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

Abstract

Recently Blondel, Nesterov and Protasov proved 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 earlier used 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.

17 pages, 38 bibliography references, minor corrections, accepted for publication in Linear Algebra and its Applications

References in corpus (4)

• Continuity properties of the lower spectral radius
• Rank-one Characterization of Joint Spectral Radius of Finite Matrix Family
• Entropy Games and Matrix Multiplication Games
• Rank one matrices do not contribute to the failure of the finiteness property

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• Log-sum-exp neural networks and posynomial models for convex and log-log-convex data
• The greedy strategy in optimizing the Perron eigenvalue
• Constructive stability and stabilizability of positive linear discrete-time switching systems
• The finiteness conjecture holds in SL(2,Z>=0)^2
• Minimax theorem for the spectral radius of the product of non-negative matrices