AbstractIn this note we study the concepts of generalized spectral subradius and joint spectral subradius of a family of matrices. We show that both these numbers are equal and we present two different formulas for them. We also explain the relation between them and maximal Lyapunov exponent of discrete linear time varying system. Finally we show that the spectral subradius is less than one if and only if a discrete linear inclusion is stable in a certain sense

Czornik, Adam

Elsevier - Publisher Connector 

On the generalized spectral subradius 

Linear Algebra and its Applications 407 (2005) 242–248
www.elsevier.com/locate/laa
On the generalized spectral subradius
Adam Czornik ∗
Department of Automatic Control, Silesian Technical University, ul. Akademicka 16,
44-101 Gliwice, Poland
Received 17 May 2004; accepted 18 May 2005
Available online 14 July 2005
Submitted by H. Schneider
Abstract
In this note we study the concepts of generalized spectral subradius and joint spectral
subradius of a family of matrices. We show that both these numbers are equal and we present
two different formulas for them. We also explain the relation between them and maximal
Lyapunov exponent of discrete linear time varying system. Finally we show that the spectral
subradius is less than one if and only if a discrete linear inclusion is stable in a certain sense.
© 2005 Elsevier Inc. All rights reserved.
Keywords: Generalized spectral subradius; Discrete linear inclusions; Stability
1. Introduction
Let  denote a nonempty set of complex k-by-k matrices. For m  1,m is the
set of all products of matrices in  of length m,
m = {A1A2 . . . Am : Ai ∈ , i = 1, . . . , m}.
Denote by ρ(A) the spectral radius and by ‖A‖ a matrix norm of a matrix A. Set
αm = inf
A∈m ‖A‖,
βm = inf
A∈m ρ(A)
∗ Tel.: +48 32 237 1075; fax: +48 32 237 1165.
E-mail address: adam.czornik@polsl.pl
0024-3795/$ - see front matter  2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.laa.2005.05.006
A. Czornik / Linear Algebra and its Applications 407 (2005) 242–248 243
and define the joint spectral subradius
ρ̂∗() = inf
m1
α
1/m
m
and the generalized spectral subradius
ρ¯∗() = inf
m1
β
1/m
m .
These concepts have been introduced in [7] to present conditions for Markov asymp-
totic stability of a discrete linear inclusion. In that paper it has been also shown
that
ρ̂∗() = ρ¯∗()
for finite . Next in [2] these concepts have been related to the so-called mortality
problem. We say that the set of matrices  is mortal if the zero matrix can be ex-
pressed as the product of finitely many matrices from . It turns out that  is mortal
if and only if ρ̂∗() = 0.
The joint spectral subradius measures the minimal average norm of long products
of matrices taken from . If we are interested in maximal average norm we introduce
the following quantity:
ρ̂() = inf
m1
γ
1/m
m ,
where
γm = sup
A∈m
‖A‖,
which is called joint spectral radius. It has been introduced in [8] and since then it
has been extensively investigated see [3–6,9] for a small sample.
The objective of this note is to investigate the concept of generalized spectral
subradius. First we show that the generalized spectral subradius and the joint spectral
subradius are equal for any nonempty set  that means that equality (1) holds for any
nonempty set . Next we characterize this common value in terms of lower limits of
certain sequences and finally we give an interpretation of the inequality ρ̂∗() < 1
in terms of stability theory of time-varying linear systems.
2. Main results
We start with the generalization of Theorem B.1 from [7].
Theorem 1. For any nonempty set  of complex k-by-k matrices one has
ρ̂∗() = ρ¯∗(). (1)
Proof. Since for any matrix A, it is true that ρ(A)  ‖A‖, therefore βm  αm and
consequently
ρ¯∗()  ρ̂∗(). (2)
244 A. Czornik / Linear Algebra and its Applications 407 (2005) 242–248
