Weak MPD and DMP inverses (Q6612225)

For \(A \in \mathbb{C}^{n \times n}\), it is well known that the Moore-Penrose inverse (denoted by \(A^{\dag}\)) is the unique matrix \(X \in \mathbb{C}^{n \times n}\) which satisfies the equations \(AXA=A, XAX=X, (AX)^*=AX\) and \((XA)^*=XA.\) The Drazin inverse (denoted by \(A^D\)) is the unique matrix \(X\) which solves the equations \(XAX=X, AX=XA\) and \(A^{k+1}X=A^k,\) where \(k\) is the index of \(A\). The DMP-inverse of \(A\) is defined by \(A^{D,{\dag}}:=A^DAA^{\dag}\), whereas the MPD (or the dual DMP) inverse of \(A\) is defined by \(A^{{\dag},D}:=A^{\dag}AA^{D}\). Among other things, the author investigates weaker versions of these generalized inverses and prove many of their properties. Potential applications to linear systems are included.