Weak MPD and DMP inverses

Abstract

Based on a minimal rank weak Drazin inverse and minimal rank right weak Drazin inverse, our aim is to solve weakened systems of matrix equations than already existing systems for defining the MPD and DMP inverses. These new systems have unique solutions called the weak MPD and DMP inverses, which present new types of generalized inverses involving MPD, DMP, GDC, GC, MPWG and MP-m-WG inverses. Many characterizations and expressions for weak MPD and DMP inverses are established. Consequently, we obtain several new and some known properties of MPD, DMP, GDC, GC, MPWG and MP-m-WG inverses. Furthermore, the weak MPD and DMP inverses are applied in solving linear equations. As a consequence, we get a fundamental result related to application of the Moore–Penrose inverse in solving linear equation.