To prove the opposite inequality let us fix a positive number ε > 0. Then there exists
n0 such that
(ρ¯∗() + ε)n0 > βn0 .
The last inequality implies that
(ρ¯∗() + ε)n0 > ρ
(
Ai1Ai2 . . . Ain0
)
and
ρ¯∗() + ε >
(
ρ
(
Ai1Ai2 . . . Ain0
)) 1n0 (3)
for certain Ai1 , Ai2 , . . . , Ain0 ∈ . From the other side
ρ̂∗() 
(
αn0k
) 1
n0k  ‖
(
Ai1Ai2 . . . Ain0
)k ‖ 1n0k ,
taking the limit when k tends to infinity and having in mid that for any square matrix
lim
k→∞ ‖A
k‖ 1k = ρ(A)
and taking into account that ε > 0 was arbitrary we conclude
ρ̂∗()  ρ
1
n0
(
Ai1Ai2 . . . Ain0
)
. (4)
Combining (3) and (4) we obtain
ρ̂∗()  ρ¯∗().
This completes the proof. 
This theorem enables us to call the common value of ρ¯∗() and ρ̂∗() the gener-
alized subradius of  and we will denote it by ρ∗(). The next theorem characterizes
the generalized spectral subradius of  in terms of infimum limits of α1/mm and β1/mm .
Theorem 2. For any nonempty set  of complex k-by-k matrices one has
lim inf
m→∞ α
1/m
m = lim inf
m→∞ β
1/m
m = ρ∗(). (5)
In the proof we will use the following simple lemma.
Lemma 1. If am,m = 1, 2, . . . is a sequence of real numbers and
inf{am : m = 1, 2, . . .} /∈ {am : m = 1, 2, . . .},
then
lim inf
m→∞ am = inf{am : m = 1, 2, . . .}.
Proof of Theorem 2. First we show that
lim inf
m→∞ α
1/m
m = ρ̂∗(). (6)
A. Czornik / Linear Algebra and its Applications 407 (2005) 242–248 245
Observe that by the definition of ρ̂∗() it is clear that
lim inf
m→∞ α
1/m
m  ρ̂∗(). (7)
Now we prove the opposite inequality. Having in mind Lemma 1 we may assume
that there exists n0 such that
ρ̂∗() = α1/n0n0 .
Let us fix ε > 0. Then there exist matrices Ai1 , Ai2 , . . . , Ain0 ∈  such that
ρ̂∗() + ε > ‖Ai1Ai2 . . . Ain0 ‖
1
n0 . (8)
Consider the subsequence (αn0k)
1
n0k , k = 1, 2 . . . of the sequence α1/mm ,m = 1, 2, . . .
We have
(αn0k)
1
n0k 
∥∥(Ai1Ai2 . . . Ain0 )k∥∥ 1n0k  ∥∥Ai1Ai2 . . . Ain0∥∥ 1n0
and by (8)
(αn0k)
1
n0k < ρ̂∗() + ε.
The last inequality implies that
lim inf
m→∞ α
1/m
m  ρ̂∗().
This together with (7) gives (6).
Now we show that
lim inf
m→∞ β
1/m
m = ρ¯∗(). (9)
Observe that by the definition of ρ¯∗() it is clear that
lim inf
m→∞ β
1/m
m  ρ¯∗(). (10)
Now we prove the opposite inequality. Again by the Lemma 1 we may assume that
there exists n0 such that
ρ¯∗() = β1/n0n0 .
Let us fix ε > 0. Then there exist matrices Ai1 , Ai2 , . . . , Ain0 ∈  such that
ρ̂∗() + ε > (ρ(Ai1Ai2 . . . Ain0 ))
1
n0 = (ρ((Ai1Ai2 . . . Ain0 )k))
1
n0k  β1/n0kn0k .
(11)
Therefore
ρ̂∗()  lim inf
m→∞ β
1/m
m .
This together with (10) gives (9). Finally from (6) (9) and Theorem 1 the conclusion
of the Theorem follows. 
246 A. Czornik / Linear Algebra and its Applications 407 (2005) 242–248
It is not difficult to check that the numbers αm satisfy the following inequality
αn+m  αmαn, therefore by Fekete lemma the limit in limm→∞ α1/mm exists. Now
we present relation between the concept of the Lyapunov exponent of time varying
linear system and generalized subradius.
Consider a sequence d = (A1, A2, . . .) of matrices from . Denote by D the set
of all such sequences. For fixed d ∈ D define
T dm = AmAm−1 . . . A1
and
ρ(d) = lim inf
m→∞ ‖T
d
m‖
1
m .
The quantity ln ρ(d) is known in the control theory literature (see [1]) as the greatest
Lyapunov exponent of the following linear time varying system
x(m + 1) = Amx(m).
We have the following.
Theorem 3. For any nonempty set  of complex k-by-k matrices one has
ρ∗() = inf
d∈D ρ(d).
Proof. From the definition of αm it is clear that
‖T dm‖
1
m  α1/mm
for each d ∈ D and therefore
inf
d∈D ρ(d)  ρ∗(). (12)
Fix a natural number n0 and matrices Ai1 , Ai2 , . . . , Ain0 ∈ . Let d0 be the se-
quence starting with Ai1 , Ai2 , . . . , Ain0 and continuing periodically. Then
inf
d∈D ρ(d) ρ(d0)  lim infm→∞ ‖(Ai1Ai2 . . . , Ain0 )
m‖ 1n0m
 ‖Ai1Ai2 . . . , Ain0 ‖
