  This paper is aimed to identify some new characterizations and
representations of the Minkowski inverse in Minkowski space. First of all, a
few representations of {1,3m}, {1,2,3m}, {1,4m} and {1,2,4m}-inverses are given
in order to represent the Minkowski inverse. Secondly, some famous
characterizations of the Moore-Penrose inverse are extended to that of the
Minkowski inverse. Thirdly, using the Hartwig-Spindelb\"ock 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.
