Products of generalized stochastic Sarymsakov matrices

Weiguo Xia, Ji Liu, Ming Cao, Karl H. Johansson, Tamer Basar

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of stochastic Sarymsakov matrices is the largest known subset (i) that is closed under matrix multiplication and (ii) the infinitely long left-product of the elements from a compact subset converges to a rank-one matrix. In this paper, we show that a larger subset with these two properties can be derived by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved either by introducing an SIA index, whose value is one for Sarymsakov matrices, and then looking at those stochastic matrices with larger SIA indices, or by considering matrices that are not even SIA. Besides constructing a larger set, we give sufficient conditions for generalized Sarymsakov matrices so that their products converge to rank-one matrices. The new insight gained through studying generalized Sarymsakov matrices and their products has led to a new understanding of the existing results on consensus algorithms and will be helpful for the design of network coordination algorithms.

Original languageEnglish (US)
Title of host publication54rd IEEE Conference on Decision and Control,CDC 2015
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781479978861
StatePublished - Feb 8 2015
Event54th IEEE Conference on Decision and Control, CDC 2015 - Osaka, Japan
Duration: Dec 15 2015Dec 18 2015

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume54rd IEEE Conference on Decision and Control,CDC 2015
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Other54th IEEE Conference on Decision and Control, CDC 2015

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization


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