- Email: [email protected]
A characterization of the generalized spectral radius with Kronecker powers
Automatica 47 (2011) 1530–1533
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
A characterization of the generalized spectral radius with Kronecker powers✩ Jianhong Xu, Mingqing Xiao ∗ Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA
article
abstract
info
Article history: Received 22 July 2010 Received in revised form 25 February 2011 Accepted 26 February 2011 Available online 6 May 2011
Based on Turán’s power sum theory, we extend a recent result obtained by Blondel and Nesterov [Blondel, V. D., & Nesterov, Y. (2005). Computationally efficient approximations of the joint spectral radius, SIAM Journal on Matrix Analysis and Applications, 27, 256–272], by deriving a new characterization of the generalized spectral radius in terms of Kronecker powers. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Generalized spectral radius Matrix trace Kronecker power Power sum
1. Introduction According to classical matrix theory, the spectral radius of a single square matrix is defined by
ρ(A) = max |λ|: λ is an eigenvalue of A . For a finite set Σ = {A1 , . . . , Am } of square matrices of the same size, the generalized spectral radius of Σ was introduced by Daubechies and Lagarias (1992) and defined as
ρ(Σ ) = lim sup ρj (Σ )1/j , j→∞
where ρj (Σ ) := max ρ(Ai1 · · · Aij ) : each Ai ∈ Σ ; this implies that the maximization is taken over all sequences {i1 , . . . , ij } for integer j > 0. The generalized spectral radius plays a critical role in a variety of applications such as switched systems (Dai, Huang, & Xiao, 2008; Kozyakin, 2007), differential equations (Guglielmi & Zennaro, 2001), coding theory (Moision, Orlitsky, & Siegel, 2007), and wavelets (Protasov, 2006). For example, given a discrete-time switched linear system in Rn (see Dai et al., 2008 and references therein),
x(k + 1) = Aσ (k) x(k), Aσ (k) ∈ Σ = {A1 , . . . , Am },
(1) k = 0, 1, . . . ,
✩ The research of the second author is supported in part by NSF grants 0605181 and 1021203. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Mikael Johansson under the direction of Editor André L. Tits. ∗ Corresponding author. Tel.: +1 618 453 6572; fax: +1 618 453 5300. E-mail addresses: [email protected] (J. Xu), [email protected] (M. Xiao).
0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.04.007
where σ : N ∪ {0} → {1, 2, . . . , m} represents a switching signal; the generalized spectral radius ρ = ρ(Σ ) is the smallest number such that for every trajectory there is a constant C for which
|x(k)| ≤ C ρ k ,
(2)
assuming that Σ is irreducible (also see Section 2.1.1, Jungers, 2009). Recall that a subset K ⊂ Rn is called a cone in Rn if λv ∈ K for all λ > 0 and v ∈ K . A cone K is said to be proper if it is closed, convex, has a nonempty interior, and contains no straight line. A matrix S is said to leave a proper cone K invariant if SK ⊆ K . If we denote by Rn+ the set of n-tuples with nonnegative entries, then it is straightforward to verify that Rn+ is a proper cone of Rn , and matrices with real nonnegative entries leave the proper cone Rn+ invariant. Recently, Blondel and Nesterov (2005) has showed that for a given set of matrices Σ = {A1 , . . . , Am } ⊂ Rn×n , if all Ai ∈ Σ (i = 1, . . . , m) leave a proper cone K ⊂ Rn invariant, then the generalized spectral radius can be expressed as k ⊗k ρ(Σ ) = lim ρ 1/k (A⊗ 1 + · · · + Am ), k→∞
(3)
where A⊗k is the kth Kronecker power of A. An interesting characteristic of this expression is that the limit in (3) does not directly involve any mixed product of Ai ’s. Blondel and Nesterov’s result relies on the existence of a proper cone K for which all Ai leave it invariant. For example, since Rn+ is a proper cone of Rn , if all Ai ∈ Σ are matrices with real nonnegative entries, then expression (3) holds. However, as we know, Rn itself is not a proper cone and the question of how the expression (3) should be adapted for real matrices remains unanswered in the current literature.
