Generalized Sarymsakov Matrices

Weiguo Xia, Ji Liu, Ming Cao, Karl Henrik Johansson, Tamer Basar

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Within the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of Sarymsakov matrices is the largest known subset (i) that is closed under matrix multiplication, and more critically (ii) whose compact subsets are all consensus sets. This paper shows that a larger subset with these two properties can beoobtained by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved by introducing the notion of the SIA index of a stochastic matrix, whose value is 1 for Sarymsakov matrices, and then exploring
those stochastic matrices with larger SIA indices. In addition to constructing the larger set, the paper introduces another class of generalized Sarymsakov matrices, which contains matrices that are not SIA, and studies their products. Sufficient conditions are provided for an infinite product of matrices from this class, converging to a rank-one matrix. Finally, as an application of the results just described and to confirm their usefulness, a necessary and sufficient combinatorial condition, the “avoiding set condition”, for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited.
Original languageEnglish
Pages (from-to)3085-3100
Number of pages16
JournalIEEE-Transactions on Automatic Control
Issue number8
Early online date1-Aug-2019
Publication statusPublished - Aug-2019

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