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In this paper, some useful necessary and sufficient conditions for the uniquesolution 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 linearsystem $Ax=b$ with $A\in \mathbb{R}^{n\times n}$. Not only that, some newsufficient conditions for the unique solution of the GAVE are obtained, whichare weaker than the previous published works.

On the unique solution of the generalized absolute value equation