Further results for the dual Hartwig-Spindelb{ö}ck decomposition and its applications

In this paper, we introduce two new forms of the dual Hartwig-Spindelb{ö}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further explore and characterize the relationships and properties of these inverses, investigate the dual composite generalized inverses, and verify the applicability of dual partial orders. The proposed decomposition provides a systematic and convenient framework for the study of dual matrices.