The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains

For an m-state homogeneous Markov chain whose one-step transition matrix is T, the group inverse, A^#, of the matrix A = I - 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 A^#. Furthermore, it is shown that the introduction of A^# 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.