On the finiteness property for rational matrices

Jungers, Raphaël M.; Blondel, Vincent D.

doi:10.1016/j.laa.2007.07.007

Cited by 35 publications

(20 citation statements)

References 20 publications

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“…Finally, in the recent paper [GCCZ], we have proved that every pair of 2 × 2 binary sign-matrices, that is, with entries in {−1, 0, 1}, has the finiteness property. This supports the conjecture by Blondel and Jüngers [JB08] that this holds in general for all pairs of sign-matrices of any dimension. In turn, this fact would have the consequence that the finiteness property holds true for all families of rational matrices.…”

Section: Proposition 36 Let the Finite Family F Of Real N×n-matricesupporting

confidence: 90%

Finding Extremal Complex Polytope Norms for Families of Real Matrices

Guglielmi¹,

Zennaro²

2009

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper we consider finite families F of real n×n-matrices. In particular, we focus on the computation of the joint spectral radius ρ(F ) via the detection of an extremal norm in the class of complex polytope norms, whose unit balls are balanced complex polytopes with a finite essential system of vertices. Such a finiteness property is very useful in view of the construction of efficient computational algorithms. More precisely, we improve the results obtained in our previous paper [GWZ05], where we gave some conditions on the family F which are sufficient to guarantee the existence of an extremal complex polytope norm. Unfortunately, they are such to exclude unnecessarily many interesting cases of real families. Therefore, here we relax somehow the conditions given in [GWZ05], in order to provide a more satisfactory treatment of the real case.

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Section: Proposition 36 Let the Finite Family F Of Real N×n-matricesupporting

confidence: 90%

Finding Extremal Complex Polytope Norms for Families of Real Matrices

Guglielmi¹,

Zennaro²

2009

SIAM J. Matrix Anal. & Appl.

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“…In [23] it was demonstrated that the finiteness conjecture is valid for any pair of 2 × 2 binary matrices, i.e. matrices with the elements {0, 1}, and in [24] a similar result was proved for any pair of 2 × 2 sign-matrices, i.e.…”

Section: Introductionmentioning

confidence: 81%

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

Kozyakin

2016

Linear Algebra and its Applications

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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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“…, } such that ρ(A i1 · · · A in ) = (A) n ? This question was answered negatively in 2002 by Bousch and Mairesse [13], but continues to stimulate research [7,14,26,31,35,40].…”

Section: Applications To the Study Of The 1-ratio Of Pairs Of Matricesmentioning

confidence: 99%

Mather sets for sequences of matrices and applications to the study of joint spectral radii

Morris

2013

Proceedings of the London Mathematical Society

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The joint spectral radius of a compact set of d×d matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article, we investigate the ergodic‐theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences, which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise matrix sequences of maximum growth. We give applications of this theorem to the study of joint spectral radii and to the stability theory of discrete linear inclusions. These results rest on some general theorems on the structure of orbits of maximum growth for subadditive observations of dynamical systems, including an extension of the semiuniform subadditive ergodic theorem of Schreiber, Sturman and Stark, and an extension of a noted lemma of Y. Peres. These theorems are presented in the appendix.

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On the finiteness property for rational matrices

Cited by 35 publications

References 20 publications

Finding Extremal Complex Polytope Norms for Families of Real Matrices

Finding Extremal Complex Polytope Norms for Families of Real Matrices

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

Mather sets for sequences of matrices and applications to the study of joint spectral radii

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