In　this　work,　we　investigate　the　problem　that　maximizes　a　weakly　k-submodular　function　under　the　matroid　constraint.　Different　from　traditional　submodular　function　maximization,　there　are　k　disjoint　subsets　in　k-submodular　function　optimization,　instead　of　a　single　set　in　the　submodular　maximization.　For　the　weakly　k-submodular　maximization　problem,　we　provide　a　greedy　algorithm　whose　approximation　ratio　is　alpha/(1　+　alpha),　where　parameter　0　＜　alpha　＜=　1　is　the　orthant　submodularity　ratio.　Then　we　extend　to　cardinality　constraint　which　maintains　the　same　performance　ratio.