Minkowski reduction of integral matrices

Abstract

In 1905 Hermann Minkowski introduced his theory of reduction of positive definite quadratic forms. Recently, Hans J. Zassenhaus has suggested that this theory can be applied to the problem of row reduction of matrices of integers. Computational investigations have shown that for matrices with more columns than rows, the number of steps required for reduction decreases drastically. In this paper it is proved that as the number of columns increases, the probability that a matrix is Minkowski reduced approaches one. This fact is the motivation behind the introduction of a modified version of Minkowski reduction, resulting in a reduction procedure more suitable for computation.