Extended integer rank reduction formulas and Smith normal form

We present an integer rank reduction formula for transforming the rows and columns of an integer matrix A. By repeatedly applying the formula to reduce rank, an extended integer rank reducing process is derived. The process provides a general finite iterative approach for constructing factorizations of A and A T under a common framework of a general decomposition V T AP = Ω. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process. Both the integer biconjugation process associated with the Wedderburn rank reduction process and the scaled extended integer Abaffy–Broyden–Spedicato (ABS) class of algorithms are shown to be in the integer rank reducing process. We also show that the integer biconjugation process can be derived from the scaled integer ABS class of algorithms applied to A or A T . Finally, we show that the integer biconjuagation process is a special case of our proposed ABS class of algorithms for computing the Smith normal form.