M. Khovanov and L. Rozansky constructed a homology for a link diagram whose Euler characteristic is the quantum link invariant associated to the quantum group $\mathrm{Uq}({\mathrm{sl}}_n)$ and its vector representation $V_n$ by using matrix factorizations. In my thesis, I study a generalization of the Khovanov-Rozansky homology for the quantum link invariant associated to $\mathrm{Uq}({\mathrm{sl}}_n)$ and its fundamental representations $\wedge V_n$. In this talk, I will define a new link invariant derived from the generalization of Khovanov-Rozansky homology.