Full-rank and determinantal representation of the Drazin inverse

AbstractIn 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. Using this general representation, we introduce a determinantal representation of the Drazin inverse. More precisely, we represent elements of the Drazin inverse AD as a fraction 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 the Drazin inverse AD, powers of the Drazin inverse and the minors of the matrix Ak, k=ind(A), are explicitly derived