Generalized Sarymsakov Matrices

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

Research output: Contribution to journalArticlepeer-review


Within the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of Sarymsakov matrices is the largest known subset that is closed under matrix multiplication, and more critically whose compact subsets are all consensus sets. This paper shows that a larger subset with these two properties can be obtained 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, this 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. In addition, a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set.

Original languageEnglish (US)
Article number8513840
Pages (from-to)3085-3100
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number8
StatePublished - Aug 2019


  • Cooperative control
  • Sarymsakov matrices
  • doubly stochastic matrices
  • multi-agent systems
  • products of stochastic matrices

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

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