Extension and generalization properties of the weighted Minkowski inverse in a Minkowski space for an arbitrary matrix

The weighted Minkowski inverse AM,N⊕∈Mn,m ​related to the positive definite matrices M∈Mm and N∈Mn of an arbitrary matrix A∈Mm,n (including singular and rectangular) is one of the important generalized inverses for solving matrix equations in Minkowski space μ. In this paper, the results are introduced in the following three ways. First, we establish some new and attractive properties of the weighted Minkowski inverse AM,N⊕ in a Minkowski space μ. Second, new representations and conditions for the continuity of the weighted Minkowski inverse AM,N⊕ in a Minkowski space μ are discussed. Finally, some illustrated counterexamples are also studied to show that some well-known properties of the weighted Moore-Penrose inverse AM,N+ in a Hilbert space H are not valid in the Minkowski inverse AM,N⊕ in a Minkowski space μ.