Consensus Sets Based on Sarymsakov Matrices

In the study of convergence property of infinite products of stochastic matrices, a pivotal concern revolves around the characterization of a subset from the family of stochastic, indecomposable and aperiodic (SIA) matrices such that the subset is closed under matrix multiplication. Historically, the collection of Sarymsakov matrices stood as the most expansive subset known to possess such a closure property. Consequently, an infinite product involving Sarymsakov matrices guarantees convergence toward a rank-one matrix. In recent times, a subset larger than the Sarymsakov matrix collection has been proposed, demonstrating an equivalent closure property. In this exposition, we establish that an even more extensive collection possessing the aforementioned property can be devised utilizing a similar yet generalized approach. During the formulation of our subset, we also address the unresolved quandary of verifying a scrambling matrix by scrutinizing the powers of an SIA matrix. We offer a solution to this quandary by providing a sharp upper bound for the powers required for verification. This particular outcome streamlines the construction process of our extended set. Numerical examples are provided to illustrate our work.