The problem of principal subspace tracking under a sparsity constraint on the weight matrix is considered. The sparsity constraint is added to resolve the interpretability problem encountered after the estimation of the subspace weight matrix in the context of data analysis. This is also important in blind system identification context when the unknown mixing matrix has a certain sparse structure. Most of the literature methods suffer from a trade-off between the subspace performance and the targeted sparsity level. Therefore, a two-step approach is proposed, where the first one uses the Fast Approximated Power Iteration subspace tracking algorithm FAPI for the adaptive extraction of an orthonormal basis of the principal subspace. Then, an estimation of the desired sparse weight matrix is done in the second step using different optimization techniques. The four resulting algorithms give the user the necessary flexibility concerning the consideration of the orthogonality constraint and the target trade-off between sparsity and computational cost. Under some mild conditions, a theoretical convergence analysis shows that the proposed approach allows us to recover the sparse ground truth mixing matrix. Compared to the state-of-art solutions, our algorithms have low computational complexity and they achieve both good convergence and estimation performance.