J. Xu, M. Xiao / Automatica 47 (2011) 1530–1533
In this short note, based on Turán’s power sum theory (Turán, 1984), we extend the above result to a more general case. More specifically, let Cn×n be the set of complex n × n matrices. Then, for Σ = {A1 , . . . , Am } ⊂ Cn×n , the generalized spectral radius has the following expression:
Next, consider any ν ≥ 1; according to (7), there is some k ≥ ν + 1 such that
|sk |1/k ≥ |z1 |
(4)
Notice here that Ai ∈ Σ is allowed to be complex1 and there is no requirement for the existence of a proper cone K such that all Ai leave it invariant. Our approach can be summarized as follows. We first show that for Σ = {A1 , . . . , Am } ⊂ Cn×n , the generalized spectral radius is given by
ρ(Σ ) = lim sup max |tr(A)|1/k ,
(5)
A∈Πk
where Πk is the set of products of Ai of length k whose factors are in Σ . Then, based on (5) and Turán’s power sum theorem, we provide a proof of (4). It should be pointed out here that Chen and Zhou have shown the formula (5) in Chen and Zhou (2000) but their proof is much more complicated than the one given in this paper. 2. Main results Our proof of (5) is based upon Turán’s power sum theory, which has a profound impact on the study of prime and algebraic numbers, polynomials, differential equations, and entire functions; see Turán (1984). Let us begin with the following lemma (Sós & Turán, 1955): Lemma 2.1. Let bj , zj ∈ C, j = 1, . . . , n. Suppose that |z1 | ≥ · · · ≥ |zn |. Then, for any integer ν ≥ 0, there exists integer k, with ν + 1 ≤ k ≤ ν + n, such that
[ ]n j n − − n k k bj zj ≥ |z1 | b min l . j=1 24e2 (ν + 2n) j=1,...,n l=1
(6)
In particular, when bj = 1 for all j, (6) reduces to [ ]n n − n k k z ≥ |z1 | . j=1 j 24e2 (ν + 2n)
(7)
∑n
24e2 (k + 2n − 1)
→ |z1 |
as k → ∞ by Lemma 2.1. In other words, (7) implies that the upper limit of |sk |1/k is greater than or equal to |z1 | for some subsequence of {k}∞ 1 , which is specified by Turán’s lemma. This leads to lim sup |sk |1/k ≥ |z1 |.
(11)
k→∞
Therefore, combining (10) and (11) yields the desired result. This completes the proof. In the context of matrices, Lemma 2.2 can be rephrased as: Lemma 2.3. For any A ∈ Cn×n , its spectral radius can be formulated as
ρ(A) = lim sup |tr(Ak )|1/k .
(12)
k→∞
Proof. Suppose that A has spectrum σ (A) = {z1 , . . . , zn } with ρ(A) = |z1 |. The conclusion follows immediately from the fact tr(Ak ) = sk and Lemma 2.2. Before proceeding, we mention that for A ∈ Cn×n , the inequality (9) leads to
|tr(Al )|1/l ≤ n1/l ρ(A)
(13)
for any integer l ≥ 1. We shall refer to this inequality again later. Lemma 2.3 was proved in a more general setting, for matrix sets, in Chen and Zhou (2000) (Theorem 1.3 for arbitrary real matrices). It appears that the proof can be easily extended to arbitrary complex matrices. However, as shown in our next result, Lemma 2.3 does translate to the more general case involving a set of complex matrices for the expression of its generalized spectral radius.
Proof. It is well known (Berger & Wang, 1992) that the generalized spectral radius can be written as
ρ(Σ ) = lim sup max ρ 1/k (A). k→∞
A∈Πk
Let A ∈ Πk . By setting l = 1 in (13), we obtain
Lemma 2.2. For any zj ∈ C such that |z1 | ≥ · · · ≥ |zn |, we have (8)
|tr(A)|1/k ≤ n1/k ρ 1/k (A), and therefore,
k→∞
lim sup max |tr(A)|1/k ≤ lim sup max ρ 1/k (A) = ρ(Σ ).
Proof. First, we observe
k→∞
(9)
The right-hand side above approaches |z1 | as k → ∞. Hence, (9) implies the upper limit lim sup |sk |1/k ≤ |z1 |.
]n/k
n
Lemma 2.4. Given Σ = {A1 , . . . , Am } ⊂ Cn×n , the generalized spectral radius of Σ can be formulated as in (5).
k For convenience, we write sk = j=1 zj , the so-called kth power sum. Lemma 2.1 leads to the next useful conclusion.
|sk |1/k ≤ n1/k |z1 |.
