Summary: We study the Lie structure of the set of infinite matrices associated with bounded operators on \(\ell_\infty\) with the property that their row sums are constant. That includes, in particular, infinite row stochastic and zero-row-sum matrices. We also consider the compactness of these operators as related to the Krein-Rutman theorem, we discuss their Abel limits and we consider their connection to the convergence of Markov chains as well as to sequence transformations and generalized limits.