On the unique solution of the generalized absolute value equation

In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) Ax-B|x|=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax-B|x|=b$$\end{document} with A,B∈Rn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A, B\in \mathbb {R}^{n\times n}$$\end{document} from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system Ax=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax=b$$\end{document} with A∈Rn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\in \mathbb {R}^{n\times n}$$\end{document}. Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.