]n/k
n 24e2 (ν + 2n)
≥ |z1 |
k→∞
|z1 | = lim sup |sk |1/k .
[ [
k ⊗k ρ(Σ ) = lim sup ρ 1/k (A⊗ 1 + · · · + Am ).
k→∞
1531
(10)
k→∞
A∈Πk
k→∞
A∈Πk
Conversely, we fix an arbitrary A ∈ Πk . From Lemma 2.3, there exists some sequence {ki } with ki → ∞, as i → ∞, such that
|tr(Aki )|1/ki ≥ ρ(A)/2, i.e.
ρ 1/k (A) ≤ 21/k |tr(Aki )|1/ki k . Denote mi = ki k. Also note that Aki ∈ Πmi . Hence,
1 The complex case may be identified as the one involving real matrices (p. 15, Sós & Turán, 1955) at the expense of doubling the size of matrices.
ρ 1/k (A) ≤ 21/k max |tr( A)|1/mi , A∈Πmi
1532
J. Xu, M. Xiao / Automatica 47 (2011) 1530–1533
which implies
ρ
1/k
(A) ≤ 2
1/k
Hence, using Lemmas 2.3 and 2.4, we find lim sup max |tr( A)|1/m .
k ⊗k 1/k ρ 1/k (A⊗ lim sup max |tr(A)|1/l 1 + · · · + Am ) ≤ m
m→∞ A∈Πm
=m
This further leads to
ρ(Σ ) = lim sup max ρ k→∞
1/k
A∈Πk
(A) ≤ lim sup max |tr( A)|1/m .
k ⊗k lim sup ρ 1/k (A⊗ 1 + · · · + Am ) ≤ ρ(Σ ).
k→∞
The preceding proof also demonstrates that (12) is not merely a special case of (5); these two formulae are indeed equivalent to each other since Lemma 2.3 also implies (5) as shown in the proof. In order to establish our main result, we recall here the notion of Kronecker product of two matrices. Let A = (aij )m×n and B be arbitrary matrices; then the Kronecker product of A and B is defined as
.
.. .
··· ··· .. .
am1 B
am2 B
···
a11 B a21 B
A⊗B= ..
a12 B a22 B
a1n B a2n B amn B
On the other hand, in view of (13) and (16), we have
k ⊗k ρ 1/k (A⊗ 1 + · · · + Am ) ≥
=
.. , .
(14)
and the kth Kronecker power of matrix A is defined to be
A ⊗ · · · ⊗ A, and denoted by A⊗k . Let {Ai }m i=1 be a set of arbitrary square matrices with same size. Then the Kronecker product has the following properties: (i) (Ai ⊗ Aj )(As ⊗ At ) = (Ai As ) ⊗ (Aj At );
⊗k
trk Ai1 Ai2 · · · Aiℓ
=
for any positive
1/kl k ⊗k l tr A⊗ 1 + · · · + Am n1/l
1/kl ∑ trk (A) A∈Πl n1/l
k ⊗k ρ 1/k (A⊗ 1 + · · · + Am )
≥
1
max |tr(A)|1/l n1/l A∈Πl
[
]N /kl
N 24e2 (ν + 2N )
.
(17)
In particular, setting ν = N in (17), we obtain some sequence {kl }, with kl → ∞ as l → ∞, such that 1/l kl ρ 1/kl (A1 l + · · · + A⊗ m ) ≥ max |tr(A)|
⊗k
A∈Πl
For more background material on Kronecker products, please see, for example, Graham (1981) and Horn and Johnson (1994). Continuing, we develop the following extension of (3), which provides a new characterization of the generalized spectral radius:
.
This, by Lemma 2.1, implies that for any ν ≥ 1, there exists k ≥ ν + 1 so that
k-times
(ii) tr Ai1 Ai2 · · · Aiℓ integer k.
ρ(Σ ),
which then yields
m→∞ A∈Πm
The proof is now complete.
A∈Πl
l→∞
1/k
1 72e2 n
1/l
,
which, according to Lemma 2.4, leads to k ⊗k 1/l lim sup ρ 1/k (A⊗ 1 + · · · + Am ) ≥ lim sup max |tr(A)| k→∞
l→∞
A∈Πl
= ρ(Σ ).
