Falseness of the Finiteness Property of the Spectral Subradius

Czornik, Adam; Jurgaś, Piotr

doi:10.2478/v10006-007-0016-1

Cited by 7 publications

(4 citation statements)

References 7 publications

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“…It is well-known that the condition ρ (A i ) < 1 for i ∈ I is only the necessary but not sufficient condition for absolute stability of DLI (Σ) [12]. In our further considerations we will show that this condition together with some bounds on A i − A j for i, j ∈ I implies the absolute stability of DLI (Σ).…”

Section: Resultsmentioning

confidence: 99%

Estimation of the Joint Spectral Radius

Czornik

Jurgaś

Niezabitowski

2015

Advances in Intelligent Systems and Computing

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The joint spectral radius of a set of matrices is a generalization of the concept of spectral radius of a matrix. Such notation has many applications in the computer science, and more generally in applied mathematics. It has been used, for example in graph theory, control theory, capacity of codes, continuity of wavelets, overlap-free words, trackable graphs. It is impossible to provide analytic formulae for this quantity and therefore any estimation are highly desired. The main result of this paper is to provide an estimation of the joint spectral radius in the terms of matrices norms and spectral radii.

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Section: Resultsmentioning

confidence: 99%

Estimation of the Joint Spectral Radius

Czornik

Jurgaś

Niezabitowski

2015

Advances in Intelligent Systems and Computing

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“…The lower radius in a sense is more complex object for analysis than the generalized spectral radius because it generally depends on A not continuously [19,20]. However, for the lower spectral radius, disproof of the finiteness conjecture was found to be even easier [12,21] than for the generalized spectral radius.…”

Section: Introductionmentioning

confidence: 98%

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Kozyakin

2015

Preprint

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Recently Blondel, Nesterov and Protasov proved [1,2] that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used in [3], by the author and his companions, to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii.

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“…This is indeed the case from the theoretical point of view. However, in practice it is rather difficult, if at all possible, to calculate in a closed formula form the limits in (3) and (5), see, e.g., numerous negative results in [14][15][16][17][18][19]. This implies the need to make use of approximate computational methods.…”

Section: Introductionmentioning

confidence: 99%

Constructive stability and stabilizability of positive linear discrete-time switching systems

Kozyakin

2017

J. Commun. Technol. Electron.

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We describe a new class of positive linear discrete-time switching systems for which the problems of stability or stabilizability can be resolved constructively. The systems constituting this class can be treated as a natural generalization of systems with the so-called independently switching state vector components. Distinctive feature of such systems is that their components can be arbitrarily 're-connected' in parallel or in series without loss of the 'constructive resolvability' property for the problems of stability or stabilizability of a system. It is shown also that, for such systems, the individual positive trajectories with the greatest or the lowest rate of convergence to the zero can be built constructively.

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Falseness of the Finiteness Property of the Spectral Subradius

Cited by 7 publications

References 7 publications

Estimation of the Joint Spectral Radius

Estimation of the Joint Spectral Radius

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Constructive stability and stabilizability of positive linear discrete-time switching systems

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