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

TL;DR

This paper demonstrates that the finiteness conjecture for spectral radii of certain non-negative matrix sets can be derived using the hourglass alternative, introducing a new class of matrices with the finiteness property.

Contribution

The paper introduces an axiomatization of the hourglass alternative, defining a new class of positive matrix sets with the finiteness property, expanding the understanding of spectral radius behavior.

Findings

Finiteness conjecture holds for these matrix sets.

A new class of matrices with the finiteness property is defined.

Constructs a semiring of matrices with Minkowski operations.

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.