The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains
Carl Dean Meyer · SIAM Review · 1975
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.