1
n0
and taking into account that the matrices Ai1 , Ai2 , . . . , Ain0 ∈  and number n0 were
arbitrary have
inf
d∈D ρ(d)  α
1
n0
n0 .
Finally taking into account that n0 was arbitrary we have
inf
d∈D ρ(d)  ρ∗().
Combining the last inequality with (12) we obtain the conclusion of the theorem. 
A. Czornik / Linear Algebra and its Applications 407 (2005) 242–248 247
Finally we explain the meaning of the inequality ρ∗() < 1 in terms of stability of
discrete linear inclusion. The discrete linear inclusion generated by  (DLI()) is the
set of all sequences {x(m) : m  0} such that x(m + 1) = Amx(m) for some Am ∈
. In [7] it has been shown that for finite  the inequality ρ∗() < 1 holds if and
only if DLI() is Markov asymptotically stable. The concept of Markov asymptotic
stability is a probabilistic concept. Now we propose a new definition of stability
which is purely deterministic and turns out to be equivalent to ρ∗() < 1.
Definition 1. We say that DLI() is selectably stable if there exists a sequence d ∈
D such that for all z0 ∈ Ck
lim
m→∞ T
d
mz0 = 0. (13)
We have the following.
Theorem 4. DLI() is selectably stable if and only if
ρ∗() < 1.
Proof. Suppose that DLI() is selectably stable. Denote by d0 = (A1, A2, . . .) the
sequence d0 ∈ D such that (13) holds. It is clear that
lim
m→∞ ‖T
d
m‖ = 0
and therefore
‖An0An0−1 . . . A1‖ <
1
2
for certain n0. This implies α
1
n0
n0 <
( 1
2
) 1
n0 and consequently
ρ∗() <
(
1
2
) 1
n0
< 1.
Conversely, if ρ∗() < 1, then for fixed q ∈ (ρ∗(), 1) there exist matrices
Ai1 , Ai2 , . . . , Ain0
∈  such that
‖Ai1Ai2 . . . Ain0 ‖ < q.
Let d0 be the sequence starting with Ai1 , Ai2 , . . . , Ain0 and continuing periodically
then limm→∞ ‖T d0m ‖ = 0. 
3. Conclusions
In this note we have generalized the formula from [7] for the case of infinite set
of matrices. The formula states that joint spectral subradius and generalized spectral
248 A. Czornik / Linear Algebra and its Applications 407 (2005) 242–248
subradius are equal. We have also showed that these common values are equal to
infimum limits of infA∈m ‖A‖ 1m and infA∈m ρ 1m (A). Moreover we have proved
that the infimum of largest Lyapunov exponents of all linear time varying systems
having coefficients in  is equal to the generalized spectral subradius of . Finally
we propose a new definition of stability of discrete linear inclusions which turns out
to be equivalent to the fact that generalized spectral subradius is less than one.
Acknowledgment
This work was supported by Polish KBN under grant 3T11A 029 28.
References
[1] L. Arnold, H. Crauel, J.P. Eckmann (Eds.), Lyapunov exponents, Lecture Notes in Mathematics, vol.
1486, Springer-Verlag, Berlin, 1991.
[2] V.D. Blondel, J.N. Tsitsiklis, A survey of computational complexity results in systems and control,
Automatica 36 (2000) 1247–1249.
[3] I. Daubechies, J.C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra
Appl. 161 (1992) 227–263.
[4] I. Daubechies, J.C. Lagarias, Corrigendum/addendum to: sets of matrices all infinite products of
which converge, Linear Algebra Appl. 327 (2001) 69–83.
[5] L. Elsner, The generalized spectral radius theorem: an analytic–geometric proof, Linear Algebra Appl.
220 (1995) 151–159.
[6] N. Guglielmi, M. Zennaro, On the asymptotic properties of a family of matrices, Linear Algebra Appl.
322 (2001) 169–192.
[7] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl. 231 (1995) 47–85.
[8] G.C. Rota, G. Strang, A note on the joint spectral radius, Inag. Math. 22 (1960) 379–381.
[9] M.H. Shih, Simultaneous Schur stability, Linear Algebra Appl. 287 (1999) 323–336.


file:///data/remote/core/dit/data/elsevier/pdf/869/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAyNDM3OTUwNTAwMjU4Mg==.pdf

On the generalized spectral subradius

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector