On generalized Schur complement of matrices and its applications to real and integer matrix factorizations

We provide a general finite iterative approach for constructing factorizations of a matrix A under a common framework of a general decomposition A = BC based on the generalized Schur complement. The approach applies a zeroing process using two index sets. Different choices of the index sets lead to different real and integer matrix factorizations. We also provide the conditions under which this approach is well-defined.