ROLE OF GROUP GENERALIZED INVERSE IN THEORY OF FINITE MARKOV CHAINS

For an m-state homogeneous Markov chain whose one-step transition matrix is T, the group inverse of the matrix A equals I minus T is shown to play a central role. For an ergodic chain, it is demonstrated that virtually everything that one would want to know about the chain can be determined by computing the group inverse. Furthermore, it is shown that the introduction of the group inverse into the theory of ergodic chains provides not only a theoretical advantage, but it also provides a definite computational advantage that is not realized in the traditional framework of the theory.