Computation of matrix splittings and their applications (Q2770152)

An index splitting \(A= U-V\) of \(A\in\mathbb{C}^{n\times n}\) satisfies \(\text{Rg}(U)= \text{Rg}(A^k)\), \(\text{Ker}(U)= \text{Ker}(A^k)\), where \(k= \min\{p: \text{rank}(A^p)= \text{rank}(A^{p+1})\}\) is the index of \(A\). Formulae for constructing index splittings are presented and representations of the Drazin and Moore-Penrose inverses of \(A\) are given. Thus previous results of \textit{A. Berman} and \textit{M. Neumann} [SIAM J. Appl. Math. 31, 307--312 (1976; Zbl 0352.65017)], \textit{A. Berman} and \textit{R. J. Plemmons} [SIAM J. Numer. Anal. 11, 145--154 (1974; Zbl 0273.65029)], \textit{Y. Wei} [Appl. Math. Comput. 95, No. 2--3, 115--124 (1998; Zbl 0942.15003)], and \textit{G. Wang} and \textit{Y. Wei} [Numer. Math., J. Chin. Univ. 7, No. 1, 1--13 (1998; Zbl 0906.65041)] are generalized. An iterative scheme for the solution of the singular linear equation \(Ax= b\) is also considered.