We propose recursive algorithms to update an orthogonal numerical basis matrix of the null space of the block row, (banded) block Toeplitz, and block Macaulay matrix, which is the multivariate generalization of the (banded) block Toeplitz matrix. These structured matrices are often constructed in an iterative way, and, for some applications, a basis matrix of the null space is required in every iteration. Consequently, recursively updating a numerical basis matrix of the null space, while exploiting the inherent structure of the matrices involved, induces large savings in the computation time. Moreover, we also develop a sparse adaptation of one of the recursive algorithms that avoids the explicit construction of the block Macaulay matrix and results in a considerable reduction of the required memory. We provide several numerical experiments to illustrate the proposed algorithms: for example, we solve four multiparameter eigenvalue problems via the null space of the block Macaulay matrix and notice that the recursive and sparse approach are, on average, 450 and 1300 times faster than the standard approach, respectively.