Existence and characterizations of hyper-dual group inverse

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 $n$.