; however, for convenience we include a shorter and more direct proof here.
Theorem 4. The product
Q∞
(I + An ) converges invertibly if
P∞
|An |¡∞.
Proof. Let N be an integer such that ∞
n=N |An |¡1, and let BN be the Banach space of all bounded
sequences X = {Xn }∞
of
k
×
k
matrices,
with norm kXk = supn¿N |Xn |. Then the transformation
n=N
P
257
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
Y = T X de ned by
Yn = I −
∞
X
Am Xm ;
n¿N;
m=n
is a contraction mapping of BN into itself. Let X̂ be the xed point of this mapping; thus,
X̂ n = I −
∞
X
Am X̂ m ;
(2)
n¿N;
m=n
so X̂ n+1 = (I + An )X̂ n ; n¿N , and therefore
X̂ n =
n−1
Y
!
(I + An ) X̂ N ;
m=N
(3)
n¿N:
From (2), limn→∞ X̂ n = I . Therefore (3) implies that I = (
Q∞
m=N (I
+ An ))X̂ N , so
Q∞
m=N (I
The following theorem is a weaker sucent condition for invertible convergence of
−1
+ An ) = X̂ N .
Q∞
(I + An ).
Theorem 5. If there is a sequence {Rn } of k × k matrices such that
lim Rn = I
(4)
n→∞
and
then
∞
X
|(I + An )Rn − Rn+1 |¡∞
Q∞
(5)
(I + An ) converges invertibly.
Proof. Let Gn = (I + An )Rn − Rn+1 . Then ∞ |Gn |¡∞ from (5), so limn→∞ Gn = 0 and therefore
Choose N so that Rn ; I + An and I + R−1
limn→∞ An = 0 by (4).
n+1 Gn are invertible if n¿N . Now de ne
Qn
−1
−1
PN −1 = I and Pn = m=N (I + Am ); n¿N . If n¿N then I + An = Pn Pn−1
, so Gn = Pn Pn−1
Rn − Rn+1 ,
−1
−1
and therefore Pn = Rn+1 (I + Rn+1 Gn )Rn Pn−1 , which implies that
Pn = Rn+1
P
"
n
Y
#
−1
(I + R−1
m+1 Gm ) RN :
m=N
(6)
Since
(4) and (5) imply that ∞ |R−1
m+1 Gm |¡∞, Theorem 4 implies that the in nite product Q =
Q∞
−1
(
I
+
R
G
)
converges
invertibly;
moreover Q is invertible because I + R−1
m+1 m
m+1 Gm is invertible
m=N
if m¿N . Now (4) and (6) imply that limn→∞ Pn = QR−1
is
nite
and
invertible.
N
P
To apply this theorem nontrivially we must exhibit sequences {Rn } that yield results even if
|An | = ∞. The following theorem provides ways to do this.
P∞
258
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
Theorem 6. Suppose that for some positive integer q the sequences
An(k) =
∞
X
Am Am(k−1) ;
k = 1; : : : ; q (with A(0)
m = I );
m=n
are all de ned, and
Then
∞
X
|An An(q) |¡∞:
Q∞
(7)
(I + An ) converges invertibly.
Proof. De ne
Rn(l) = I +
l
X
(−1) j A(nj) ;
16l6q:
j=1
We show by nite induction on l that
(l)
(I + An )Rn(l) − Rn+1
= (−1)l An An(l)
(8)
for 16l6q. Since limn→∞ Rn(q)
= I we can then set l = q in (8) and conclude from (7) and
Q
Theorem 5 with Rn = Rn(q) that ∞ (I + An ) converges invertibly.
(1)
Since R(1)
n = I − An the left-hand side of (8) with l = 1 is
(1)
(1)
(1)
(1)
(1)
(I + An )(I − A(1)
n ) − (I − An+1 ) = An − An − An An + An+1 = − An An ;
(1)
since A(1)
n+1 + An = An . This proves (8) for l = 1.
Now suppose that (8) holds if 16l¡q − 1. Since
Rn(l) = Rn(l+1) + (−1)l An(l+1) ;
(8) implies that
(l+1)
(l+1)
− (−1)l An+1
(I + An )(Rn(l+1) + (−1)l An(l+1) ) − Rn+1
= (−1)l An An(l) :
Therefore,
(l+1)
(l+1)
)
= (−1)l (An An(l) − An(l+1) − An An(l+1) + An+1
(I + An )Rn(l+1) − Rn+1
= (−1)(l+1) An An(l+1) ;
since
(l+1)
An+1
+ An An(l) = An(l+1) :
This completes the induction.
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
259
3. Preparation for applications of Theorem 6
We now prepare for speci c applications of Theorem 6. Henceforth let
E (t ) = diag (ei 1 t ; ei 2 t ; : : : ; ei k t );
where 1 ; 2 ; : : : ; k and t are real numbers. Let be the forward di erence operator;
thus, Gm = Gm+1 − Gm .
