group inverse

Let $A$ be an $n\times n$ matrix over $ℝ$.A group inverse for $A$ is an $n\times n$ matrix$X$ such that

$AXA$$=A$(1)

$XAX$$=X$(2)

$AX$$=XA.$(3)

Such a matrix, when it exists, is unique and is denoted by $A^{\mathrm{\#}}$.A group inverse is a special case of a Drazin inverse.