Theorem 2.1. Let Σ = {A1 , . . . , Am } ⊂ Cn×n . The generalized spectral radius of Σ is characterized by
This completes the proof.
k ⊗k ρ(Σ ) = lim sup ρ 1/k (A⊗ 1 + · · · + Am ).
Before ending this note, we give one simple, yet illustrative, example for Theorem 2.1. Consider Σ = {A1 , A2 }, where
(15)
k→∞
[
Proof. For any integers k, l ≥ 1,
A1 =
⊗k l
k tr A⊗ 1 + · · · + Am
−
= tr
1≤i1 ,...,il ≤m
−
= tr
Ai1 · · · Ail
⊗k
−
tr Ai1 · · · Ail
−
trk Ai1 · · · Ail ,
1≤i1 ,...,il ≤m
i.e. k ⊗k tr A⊗ 1 + · · · + Am
l
=
−
[ and A2 =
−1 0
]
0 . −0.5
2
0T
k ⊗k A⊗ 1 + A2 =
0
D
,
⊗k
1≤i1 ,...,il ≤m
=
]
1≤i1 ,...,il ≤m
=
0 0.5
It is easy to see that ρ(Σ ) = ρ(A1 ) = ρ(A2 ) = 1. For any k ≥ 1, however, one can verify that if k is even,
k ⊗k A⊗ i1 · · · Ail
1 0
trk (A),
A∈Πl
(16)
where D is a diagonal matrix whose nonzero entries are less than k k or equal to 1; whereas A⊗ + A⊗ = 0 if k is odd. Notice that 1 2 ⊗k ⊗k ρ(A1 +A2 ) is either 0 or 2, depending on whether k is odd or even, k ⊗k and thus the limit limk→∞ ρ 1/k (A⊗ 1 + A2 ) does not exist. However, k ⊗k our Theorem 2.1 gives that lim supk→∞ ρ 1/k (A⊗ 1 + A2 ) = 1 = ρ(Σ ).
with the cardinality of Πl being N = ml . On one hand, (16) implies
Acknowledgements
1/kl ⊗k ⊗k l tr A1 + · · · + Am ≤ m1/k max |tr(A)|1/l . A∈Πl
The authors would like to thank the anonymous reviewers for their very helpful suggestions which led to the improvement of this manuscript.
J. Xu, M. Xiao / Automatica 47 (2011) 1530–1533
References Berger, M. A., & Wang, Y. (1992). Bounded semigroups of matrices. Linear Algebra and its Applications, 166, 21–27. Blondel, V. D., & Nesterov, Y. (2005). Computationally efficient approximations of the joint spectral radius. SIAM Journal on Matrix Analysis and Applications, 27, 256–272. Chen, Q., & Zhou, X. (2000). Characterization of joint spectral radius via trace. Linear Algebra and its Applications, 315, 175–188. Dai, X., Huang, Y., & Xiao, M. (2008). Almost sure stability of discrete-time switching linear systems: a topological point of view. SIAM Journal on Control and Optimization, 47, 2137–2156. Daubechies, I., & Lagarias, J. C. (1992). Sets of matrices all infinite products of which converge. Linear Algebra and its Applications, 161, 227–263. Graham, A. (1981). Kronecker products and matrix calculus: with applications. Ellis Horwood Limited.
