Existence and characterizations of hyper-dual group inverse

TL;DR

This work introduces the hyper-dual group generalized inverse (HDGGI) for hyper-dual matrices and establishes precise existence and uniqueness criteria, notably via the condition $(I-\widehat{A}\widehat{A}^{\#})\widehat{A}_{0}(I-\widehat{A}\widehat{A}^{\#})=0$ and a compact formula for $\tilde{A}^{\#}$. It then develops the HDGGI-based solution framework for linear hyper-dual systems, including solvability conditions and the general solution, and analyzes least-squares properties. The authors also derive sufficient conditions under which reverse and forward order laws hold for HDGGI and its Moore–Penrose counterpart, and extend the dual-matrix theory to $n$-order dual matrices, providing existence criteria and a closed-form expression for the group inverse. Together, these results broaden the theory and computation of generalized inverses in multi-layer dual-number algebras, with practical impact on hyper-dual linear problems and related applications.

Abstract

Motivated by the recent work of Xiao and Zhong [AIMS Math. 9 (2024), 35125--35150: MR4840882], we propose a generalized inverse for a hyper-dual matrix called hyper-dual group generalized inverse (HDGGI). Under certain necessary and sufficient conditions, we establish the existence of the HDGGI of a hyper-dual matrix. We then show that the HDGGI is unique (whenever exists). The HDGGI is then used to solve a linear hyper-dual system. We also exploit some sufficient conditions under which the reverse and forward-order laws for a particular form of the HDGGI and HDMPGI hold. We also discuss the least-squares properties of hyper-dual group inverse. Using the definition of dual matrix of order $n$, we finally establish necessary and sufficient condition for the existence of the group inverse of a dual matrix of order $n$.

Theorems & Definitions (31)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3