The Finiteness Conjecture for the Joint Spectral Radius of a Pair of Matrices

Wang, Shuoting; Wen, Junhai

doi:10.1109/cis.2013.174

Cited by 2 publications

(2 citation statements)

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“…matrices with the elements {−1, 0, 1}. Finally, in [25][26][27][28][29] it was shown that the finiteness conjecture holds for any bounded family of matrices A , whose matrices, except perhaps one, have rank 1. There are also a number of works with less constructive sufficient conditions for attainability of the generalized spectral radius on a finite product of matrices, see, e.g., the references in [30].…”

Section: Introductionmentioning

confidence: 98%

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

Kozyakin

2015

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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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