1533
Guglielmi, N., & Zennaro, M. (2001). On the zero-stability of variable stepsize multistep methods: the spectral radius approach. Numerische Mathematik, 88, 445–458. Horn, R. A., & Johnson, C. R. (1994). Topics in matrix analysis. Cambridge University Press. Jungers, R. M. (2009). The joint spectral radius, theory and applications. In Lecture notes in control and information sciences: Vol. 385. Berlin: Springer-Verlag. Kozyakin, V. S. (2007). Structure of extremal trajectories of discrete linear systems and the finiteness conjecture. Automation and Remote Control, 68, 174–209. Moision, B., Orlitsky, A., & Siegel, P. H. (2007). On codes with local joint constraints. Linear Algebra and its Applications, 422, 442–454. Protasov, V. Y. (2006). Fractal curves and wavelets. Izvestiya Mathematics, 70, 123–162. Sós, V. T., & Turán, P. (1955). On some new theorems in the theory of Diophatine approximations. Acta Mathematica Hungarica, 6, 241–257. Turán, P. (1984). On a new method of analysis and its applications. John Wiley & Sons.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
A characterization of the generalized spectral radius with Kronecker powers✩ Jianhong Xu, Mingqing Xiao ∗ Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA
article
abstract
info
Article history: Received 22 July 2010 Received in revised form 25 February 2011 Accepted 26 February 2011 Available online 6 May 2011
Based on Turán’s power sum theory, we extend a recent result obtained by Blondel and Nesterov [Blondel, V. D., & Nesterov, Y. (2005). Computationally efficient approximations of the joint spectral radius, SIAM Journal on Matrix Analysis and Applications, 27, 256–272], by deriving a new characterization of the generalized spectral radius in terms of Kronecker powers. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Generalized spectral radius Matrix trace Kronecker power Power sum
1. Introduction According to classical matrix theory, the spectral radius of a single square matrix is defined by
ρ(A) = max |λ|: λ is an eigenvalue of A . For a finite set Σ = {A1 , . . . , Am } of square matrices of the same size, the generalized spectral radius of Σ was introduced by Daubechies and Lagarias (1992) and defined as
ρ(Σ ) = lim sup ρj (Σ )1/j , j→∞
where ρj (Σ ) := max ρ(Ai1 · · · Aij ) : each Ai ∈ Σ ; this implies that the maximization is taken over all sequences {i1 , . . . , ij } for integer j > 0. The generalized spectral radius plays a critical role in a variety of applications such as switched systems (Dai, Huang, & Xiao, 2008; Kozyakin, 2007), differential equations (Guglielmi & Zennaro, 2001), coding theory (Moision, Orlitsky, & Siegel, 2007), and wavelets (Protasov, 2006). For example, given a discrete-time switched linear system in Rn (see Dai et al., 2008 and references therein),
x(k + 1) = Aσ (k) x(k), Aσ (k) ∈ Σ = {A1 , . . . , Am },
(1) k = 0, 1, . . . ,
✩ The research of the second author is supported in part by NSF grants 0605181 and 1021203. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Mikael Johansson under the direction of Editor André L. Tits. ∗ Corresponding author. Tel.: +1 618 453 6572; fax: +1 618 453 5300. E-mail addresses: [email protected] (J. Xu), [email protected] (M. Xiao).
0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.04.007
where σ : N ∪ {0} → {1, 2, . . . , m} represents a switching signal; the generalized spectral radius ρ = ρ(Σ ) is the smallest number such that for every trajectory there is a constant C for which
|x(k)| ≤ C ρ k ,
(2)
assuming that Σ is irreducible (also see Section 2.1.1, Jungers, 2009). Recall that a subset K ⊂ Rn is called a cone in Rn if λv ∈ K for all λ > 0 and v ∈ K . A cone K is said to be proper if it is closed, convex, has a nonempty interior, and contains no straight line. A matrix S is said to leave a proper cone K invariant if SK ⊆ K . If we denote by Rn+ the set of n-tuples with nonnegative entries, then it is straightforward to verify that Rn+ is a proper cone of Rn , and matrices with real nonnegative entries leave the proper cone Rn+ invariant. Recently, Blondel and Nesterov (2005) has showed that for a given set of matrices Σ = {A1 , . . . , Am } ⊂ Rn×n , if all Ai ∈ Σ (i = 1, . . . , m) leave a proper cone K ⊂ Rn invariant, then the generalized spectral radius can be expressed as k ⊗k ρ(Σ ) = lim ρ 1/k (A⊗ 1 + · · · + Am ), k→∞
(3)
where A⊗k is the kth Kronecker power of A. An interesting characteristic of this expression is that the limit in (3) does not directly involve any mixed product of Ai ’s. Blondel and Nesterov’s result relies on the existence of a proper cone K for which all Ai leave it invariant. For example, since Rn+ is a proper cone of Rn , if all Ai ∈ Σ are matrices with real nonnegative entries, then expression (3) holds. However, as we know, Rn itself is not a proper cone and the question of how the expression (3) should be adapted for real matrices remains unanswered in the current literature.
