An alternative limit expression of Drazin inverse and its application (Q1322895)

Let \(A\) be an \(n\times n\) matrix and \(\text{ind}(A)\) the index of \(A\), the smallest nonnegative integer for which \(\text{rank}(A^ k)=\text{rank}(A^{k+1})\). The author presents a new limit expression for the Drazin inverse (1) \(A^ D=\lim_{\lambda\to 0} (\lambda+A)^{-(\ell+1)} A^ \ell\), where \(\ell\geq k=\text{ind}(A)\). It is assumed that \(- \lambda\not\in\sigma(A)\), the set of all eigenvalues of \(A\). The limit expression (1) provides a new proof of a finite algorithm for the Drazin inverse given by \textit{T. N. E. Greville} [Linear Algebra Appl. 6, 205-208 (1973; Zbl 0247.15004)].