Algebraic and Spectral Decomposition in the Rectangular Setting: Towards a Drazin-Type Inverse

This construction follows the fundamental philosophy of the Drazin inverse—invert the regular part and annihilate the singular part—while respecting the fact that a rectangular matrix maps between two different vector spaces. The paper has explicitly distinguished these structural re-sults from the established rectangular Drazin theory initiated by Cline and Greville and subsequently extended through weighted Drazin, weighted G-Drazin, DMP, core-EP, weak-group, and minimal-rank weighted weak Drazin constructions. The framework provides a basis for further investigation of rectangular Drazin-type inverses and their algebraic and spectral decompositions.

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