J. Xu, M. Xiao / Automatica 47 (2011) 1530–1533
In this short note, based on Turán’s power sum theory (Turán, 1984), we extend the above result to a more general case. More specifically, let Cn×n be the set of complex n × n matrices. Then, for Σ = {A1 , . . . , Am } ⊂ Cn×n , the generalized spectral radius has the following expression:
Next, consider any ν ≥ 1; according to (7), there is some k ≥ ν + 1 such that
|sk |1/k ≥ |z1 |
(4)
Notice here that Ai ∈ Σ is allowed to be complex1 and there is no requirement for the existence of a proper cone K such that all Ai leave it invariant. Our approach can be summarized as follows. We first show that for Σ = {A1 , . . . , Am } ⊂ Cn×n , the generalized spectral radius is given by
ρ(Σ ) = lim sup max |tr(A)|1/k ,
(5)
A∈Πk
where Πk is the set of products of Ai of length k whose factors are in Σ . Then, based on (5) and Turán’s power sum theorem, we provide a proof of (4). It should be pointed out here that Chen and Zhou have shown the formula (5) in Chen and Zhou (2000) but their proof is much more complicated than the one given in this paper. 2. Main results Our proof of (5) is based upon Turán’s power sum theory, which has a profound impact on the study of prime and algebraic numbers, polynomials, differential equations, and entire functions; see Turán (1984). Let us begin with the following lemma (Sós & Turán, 1955): Lemma 2.1. Let bj , zj ∈ C, j = 1, . . . , n. Suppose that |z1 | ≥ · · · ≥ |zn |. Then, for any integer ν ≥ 0, there exists integer k, with ν + 1 ≤ k ≤ ν + n, such that
[ ]n j n − − n k k bj zj ≥ |z1 | b min l . j=1 24e2 (ν + 2n) j=1,...,n l=1
(6)
In particular, when bj = 1 for all j, (6) reduces to [ ]n n − n k k z ≥ |z1 | . j=1 j 24e2 (ν + 2n)
(7)
∑n
24e2 (k + 2n − 1)
→ |z1 |
as k → ∞ by Lemma 2.1. In other words, (7) implies that the upper limit of |sk |1/k is greater than or equal to |z1 | for some subsequence of {k}∞ 1 , which is specified by Turán’s lemma. This leads to lim sup |sk |1/k ≥ |z1 |.
(11)
k→∞
Therefore, combining (10) and (11) yields the desired result. This completes the proof. In the context of matrices, Lemma 2.2 can be rephrased as: Lemma 2.3. For any A ∈ Cn×n , its spectral radius can be formulated as
ρ(A) = lim sup |tr(Ak )|1/k .
(12)
k→∞
Proof. Suppose that A has spectrum σ (A) = {z1 , . . . , zn } with ρ(A) = |z1 |. The conclusion follows immediately from the fact tr(Ak ) = sk and Lemma 2.2. Before proceeding, we mention that for A ∈ Cn×n , the inequality (9) leads to
|tr(Al )|1/l ≤ n1/l ρ(A)
(13)
for any integer l ≥ 1. We shall refer to this inequality again later. Lemma 2.3 was proved in a more general setting, for matrix sets, in Chen and Zhou (2000) (Theorem 1.3 for arbitrary real matrices). It appears that the proof can be easily extended to arbitrary complex matrices. However, as shown in our next result, Lemma 2.3 does translate to the more general case involving a set of complex matrices for the expression of its generalized spectral radius.
Proof. It is well known (Berger & Wang, 1992) that the generalized spectral radius can be written as
ρ(Σ ) = lim sup max ρ 1/k (A). k→∞
A∈Πk
Let A ∈ Πk . By setting l = 1 in (13), we obtain
Lemma 2.2. For any zj ∈ C such that |z1 | ≥ · · · ≥ |zn |, we have (8)
|tr(A)|1/k ≤ n1/k ρ 1/k (A), and therefore,
k→∞
lim sup max |tr(A)|1/k ≤ lim sup max ρ 1/k (A) = ρ(Σ ).
Proof. First, we observe
k→∞
(9)
The right-hand side above approaches |z1 | as k → ∞. Hence, (9) implies the upper limit lim sup |sk |1/k ≤ |z1 |.
]n/k
n
Lemma 2.4. Given Σ = {A1 , . . . , Am } ⊂ Cn×n , the generalized spectral radius of Σ can be formulated as in (5).
k For convenience, we write sk = j=1 zj , the so-called kth power sum. Lemma 2.1 leads to the next useful conclusion.
|sk |1/k ≤ n1/k |z1 |.
