Further characterizations and representations of the Minkowski inverse in Minkowski space

Abstract

<abstract><p>This paper serves to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of $ \{1, 3^{\mathfrak{m}}\} $-, $ \{1, 2, 3^{\mathfrak{m}}\} $-, $ \{1, 4^{\mathfrak{m}}\} $- and $ \{1, 2, 4^{\mathfrak{m}}\} $-inverses are given in order to represent the Minkowski inverse. Second, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Third, using the Hartwig-Spindelböck decomposition, we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.</p></abstract>