From projectors to 1MP and MP1 generalized inverses and their induced partial orders (Q2050219)

Let \(A\) be an \(m \times n\) complex matrix. For each \(A^{ - } \in \mathcal{A} \{1 \}\), the matrix \[A^{ - \, \dag }:= A^{ - } A A^{ \dag } \in \mathbb{C}^{n \times m}\] is called a 1MP-inverse of \(A\). For each \(A^{ - } \in \mathcal{A} \{1 \}\), the MP1-inverse of \(A\), denoted by \(A^{ \dag \, - }\), is the \(n \times m\) matrix \[A^{ \dag \, - }:= A^{ \dag } A A^{ - }.\] This paper is devoted to the study of the above generalized inverses of rectangular complex matrices, 1MP and MP1-inverses. The 1MP-inverses of a matrix \(A\) are characterized from a singular value decomposition of \(A\). The authors give some characterizations of 1MP-inverses of a matrix \(A\) as \(\{1, 2, 3 \}\)-inverses of \(A\), as the solutions of a matrix equation system, and as a 1-parametrized set. As an application, they analyze the restriction of 1MP-inverses to the set of partial isometries. They also introduce and study a partial order associated to 1MP-inverses as other application of 1MP-inverses. Indeed, the binary relations induced by these new generalized inverses are proved to be partial orders. The dual case (called MP1-inverse) is studied too.