Somewhat stochastic matrices, S, are real square matrices whose individual row sum is 1 for each row. We define and develop an eigenvalue characterization of when Sk converges to a steady state distribution π, as k converges to ∞. This result generalizes previously known linear algebraic properties about n × n stochastic matrices to n × n somewhat stochastic matrices. As an application, an alternative method to calculate infinite-time gambler’s ruin on a finite state space using algebraic duality is described. Examples are presented. When the eigenvalues of certain classes of matrices are known, then the k or less steps gambler’s ruin probabilities can often be determined in terms of these eigenvalues using Siegmund duality. A corresponding approach to determine the absorbing probabilities in an infinite state space setting is still being explored.