Generalized inverse \(A^{(2)}_{T,S}\) and a rank equation (Q1883547)

The paper establishes the connection between the matrices \(A\in \mathbb{R} ^{m\times n},B\in \mathbb{R} ^{n\times n},C\in \mathbb{R} ^{m\times m}\) in order \(X=A_{T,S}^{(2)}\) (the outer inverse of \(A\) with prescribed range \(T\) and null space \(S\)) be the solution of the rank equation : rank\(\left( \begin{smallmatrix} A & B \\ C & X \end{smallmatrix} \right) \)= rank\((A) \). As special cases, the corresponding results for the Moore-Penrose inverse \(A^{\dag }\), the weighted Moore-Penrose inverse \(A_{M,N}^{+}\), the Drazin inverse \(A^{D}\) and the group inverse \(A^{\#}\) are also derived.