Dual Core-EP Generalized Inverse and Decomposition
Bibekananda Sitha, Jajati Keshari Sahoo, Nestor Thome
TL;DR
This work extends generalized inverse theory to dual-number matrices by introducing the dual core-EP generalized inverse (DCEPGI) and a corresponding dual core-EP decomposition. Building on dual Drazin and dual Moore-Penrose inverses, it establishes existence and uniqueness for square dual matrices and derives a compact computation formula, $A^D A^m (A^m)^dagger$, along with a decomposition into a core and nilpotent part. Key results include characterization theorems, explicit computation relations, and connections to other dual inverses, with an application to solving inconsistent dual linear systems. The study advances dual matrix theory and opens avenues for perturbation analysis and iterative methods in dual settings.
Abstract
In this work, we introduce a new type of generalized inverse called dual core-EP generalized inverse (in short DCEPGI) for dual square matrices. We analyze the existence and uniqueness of the DCEPGI inverse and its compact formula using dual Drazin and dual MP inverse. Moreover, some characterizations using core-EP decomposition are obtained. We present a new dual matrix decomposition named the dual core-EP decomposition for square dual matrices. In addition, some relationships with other dual generalized inverses are established. As an application, solutions to some inconsistent system of linear dual equations are derived.