]n/k
n 24e2 (ν + 2n)
≥ |z1 |
k→∞
|z1 | = lim sup |sk |1/k .
[ [
k ⊗k ρ(Σ ) = lim sup ρ 1/k (A⊗ 1 + · · · + Am ).
k→∞
1531
(10)
k→∞
A∈Πk
k→∞
A∈Πk
Conversely, we fix an arbitrary A ∈ Πk . From Lemma 2.3, there exists some sequence {ki } with ki → ∞, as i → ∞, such that
|tr(Aki )|1/ki ≥ ρ(A)/2, i.e.
ρ 1/k (A) ≤ 21/k |tr(Aki )|1/ki k . Denote mi = ki k. Also note that Aki ∈ Πmi . Hence,
1 The complex case may be identified as the one involving real matrices (p. 15, Sós & Turán, 1955) at the expense of doubling the size of matrices.
ρ 1/k (A) ≤ 21/k max |tr( A)|1/mi , A∈Πmi
1532
J. Xu, M. Xiao / Automatica 47 (2011) 1530–1533
which implies
ρ
1/k
(A) ≤ 2
1/k
Hence, using Lemmas 2.3 and 2.4, we find lim sup max |tr( A)|1/m .
k ⊗k 1/k ρ 1/k (A⊗ lim sup max |tr(A)|1/l 1 + · · · + Am ) ≤ m
m→∞ A∈Πm
=m
This further leads to
ρ(Σ ) = lim sup max ρ k→∞
1/k
A∈Πk
(A) ≤ lim sup max |tr( A)|1/m .
k ⊗k lim sup ρ 1/k (A⊗ 1 + · · · + Am ) ≤ ρ(Σ ).
k→∞
The preceding proof also demonstrates that (12) is not merely a special case of (5); these two formulae are indeed equivalent to each other since Lemma 2.3 also implies (5) as shown in the proof. In order to establish our main result, we recall here the notion of Kronecker product of two matrices. Let A = (aij )m×n and B be arbitrary matrices; then the Kronecker product of A and B is defined as
.
.. .
··· ··· .. .
am1 B
am2 B
···
a11 B a21 B
A⊗B= ..
a12 B a22 B
a1n B a2n B amn B
On the other hand, in view of (13) and (16), we have
k ⊗k ρ 1/k (A⊗ 1 + · · · + Am ) ≥
=
.. , .
(14)
and the kth Kronecker power of matrix A is defined to be
A ⊗ · · · ⊗ A, and denoted by A⊗k . Let {Ai }m i=1 be a set of arbitrary square matrices with same size. Then the Kronecker product has the following properties: (i) (Ai ⊗ Aj )(As ⊗ At ) = (Ai As ) ⊗ (Aj At );
⊗k
trk Ai1 Ai2 · · · Aiℓ
=
for any positive
1/kl k ⊗k l tr A⊗ 1 + · · · + Am n1/l
1/kl ∑ trk (A) A∈Πl n1/l
k ⊗k ρ 1/k (A⊗ 1 + · · · + Am )
≥
1
max |tr(A)|1/l n1/l A∈Πl
[
]N /kl
N 24e2 (ν + 2N )
.
(17)
In particular, setting ν = N in (17), we obtain some sequence {kl }, with kl → ∞ as l → ∞, such that 1/l kl ρ 1/kl (A1 l + · · · + A⊗ m ) ≥ max |tr(A)|
⊗k
A∈Πl
For more background material on Kronecker products, please see, for example, Graham (1981) and Horn and Johnson (1994). Continuing, we develop the following extension of (3), which provides a new characterization of the generalized spectral radius:
.
This, by Lemma 2.1, implies that for any ν ≥ 1, there exists k ≥ ν + 1 so that
k-times
(ii) tr Ai1 Ai2 · · · Aiℓ integer k.
ρ(Σ ),
which then yields
m→∞ A∈Πm
The proof is now complete.
A∈Πl
l→∞
1/k
1 72e2 n
1/l
,
which, according to Lemma 2.4, leads to k ⊗k 1/l lim sup ρ 1/k (A⊗ 1 + · · · + Am ) ≥ lim sup max |tr(A)| k→∞
l→∞
A∈Πl
= ρ(Σ ).
