Formulas for the Drazin Inverse of Matrices over Skew Fields

Abstract

Let $\mathbb{K}$$^{m \times n}$ be the set of all the $m\times n$ matrices over the skew field. For the matrices $P,~Q \in $$\mathbb{K}$$^{n \times n}$, the explicit formulas for the Drazin inverse of $P + Q$ are given when $PQ^2 = 0$, $P^2QP = 0$, $(QP)^2 = 0$ and $P^2 QP = 0$, $P^3 Q = 0$, $Q^2 = 0$, respectively. Using these formulas, the representations for the Drazin inverse of the block matrices
$
\left( {\begin{array}{*{20}c}
A & B \\
C & D \\

\end{array} } \right)\in\mathbb{K}^{n \times n}$ are showed with some conditions, where $A$ and $D$ are square.