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 $A, Bin mathbb{R}^{ntimes n}$ from the optimization field are first presented, which cover the f
undamental theorem for the unique solution of the linear system $Ax=b$ with $Ain mathbb{R}^{ntimes n}$. Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.
