  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.
