Generalized elimination method and its applications to matrix factorizations

An elimination method is a sequence of elementary row or column operations to divide a matrix into parts with partial zeroing of columns or rows of the matrix leading to a matrix factorization implicitly. We present a generalized elimination algorithm for solving real and Diophantine linear systems. The algorithm uses a zeroing process based on two index sets and provides a mechanism for constructing factorizations such as block LU factorization as well as the WZ and the ZW factorizations under a common framework. We also introduce the octant interlocking factorization method and present two new factorizations named as the ODS and the SDO factorizations. Necessary and sufficient conditions under which the generalized elimination method is carried out are proposed.