Full-rank and Determinantal Representation of the Drazin Inverse

In this article we introduce a full-rank representation of the Drazin inverse AD of a given complex matrix A, which is based on an arbitrary full-rank decomposition of Al, l ≥ k, where k is the index of A. We show that the known representation of the Drazin inverse of A, devised in [7], represents a partial case of this result. Using this general representation, we introduce a determinantal representation of the Drazin inverse. More precisely, we represent elements of the Drazin inverse AD as ratios of two expressions involving minors of the order rank(Ak), k = ind(A), taken from the matrices A and rank invariant powers Al, l ≥ k. Also, we examine conditions for the existence of the Drazin inverse for matrices whose elements are taken from an integral domain. Finally, a few correlations between the minors of powers of the Drazin inverse AD and the minors of the matrix Ak are explicitly derived.