标题： 双超群逆的存在性与刻画 显示英文标题

标题： Existence and characterizations of hyper-dual group inverse

摘要： 受肖和钟最近工作的启发 [AIMS Math. 9 (2024), 35125--35150: MR4840882]，我们提出了超对偶矩阵的一种广义逆，称为超对偶群广义逆（HDGGI）。在某些充分必要条件下，我们证明了超对偶矩阵的HDGGI的存在性。然后我们证明了当其存在时，HDGGI是唯一的。接着我们将HDGGI用于求解线性超对偶系统。我们还探讨了HDGGI和HDMPGI的逆序律和顺序律成立的一些充分条件。此外，我们讨论了超对偶群逆的最小二乘性质。最后，利用 $n$阶对偶矩阵的定义，我们建立了 $n$阶对偶矩阵群逆存在的充要条件。 显示英文摘要

摘要： 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 $n$.