The forward order law for Moore-Penrose inverse of multiple matrix product

Abstract

The relationship between the forward order law product $A_1^{\dag}A_2^{\dag}\cdots A_n^{\dag}$ of the Moore-Penrose inverses of $A_1, A_2, \cdots A_n$ and the seven common types of generalized inverse of $A_1A_2\cdots A_n$ will be studied in this paper.
Especially, we will give the necessary and sufficient condition for the $n$ term forward order law
$$(A_1A_2\cdots A_n)^{\dag}=A_1^{\dag}A_2^{\dag}\cdots A_n^{\dag}.$$