Generic properties of the lower spectral radius for some low-rank pairs of matrices

Morris, Ian D.

doi:10.1016/j.laa.2017.02.023

Cited by 4 publications

(1 citation statement)

References 21 publications

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“…The stabilizability is equivalent to the condition ρ(A) < 1. See [4,14,29] for more properties of the lower spectral radius. Apart from the dynamical systems, it has found applications in the theory of wavelets, in approximation theory, in the number theory, combinatorics, the theory of formal languages, etc.…”

Section: The Lower Spectral Radiusmentioning

confidence: 99%

Antinorms on cones: duality and applications

Protasov¹

2021

Preprint

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An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality theory in convex analysis, in particular, the Fenchel -Moreau theorem, are generalized to antinorms. However, it is shown that the duality relation for antinorms is discontinuous. In every dimension there are infinitely many self-dual antinorms on the positive orthant and, in particular, infinitely many autopolar polyhedra. For the two-dimensional case, we characterise them all. The classification in higher dimensions is left as an open problem. Applications to linear dynamical systems, to the Lyapunov exponent of random matrix products, to the lower spectral radius of nonnegative matrices, and to convex trigonometry are considered.

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Section: The Lower Spectral Radiusmentioning

confidence: 99%

Antinorms on cones: duality and applications

Protasov¹

2021

Preprint

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Antinorms on cones: duality and applications

Protasov

2021

Linear and Multilinear Algebra

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Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

Altschuler

Parrilo

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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The Lyapunov exponent corresponding to a set of square matrices A = {A 1 , . . . , A n } and a probability distribution p over {1, . . . , n} is λ(A, p) := lim k→∞ 1 k E log A σ k · · · A σ2 A σ1 , where σ i are i.i.d. according to p. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate e λ(A,p) of the stochastic linear dynamical system x k+1 = A σ k x k . This paper investigates the following "design problem": given A, compute the distribution p minimizing λ(A, p). Our main result is that it is NP-hard to decide whether there exists a distribution p for which λ(A, p) < 0, i.e. it is NP-hard to decide whether this dynamical system can be stabilized.This hardness result holds even in the "simple" case where A contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius -the deterministic kindred of the Lyapunov exponent -for which the analogous optimization problem for rank-one matrices is known to be exactly computable in polynomial time.To prove this hardness result, we first observe via Birkhoff's Ergodic Theorem that the Lyapunov exponent of rank-one matrices admits a simple formula and in fact is a quadratic form in p. Hardness of the design problem is shown through a reduction from the Independent Set problem. Along the way, simple examples are given illustrating that p → λ(A, p) is neither convex nor concave in general. We conclude with extensions to continuous distributions, exchangeable processes, Markov processes, and stationary ergodic processes.where the σ i are independently and identically distributed (i.i.d.) according to p. Over the past half century, this quantity λ(A, p) has received significant interest due to its many connections and applications to diverse fields including probability, ergodic theory, dynamical systems, functional analysis, representation theory, computer image generation of fractals, control theory, and more; see e.g. the surveys [4,9,10,16,21,43] and references within.The primary application of the Lyapunov exponent that this paper focuses on is in control theory. The fundamental connection is that λ(A, p) dictates the asymptotic growth rate of the stochastic linear dynamical system( 1.2)The authors are with the

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Generic properties of the lower spectral radius for some low-rank pairs of matrices

Cited by 4 publications

References 21 publications

Antinorms on cones: duality and applications

Antinorms on cones: duality and applications

Antinorms on cones: duality and applications

Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

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