Invertibly convergent innite products of matrices
William F. Trench
413 Lake Drive West, Divide, CO 80814, USA Received 15 July 1998
Abstract
The standard denition of convergence of an innite product of scalars is extended to the innite productP= ∞
n=1Bn ofk×k matrices; that is,Pis convergent according to the denition given here if and only if there is an integerN such that Bn is invertible for n¿N and P= limn→∞
n
m=NBm is invertible. A family of sucient conditions for this kind of convergence is given, along with examples showing that they have nontrivial applications. c1999 Elsevier Science B.V. All rights reserved.
Keywords:Convergence; Innite products; Invertible
1. Introduction
A scalar innite product p=Q∞
n=1bn of complex numbers is said to converge if bn is nonzero
for n suciently large, say n¿N, and q= limn→∞
Qn
m=Nbn exists and is nonzero. If this is so then
p is dened to be p=qQN−1
n=1 bn. With this denition, a convergent innite product vanishes if and
only if one of its factors vanishes.
If {Bn} are k×k complex matrices we dene
s
Y
n=r
Bj=
BsBs−1· · ·Br if r6s;
I if r¿s;
thus, successive terms multiply on the left. The standard denition of convergence of an innite productQ∞
n=1Bn requires only thatP= limn→∞
Qn
m=1Bmexist. With this denition, Pmay be singular
even if Bn is nonsingular for all n¿1.
We gave the following denition in [1].
Denition 1. An innite productQ∞
n=1Bn ofk×k matricesconverges invertiblyif there is an integer
N such that Bn is invertible for n¿N and
Q= lim
n→∞
n
Y
m=N
Bm (1)
exists and is invertible. In this case we dene Q∞
n=1Bn=QQN
−1
n=1 Bn.
Trgo [3] denes convergence of an innite product of matrices as in Denition 1, without the adverb “invertibly”.
Denition 1 has the following obvious consequence.
Theorem 2. An invertibly convergent innite product is singular if and only if at least one of its factors is singular.
As discussed in [1], the motivation for Denition 1 stems from a question about linear systems of dierence equations: Under what conditions on {Bn}
∞
n=1 does the solution {xn}
∞
n=0 of the
sys-tem xn=Bnxn−1; n= 1;2; : : : ; approach a nite nonzero limit whenever x06= 0? A system with this
property is said to have linear aysmptotic equilibrium. It is easy to show that the system has lin-ear asymptotic equilibrium if and only if Bn is invertible for every n¿1 and Q
∞
n=1Bn 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 innite products of the form Q∞
(I+An) where limn→∞
An= 0. (Since invertible convergence of an innite 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 innite 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 innite products.
2. Sucient conditions for invertible convergence
Throughout this paper, if Ais ak×k matrix then |A|is somep-norm ofA. 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. LetN be an integer such thatP∞
n=N|An|¡1, and letBN be the Banach space of all bounded
sequences X={Xn}∞
Y=TX dened by
is a contraction mapping of BN into itself. Let ˆX be the xed point of this mapping; thus,
ˆ
The following theorem is a weaker sucent condition for invertible convergence of Q∞
(I+An).
(I+An) converges invertibly.
Proof. Let Gn= (I +An)Rn−Rn+1. Then P
m+1Gm) converges invertibly; moreover Q is invertible because I+R
−1
m+1Gm is invertible
if m¿N. Now (4) and (6) imply that limn→∞Pn=QR
−1
N is nite and invertible.
To apply this theorem nontrivially we must exhibit sequences {Rn} that yield results even if
P∞
Theorem 6. Suppose that for some positive integer q the sequences
A(nk)=
∞
X
m=n
AmA(k
−1)
m ; k= 1; : : : ; q (with A
(0)
m =I);
are all dened, and
∞
X
|AnA(nq)|¡∞: (7)
Then Q∞
(I +An) converges invertibly.
Proof. Dene
R(l)
n =I + l
X
j=1
(−1)jA(j)
n ; 16l6q:
We show by nite induction on l that
(I +An)R(nl)−R
(l)
n+1= (−1)lAnA(nl) (8)
for 16l6q. Since limn→∞ Rn(q)=I we can then set l=q in (8) and conclude from (7) and
Theorem 5 with Rn=R(nq) that
Q∞
(I +An) converges invertibly.
Since R(1)
n =I −A(1)n the left-hand side of (8) with l= 1 is
(I +An)(I −A(1)n )−(I −A
(1)
n+1) =An−An(1)−AnA(1)n +A
(1)
n+1= −AnA(1)n ;
since A(1)n+1+An=A(1)n . This proves (8) for l= 1.
