On some matrix equalities for generalized inverses with applications (Q1017632)

A matrix \(X\) is called a \(g\)-inverse of \(A\), denoted by \(X = A^-\), if it satisfies \(AXA = A\). The collection of all possible \(g\)-inverses of \(A\) is denoted by \(\{A^-\}\). The authors obtain some necessary and sufficient conditions for the matrix equality \(A^- = PN^-Q\) to hold, where \(A \in \mathbb{C}^{m\times n}\), \(N \in \mathbb{C}^{k\times l}\), \(P \in \mathbb{C}^{n\times l}, Q \in \mathbb{C}^{k\times m}\), see also \textit{J. Gross} and \textit{Y. Tian} [Linear Algebra Appl. 417, No. 1, 94--107 (2006; Zbl 1105.15005)]. In particular, they give identifying conditions for the set equalities \(\{C^-\}=\{A^- + B^-\}\) and \(\{A^-\} =\{A_1^- + \cdots+ A_k^-\}\) to hold.