A Representation of the Drazin Inverse and Characterizations of the Index

A representation for the Drazin inverse of an arbitrary square matrix in terms of the eigenprojection is established in this paper. The Laurent expansion of the resolvent of our matrix has coefficients (for the nonnegative indices) which are powers of the Drazin inverse. Using this expansion we immediately get some limit theorems concerning the index of the given matrix. The results hold for matrices over a topological Hausdorf field.