Complementation in Stochastic Matrices and the GTH Algorithm

Abstract

The Grassmann, Taksar, and Heyman (GTH) algorithm for the computation of the stationary distribution of a finite stochastic matrix is shown to apply for the general case when there is a unique stationary distribution. The approach is elementary and matrix based, with probabilistic arguments avoided, to give insight into the essential structural properties. A byproduct is a necessary and sufficient determinantal condition for regularity of a stochastic matrix.