Efficient algorithms for deciding the type of growth of products of integer matrices

Jungers, Raphaël M.; Protasov, Vladimir Yu.; Blondel, Vincent D.

doi:10.1016/j.laa.2007.08.001

Cited by 53 publications

(53 citation statements)

References 32 publications

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“…We conclude the paper with several remarks regarding the worst asymptotic polynomial behavior, and with a comparison with the results and open questions in [11,12]. In particular we estimate both the largest asymptotic growth of single trajectories as the time goes to infinity and the asymptotic growth, as t goes to infinity, of the largest norm of a point attainable at time t from the unit ball.…”

mentioning

confidence: 79%

On the marginal instability of linear switched systems

Chitour

Mason

Sigalotti

2012

Systems & Control Letters

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Stability properties for continuous-time linear switched systems are at first determined by the (largest) Lyapunov exponent associated with the system, which is the analogous of the joint spectral radius for the discrete-time case. The purpose of this paper is to provide a characterization of marginally unstable systems, i.e., systems for which the Lyapunov exponent is equal to zero and there exists an unbounded trajectory, and to analyze the asymptotic behavior of their trajectories. Our main contribution consists in pointing out a resonance phenomenon associated with marginal instability. In the course of our study, we derive an upper bound of the state at time t, which is polynomial in t and whose degree is computed from the resonance structure of the system. We also derive analogous results for discrete-time linear switched systems. IntroductionWe consider linear switched systems of the forṁwhere x ∈ R n and the switching law A(·) is an arbitrary measurable function taking values on a compact and convex set A of n × n matrices. In the following, a switched system of the form (1) will be often identified with the corresponding set of matrices A. This paper is concerned with stability issues for (1), where the stability properties are assumed to be uniform with respect to the switching law A(·). * Y. Chitour and P. Mason are with

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mentioning

confidence: 79%

On the marginal instability of linear switched systems

Chitour

Mason

Sigalotti

2012

Systems & Control Letters

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“…. , r, there must exist at least one sequence α of length the number |α| i of occurrences of the symbol i in which satisfies the "frequency constraints" (4). Whereas condition (6), provided that condition (5) is satisfied, is equivalent to the existence of at least one sequence α of length the number |α| i of occurrences of each symbol i = 1, 2, .…”

Section: Symbolic Sequencesmentioning

confidence: 99%

“…. , r}, see, e.g., [1][2][3][4][5][6] and the bibliography therein. The question about the rate of growth of the matrix products (1) is relatively simple (at least theoretically) in edge cases, for example, when the sequence α = (α n ) is periodic or in (1) all possible sequences α = (α n ) with symbols from A = {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Matrix products with constraints on the sliding block relative frequencies of different factors

Kozyakin

2014

Linear Algebra and its Applications

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One of fundamental results of the theory of joint/generalized spectral radius, the BergerWang theorem, establishes equality between the joint and generalized spectral radii of a set of matrices. Generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the Berger-Wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products. Recently, X. Dai [1] proved an analog of the Berger-Wang theorem for the case when factors in matrix products are formed by some Markov law.We extend the concepts of joint and generalized spectral radii to products of matrices with constraints on the sliding block relative frequencies of occurrences of different factors, and prove an analog of the Berger-Wang theorem for this case.

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“…(1) It is known that for sets of nonnegative integer matrices, if ρ 1, then either ρ = 0 and the finiteness property holds, or ρ = 1, and there is a product of matrices in with a diagonal entry equal to one [11]. Such a product A ∈ t satisfies ρ( ) = ρ(A) 1/t = 1 and so the finiteness property holds when ρ( ) 1.…”

Section: Proposition 3 For Any Set Of Matricesmentioning

confidence: 99%

“…The joint spectral radius of the first set is equal to one. Indeed, it is known that for nonnegative integer matrices, if the joint spectral radius is larger than one, then there must be a product of matrices with a diagonal entry larger than one [11]. This is impossible here, since as soon as a path leaves the first node, it cannot come back to it, and no path can leave the second node.…”

Section: Proposition 3 For Any Set Of Matricesmentioning

confidence: 99%

On the Finiteness Property for Rational Matrices

Jungers

Blondel

2009

The Joint Spectral Radius

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We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This "finiteness conjecture" is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2 × 2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.

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Efficient algorithms for deciding the type of growth of products of integer matrices

Cited by 53 publications

References 32 publications

On the marginal instability of linear switched systems

On the marginal instability of linear switched systems

Matrix products with constraints on the sliding block relative frequencies of different factors

On the Finiteness Property for Rational Matrices

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