Theorem 2.1. Let Σ = {A1 , . . . , Am } ⊂ Cn×n . The generalized spectral radius of Σ is characterized by
This completes the proof.
k ⊗k ρ(Σ ) = lim sup ρ 1/k (A⊗ 1 + · · · + Am ).
Before ending this note, we give one simple, yet illustrative, example for Theorem 2.1. Consider Σ = {A1 , A2 }, where
(15)
k→∞
[
Proof. For any integers k, l ≥ 1,
A1 =
⊗k l
k tr A⊗ 1 + · · · + Am
−
= tr
1≤i1 ,...,il ≤m
−
= tr
Ai1 · · · Ail
⊗k
−
tr Ai1 · · · Ail
−
trk Ai1 · · · Ail ,
1≤i1 ,...,il ≤m
i.e. k ⊗k tr A⊗ 1 + · · · + Am
l
=
−
[ and A2 =
−1 0
]
0 . −0.5
2
0T
k ⊗k A⊗ 1 + A2 =
0
D
,
⊗k
1≤i1 ,...,il ≤m
=
]
1≤i1 ,...,il ≤m
=
0 0.5
It is easy to see that ρ(Σ ) = ρ(A1 ) = ρ(A2 ) = 1. For any k ≥ 1, however, one can verify that if k is even,
k ⊗k A⊗ i1 · · · Ail
1 0
trk (A),
A∈Πl
(16)
where D is a diagonal matrix whose nonzero entries are less than k k or equal to 1; whereas A⊗ + A⊗ = 0 if k is odd. Notice that 1 2 ⊗k ⊗k ρ(A1 +A2 ) is either 0 or 2, depending on whether k is odd or even, k ⊗k and thus the limit limk→∞ ρ 1/k (A⊗ 1 + A2 ) does not exist. However, k ⊗k our Theorem 2.1 gives that lim supk→∞ ρ 1/k (A⊗ 1 + A2 ) = 1 = ρ(Σ ).
with the cardinality of Πl being N = ml . On one hand, (16) implies
Acknowledgements
1/kl ⊗k ⊗k l tr A1 + · · · + Am ≤ m1/k max |tr(A)|1/l . A∈Πl
The authors would like to thank the anonymous reviewers for their very helpful suggestions which led to the improvement of this manuscript.
J. Xu, M. Xiao / Automatica 47 (2011) 1530–1533
References Berger, M. A., & Wang, Y. (1992). Bounded semigroups of matrices. Linear Algebra and its Applications, 166, 21–27. Blondel, V. D., & Nesterov, Y. (2005). Computationally efficient approximations of the joint spectral radius. SIAM Journal on Matrix Analysis and Applications, 27, 256–272. Chen, Q., & Zhou, X. (2000). Characterization of joint spectral radius via trace. Linear Algebra and its Applications, 315, 175–188. Dai, X., Huang, Y., & Xiao, M. (2008). Almost sure stability of discrete-time switching linear systems: a topological point of view. SIAM Journal on Control and Optimization, 47, 2137–2156. Daubechies, I., & Lagarias, J. C. (1992). Sets of matrices all infinite products of which converge. Linear Algebra and its Applications, 161, 227–263. Graham, A. (1981). Kronecker products and matrix calculus: with applications. Ellis Horwood Limited.
1533
Guglielmi, N., & Zennaro, M. (2001). On the zero-stability of variable stepsize multistep methods: the spectral radius approach. Numerische Mathematik, 88, 445–458. Horn, R. A., & Johnson, C. R. (1994). Topics in matrix analysis. Cambridge University Press. Jungers, R. M. (2009). The joint spectral radius, theory and applications. In Lecture notes in control and information sciences: Vol. 385. Berlin: Springer-Verlag. Kozyakin, V. S. (2007). Structure of extremal trajectories of discrete linear systems and the finiteness conjecture. Automation and Remote Control, 68, 174–209. Moision, B., Orlitsky, A., & Siegel, P. H. (2007). On codes with local joint constraints. Linear Algebra and its Applications, 422, 442–454. Protasov, V. Y. (2006). Fractal curves and wavelets. Izvestiya Mathematics, 70, 123–162. Sós, V. T., & Turán, P. (1955). On some new theorems in the theory of Diophatine approximations. Acta Mathematica Hungarica, 6, 241–257. Turán, P. (1984). On a new method of analysis and its applications. John Wiley & Sons.