A characterization of the Drazin inverse (Q5947467)

Given a square complex matrix \(A\) of order \(n\), the author shows that there exists a unique Drazin inverse \(Y\) of \(A\) under certain conditions involving the index of \(A\) and the rank of a composed matrix of order \(2n\) (first row: \(A\), \(AY\); second row: \(AY\), \(Y\)) that must be equal with \(\operatorname {rank}A\). On this basis an algorithm for computing the Drazin inverse is derived. Finally an illustrative example is given.