Lemma 7. Suppose that t is not an integral multiple of any of the numbers 2= 1 ; : : : ; 2= k ; and
{Gm }∞
m=0 is a sequence of complex k × k matrices such that lim m→∞ Gm = 0 and
∞
X
| Gm |¡∞
(9)
for some positive integer . Then
∞
X
P∞
" −1
X
E (mt )Gm = (I − E (t ))−
m=0
E (mt )Gm converges and
Qs Gs + E (t )
s=0
∞
X
#
E (mt )( Gm ) ;
m=0
(10)
where
Qs =
−1
X
(−1)m−s
m=s
!
E (mt );
m−s
06s6 − 1:
(11)
Proof. Our assumptions imply that I − E (t ) is invertible. Suppose that M ¿2 and let
SM = (I − E (t ))
M
X
E (mt )Gm :
(12)
m=0
Since
(I − E (t )) E (mt ) =
X
(−1)r
r=0
!
E ((m + r )t );
r
we have
SM =
M
X
m=0
=
X
X
(−1)r
r=0
r
(−1)
r=0
=
−1
X
E (mt )
m=0
+
M
X
X
E ((m + r )t )Gm
E ((m + r )t ) Gm =
(−1)r
r m=0
r
r=0
!
+r
MX
E (mt )Gm−r
r m=r
!
m
X
(−1)
r
r=0
M
+
X
m=M +1
E (mt )
!
!
X
!
Gm−r
r
r=m−M
(−1)
r
!
+
!
M
X
E (mt )
m=
!
Gm−r :
r
X
r=0
(−1)
r
!
Gm−r
r
!
260
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
Since limm→∞ Gm = 0 the last sum on the right converges to 0 as M → ∞. The second sum on the
right is
M
X
E (mt ) ( Gm− ) = E (t )
m=
M
−
X
E (mt ) ( Gm ) ;
m=0
which converges as M → ∞ because of (9). Therefore,
lim SM = S =
M →∞
−1
X
E (mt )
m=0
m
X
(−1)
r=0
r
!
Gm−r
r
!
+ E (t )
∞
X
E (mt ) ( Gm ) ;
m=0
which can also be written as
−1
X
S=
Qs Gs + E (t )
s=0
∞
X
E (mt ) ( Gm ) ;
m=0
with Qs as in (11). This and (12) imply (10).
De nition 8. If ¿0 let F be the set of in nitely di erentiable k × k matrix functions F = (Frs )kr; s=1
on [1; ∞) such that
Frs() (x) = O(x− − );
16r; s6n;
= 0; 1; : : : :
The following lemma provides an in nite set of examples of functions belonging to F .
Lemma 9. For r; s =1; : : : ; n let Frs = urs
rs ; where
rs ¿0 and urs is a rational function with positive values on [1; ∞) and a zero of order prs ¿0 at ∞. Then F =(Frs )nr; s=1 is in F ; with = min{prs
rs }kr; s = 1 .
Proof. Applying the formula of Faa di Bruno [4] for the derivatives of a composite function to
f(u) = u
where u = u(x) yields
′
X
X
d
u()(x)
u (x) r1 u′′ (x) r2
r!
(r)
−r
·
·
·
f
(
u
(
x
))
=
(
)
u
(
x
)
d x
r1 ! · · · r !
1!
2!
!
r
r=1
where
P
r
!r
;
is over all partitions of r as a sum of nonnegative integers,
r 1 + r2 + · · · + r = r
(13)
such that
r1 + 2r2 + · · · + r =
and (
)(r) =
(
− 1) · · · (
− r + 1). Since u(l) (x) = O(x−p−l ), it follows that
u
−r (x))(u′ (x))r1 (u′′ (x))r2 · · · (u() (x))r = O(x−p
− );
(14)
261
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
since
p(
− r ) + (p + 1)r1 + (p + 2)r2 + · · · + (p + )r = p
+
because of (13) and (14). Applying this argument with f = Frs , r; s = 1; : : : ; k yields the conclusion.
Note that F is a vector space over the complex numbers. Moreover, if Fi ∈ F i ; i = 1; 2, then
F1 F2 ∈ F 1 + 2 .
Henceforth, if F is a k × k matrix function then F (x) = F (x +1) −F (x). We write F (x) = O(x− )
to indicate that |F (x)| = O(x− ).
Lemma 10. If F ∈ F then
F (x) = O(x− − );
= 0; 1; 2; : : : :
Proof. Taylor’s theorem shows that
| F (x)| 6 K max |F () ()|;
x66x+
where K is a constant independent of F . Since F () (x) = O(x− − ), this implies the conclusion.
Lemma 11. Suppose that F ∈ F . Let be a xed positive integer and t be a real number, not
an integral multiple of any of the numbers 2= 1 ; : : : ; 2= k . Then
∞
X
E (mt )F (m) = E (nt )T (n) + O(n− −+1 );
m=n
where T ∈ F (and T depends upon ).
