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

Abstract

Let $E,F\in {\Bbb C}^{n\times n}$ have Drazin inverses. If $EF^iE=0$ for all $i\in {\Bbb N}$, we give the explicit representation of the
Drazin inverse of the block complex matrix $\left(
\begin{array}{cc}
E&I\\
F&0
\end{array}
\right)$. 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].