New formulae of the Drazin inverse of anti-triangular complex block matrices

Abstract

Let E, F ? Cnxn. If EFiE = 0 for all i ? N, we give the explicit representation of the Drazin inverse of the block complex matrix (E I F 0). We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear & Multilinear Algebra, 14(1983), 195-198].