Now suppose that (8) holds if 16l¡q−1. Since
R(nl)=Rn(l+1)+ (−1)lA(nl+1);
(8) implies that
(I +An)(R(nl+1)+ (−1) lA(l+1)
n )−R
(l+1)
n+1 −(−1)lA (l+1)
n+1 = (−1)lAnA(nl):
Therefore,
(I +An)R(nl+1)−R
(l+1)
n+1 = (−1)
l(A
nA(nl)−A
(l+1)
n −AnA(nl+1)+A
(l+1)
n+1 )
= (−1)(l+1)A
nA(nl+1);
since
A(nl+1+1)+AnA(nl)=A
(l+1)
n :
3. Preparation for applications of Theorem 6
We now prepare for specic applications of Theorem 6. Henceforth let
E(t) = diag (ei1t;ei2t; : : : ;eikt);
for some positive integer . Then P∞
E(mt)Gm converges and
∞
Proof. Our assumptions imply that I −E(t) is invertible. Suppose that M ¿2 and let
Since limm→∞Gm= 0 the last sum on the right converges to 0 as M→ ∞. The second sum on the
which converges as M→ ∞ because of (9). Therefore,
lim
which can also be written as
S=
The following lemma provides an innite set of examples of functions belonging to F.
Lemma 9. Forr; s=1; : : : ; nletFrs=ursrs;wherers¿0andurs is a rational function with positive
val-ues on[1;∞)and a zero of orderprs¿0at∞. ThenF=(Frs)nr; s=1 is inF;with= min{prsrs}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
r is over all partitions of r as a sum of nonnegative integers,
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∈Fi; i= 1;2, then
F1F2∈F1+2.
Henceforth, if F is a k×k matrix function then F(x) =F(x+1)−F(x). We writeF(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)|6K max
x66x+|F
()()|;
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
m=n
E(mt)F(m) =E(nt)T(n) + O(n−−+1);
where T∈F (and T depends upon ).
Proof. We write
∞
X
m=n
E(mt)F(m) =E(nt)
∞
X
m=0
E(mt)F(n+m): (15)
From Lemma 10, F(n+m) = O((n+m)−−
); therefore,
∞
X
m=0
|F(n+m)|= O(n−−+1):
Applying Lemma 7 (specically, (10)) with Gm=F(n+m) and n xed shows that
∞
X
m=0
E(mt)F(n+m) =T(n) + O(n−−+1
);
with
T(x) = (I −E(t))− −1
X
s=0
Qs(t)F(x+s);
so T∈F
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 dene
Nq={(2)=(pi)|=integer; p= 1; : : : ; q; i= 1; : : : ; k}:
Then the innite product Q∞
(I +An) converges invertibly if =∈Nq.
Proof. We show by nite induction on p that if p= 1; : : : ; q then AnA(np)=E((p+ 1)n)Fp(n) + O(n
−(p+1)−q+p); (18)
where Fp∈F(p+1). In particular, (18) with p=q implies that AnA(nq)= 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
m=n
E(m)F(m) =E((n))T1(n) + O(n
−−q+1);
with T1∈F. Therefore
AnA(1)n =E(n)F(n)(E(n)T1(n) + O(n −−q+1
)): (19)
However,F(n)E(n)=E(n) ˆF(n), where ˆFrs(n) = ei(s
−r)nF
rs(n) for 16r; s6k, so ˆF∈F. Therefore
(19) can be rewritten as AnA(1)n =E(2n)F1(n) + O(n
−2−q+1), with F
1= ˆFT1∈F2. This establishes
(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
m=n
E((p+ 1)m)Fp(m) =E((p+ 1)n)Tp(n) + O(n
−(p+1)−q+p+1
);
where Tp∈F(p+1). This and (18) imply that
A(p+1)
n =
∞
X
m=n
AmA(mp)=E((p+ 1)n)Tp(n) + O(n
−(p+1)−q+p+1
);
so
AnA(np+1)=E(n)F(n)(E((p+ 1)n)Tp(n) + O(n
However, F(n)E((p+ 1)n) =E((p+ 1)n) ˜F(n), where ˜Frs(n) = ei(s
−r)(p+1)nF
rs(n) for 16r; s6k,
so ˆF∈F. Therefore (20) can be rewritten as
AnA(np+1)=E((p+ 2)n)Fp+1(n) + O(n
−(p+2)−q+p+1);
with Fp+1= ˜FTp∈F(p+2). This completes the induction.
In the following corollaries N∞={(2)=i| rational; i= 1; : : : ; k} .
Corollary 13. If {An}
∞
is as dened in Theorem 12 then Q∞
(I +An) converges invertibly if
=∈N∞.
Corollary 14. Let F= (urs
rs)kr; s=1 where, for 16r; s6k, rs¿0 and urs is a rational function with
positive values on [1;∞) and a zero of positive order at ∞. Then the innite product Q∞
(I + E(n)F(n)) converges invertibly if =∈N∞.
Corollary 15. If F=(n−rs)k
r; s=1 where, rs¿0 for 16r; s6n; then the innite product Q
∞
(I + E(n)F(n)) converges invertibly if =∈N∞.
References
[1] W.F. Trench, Invertibly convergent innite products of matrices, with applications to dierence equations, Comput. Math. Appl. 30 (1995) 39 – 46.
[2] W.F. Trench, Conditional convergence of innite products, Amer. Math. Monthly, to appear.
[3] A. Trgo, Monodromy matrix for linear dierence operators with almost constant coecients, J. Math. Anal. Appl. 194 (1995) 697–719.
