In this paper, we design a deterministic 1/3-approximation algorithm for the problem of maximizing non-monotone $k$-submodular function under a matroid constraint. In order to reduce the complexity of this algorithm, we also present a randomized 1/3-approximation algorithm with the probability of $1-\epsilon$, where $\epsilon$ is the probability of algorithm failure. Moreover, we design a streaming algorithm for both monotone and non-monotone objective $k$-submodular functions.