Proof. We write
∞
X
E (mt )F (m) = E (nt )
m=n
∞
X
E (mt )F (n + m):
m=0
From Lemma 10, F (n + m) = O((n + m)− − ); therefore,
∞
X
| F (n + m)| = O(n− −+1 ):
m=0
Applying Lemma 7 (speci cally, (10)) with Gm = F (n + m) and n xed shows that
∞
X
E (mt )F (n + m) = T (n) + O(n− −+1 );
m=0
with
T (x) = (I − E (t ))−
−1
X
Qs (t )F (x + s);
s=0
so T ∈ F . Now (15) implies the conclusion.
(15)
262
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
4. Applications of Theorem 6
The following theorem shows that Theorem 6 has nontrivial applications for every positive
integer q.
Theorem 12. Suppose that
An = E (n)F (n);
n = 1; 2; 3; : : : ;
(16)
where is real and F ∈ F
for some
∈ (0; 1]. Let q be the smallest integer such that
(q + 1)
¿1;
(17)
and de ne
Nq = {(2)= (p i ) | = integer;
Then the in nite product
Q∞
p = 1; : : : ; q; i = 1; : : : ; k}:
(I + An ) converges invertibly if ∈= Nq .
Proof. We show by nite induction on p that if p = 1; : : : ; q then
−(p+1)
−q+p
);
An A(p)
n = E ((p + 1)n)Fp (n) + O(n
(18)
where Fp ∈ F(p+1)
. In particular, (18) with p = q implies that An An(q) = O(n−(q+1)
), so (17) implies
(7) and P converges invertibly, by Theorem 6.
From (16) and Lemma 11 with t = , =
, and = q,
A(1)
n =
∞
X
E (m)F (m) = E ((n))T1 (n) + O(n−
−q+1 );
m=n
with T1 ∈ F
. Therefore
−
−q+1
An A(1)
)):
n = E (n)F (n)(E (n)T1 (n) + O(n
(19)
However, F (n)E (n)= E (n)F̂ (n), where F̂rs (n) = ei( s − r )n Frs (n) for 16r; s6k , so F̂ ∈ F
. Therefore
−2
−q+1
(19) can be rewritten as An A(1)
), with F1 = F̂T1 ∈ F2
. This establishes
n = E (2n)F1 (n) + O(n
(18) with p = 1, so we are nished if q = 1.
Now suppose that q¿1 and (18) holds if 16p6q. Since ∈= Nq , Lemma 11 with t = (p + 1),
F = Fp , = (p + 1)
, and = q − p implies that
∞
X
E ((p + 1)m)Fp (m) = E ((p + 1)n)Tp (n) + O(n−(p+1)
−q+p+1 );
m=n
where Tp ∈ F(p+1)
. This and (18) imply that
=
A(p+1)
n
∞
X
−(p+1)
−q+p+1
);
Am A(p)
m = E ((p + 1)n)Tp (n) + O(n
m=n
so
= E (n)F (n)(E ((p + 1)n)Tp (n) + O(n−(p+1)
−q+p+1 )):
An A(p+1)
n
(20)
263
W.F. Trench / Journal of Computational and Applied Mathematics 101 (1999) 255–263
However, F (n)E ((p + 1)n) = E ((p + 1)n)F̃ (n), where F̃rs (n) = ei( s − r )(p+1)n Frs (n) for 16r; s6k ,
so F̂ ∈ F
. Therefore (20) can be rewritten as
= E ((p + 2)n)Fp+1 (n) + O(n−(p+2)
−q+p+1 );
An A(p+1)
n
with Fp+1 = F̃Tp ∈ F(p+2)
. This completes the induction.
In the following corollaries N∞ = {(2)= i | rational; i = 1; : : : ; k} .
Corollary 13. If {An }∞ is as de ned in Theorem 12 then
∈= N∞ .
Q∞
(I + An ) converges invertibly if
Corollary 14. Let F = (urs
rs )kr; s=1 where, for 16r; s6k ,
rs ¿0 and urs is a rational function
with
Q
positive values on [1; ∞) and a zero of positive order at ∞. Then the in nite product ∞ (I +
E (n)F (n)) converges invertibly if ∈= N∞ .
Corollary 15. If F =(n−
rs )kr; s=1 where,
rs ¿0 for 1 6 r; s 6 n; then the in nite product
E (n)F (n)) converges invertibly if ∈= N∞ .
Q∞
(I +
References
[1] W.F. Trench, Invertibly convergent in nite products of matrices, with applications to di erence equations, Comput.
Math. Appl. 30 (1995) 39 – 46.
[2] W.F. Trench, Conditional convergence of in nite products, Amer. Math. Monthly, to appear.
[3] A. Trgo, Monodromy matrix for linear di erence operators with almost constant coecients, J. Math. Anal. Appl.
194 (1995) 697–719.
[4] Ch.-J. de La Vallee Poussin, Cours d’analyse in nitesimale, vol. 1, 12th ed., Libraire Universitaire Louvain, Gauthier–
Villars, Paris, 1959.