Nonhomogeneous Matrix Products

Hartfiel, D. J.

doi:10.1142/9789812810052

Cited by 31 publications

(58 citation statements)

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“…Namely, the following statement, which was repeatedly 'discovered' by many authors, is true, see, e.g., [2,[8][9][10][11].…”

Section: Path-dependent Stabilizabilitymentioning

confidence: 97%

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On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

Kozyakin

2018

Discrete Dynamics in Nature and Society

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We consider the problem of convergence to zero of matrix products ⋅ ⋅ ⋅ 1 1 with factors from two sets of matrices, ∈ A and ∈ B, due to a suitable choice of matrices { }. It is assumed that for any sequence of matrices { } there is a sequence of matrices { } such that the corresponding matrix products ⋅ ⋅ ⋅ 1 1 converge to zero. We show that, in this case, the convergence of the matrix products under consideration is uniformly exponential; that is, ‖ ⋅ ⋅ ⋅ 1 1 ‖ ≤ , where the constants > 0 and ∈ (0, 1) do not depend on the sequence { } and the corresponding sequence { }. Other problems of this kind are discussed and open questions are formulated.

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“…Namely, the following statement, which was repeatedly 'discovered' by many authors, is true, see, e.g., [2,[8][9][10][11].…”

Section: Path-dependent Stabilizabilitymentioning

confidence: 97%

“…Let us consider one more issue, which is adjacent to the topic under discussion. In the theory of matrix products, the following assertion is known [2,[8][9][10][11]: let A be a finite set such that for each sequence of matrices {A n ∈ A} the sequence of norms { A n · · · A 1 , n = 1, 2, . .…”

Section: Remarkmentioning

confidence: 99%

On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

Kozyakin

2018

Discrete Dynamics in Nature and Society

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“…Further, neither W nor W (t) (see (7)) is required to have a symmetric zero/nonzero structure, the results in [17], [18] and [21] are not applicable. The following proof uses the results presented in [22] on the convergence of infinite products of substochastic matrices. Notice that [22,Theorem 6.2] requires that the smallest rowsums of all the matrices be uniformly bounded away and below 1.…”

Section: Opinion Dynamics With One Leadermentioning

confidence: 99%

“…The following proof uses the results presented in [22] on the convergence of infinite products of substochastic matrices. Notice that [22,Theorem 6.2] requires that the smallest rowsums of all the matrices be uniformly bounded away and below 1. Here, we require milder conditions.…”

Section: Opinion Dynamics With One Leadermentioning

confidence: 99%

Opinion Dynamics with Persistent Leaders

Abed

2014

53rd IEEE Conference on Decision and Control

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This paper revisits the problem of agreement seeking in a social network under the influence of leaders. The persistence of the leader's effect on the opinions of other agents is characterized by the total weight that they place on the leader's information over time. If this weight is infinite, then the leader is called persistent. Our results describe the asymptotic behavior of network opinions towards the state of a persistent leader in both cases of networks with fixed and switching topologies. We also show that only persistent leaders are able to drive the network to the leader's constant state.

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“…A product of this kind is called a positive product and the length of the shortest positive product of a primitive set M is called its exponent and it is denoted by exp(M). The concept of primitive set was just recently formalized by Protasov and Voynov [28], but has been appearing before in different fields as in stochastic switching systems [18,27] and time-inhomogeneous Markov chains [17,31]. It has lately gained more importance due to its applications in consensus of discretetime multi-agent systems [8], cryptography [11] and automata theory [3,13,5].…”

Section: Introductionmentioning

confidence: 99%

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

Azfar

Catalano

Charlier

et al. 2019

Developments in Language Theory

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A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods.Keywords: Primitive set of matrices, synchronizing automaton, Černý conjecture.Deciding whether a set is primitive is an NP-hard problem if the set contains at least three matrices (while it is still of unknown complexity for sets of two matrices) [3], and so is computing its exponent. For sets of matrices having at least a positive entry in every row and every column (called NZ [13] or allowable matrices [16,18]), the primitivity problem becomes decidable in polynomial-time [28], although computing the exponent remains NP-hard [13]. Methods for approximating the exponent have been proposed [6] as well as upper bounds that depend just on the matrix size; in particular, if we denote with exp N Z (n) the maximal exponent among all the primitive sets of n × n NZ matrices, it is known that exp N Z (n) ≤ (15617n 3 + 7500n 2 + 56250n − 78125)/46875 [3,33]. Better upper bounds have been found for some classes of primitive sets [13,17]. The NZ condition is often met in applications and in particular in the connection with synchronizing automata.

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Nonhomogeneous Matrix Products

Cited by 31 publications

References 0 publications

On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

Opinion Dynamics with Persistent Leaders

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

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