The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4 γ^{2}}{\sqrt{γr} + 1}$, where $γ$ is the submodularity ratio of the function and $r$ is the rank of the matroid. Then, we show that the same greedy algorithm offers a constant approximation factor of $(1 + 1/(1-α))^{-1}$, where $α$ is the generalized curvature of the function. In addition, we demonstrate that these approximation guarantees are applicable to several real-world applications in which the submodularity ratio and the generalized curvature can be bounded. Finally, we show that our greedy algorithm does achieve a competitive performance in practice using a variety of experiments on synthetic and real-world data.