In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) \(Ax-B|x|=b\) with \(A, B\in \mathbb {R}^{n\times n}\) from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system \(Ax=b\) with \(A\in \mathbb {R}^{n\times n}\). Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.

中文翻译：

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关于广义绝对值方程的唯一解

在本文中，为广义绝对值方程的唯一解决方案的一些有用的充分必要条件（GAVE）\（AX-B | X | = B \）与在\（A，B \ \ mathbb {R} ^ {首先给出来自优化字段的n次\ n} \），其中涵盖了线性系统\（Ax = b \）与\（A \ in \ mathbb {R} ^ {n \次n} \）。不仅如此，还为GAVE的唯一解获得了一些新的充分条件，这些条件比以前发表的著作要弱。