Finiteness of the Cyclic Group Related to the Group Inverse of A Matrix and Finite Markov Chains - Zekraoui, Hanifa (Communications in Mathematics and Applications, 2013)
The Group inverse is one of the generalized inverses possessing the properties the closest to the normal inverse. The positive and negative powers of a given matrix $A$ (the latter being interpreted as powers of $A^{#}$, the group inverse of $A$), together with the projector $AA^{#}$ as the unit element, constitute an Abelian group. In this paper, we are going to give equivalent conditions so that this group is finite, and applying the result in finite Markov chains.