On random nonsingular Hermite Normal Form

Hermite Normal Form

The Hermite Normal Form is a canonical matrix analogue of Reduced Echelon Form, but involving matrices over more general rings. In this work we formalise an algorithm to compute the Hermite Normal Form of a matrix by means of elementary row operations, taking advantage of the Echelon Form AFP entry. We have proven the correctness of such an algorithm and refined it to immutable arrays. Furtherm...

Bricklaying and the Hermite Normal Form

A Note to Hermite and Smith Normal Form Computation

Hermite normal form and Smith normal form are canonic forms of integral matrices with respect to congruence produced by multiplication with right unimodular matrix resp. by multiplication with left and right unimodular matrix simultaneously. This lower-diagonal resp. diagonal form has many applications in various elds. Unfortunately, the computation of them is extremely resource consuming. The ...

The Dual Lattice , Integer Linear Systems and Hermite Normal Form

In the first Lecture, we saw that lattices can be viewed equivalently in two different ways, i.e. as discrete additive subgroup of R n or as an additive subgroup of R n with linearly independent generators. In this section, we show a final equivalent viewpoint which relates the discreteness of a lattice to the existence of an appropriate dual.

Hermite and Smith normal form algorithms over Dedekind domains

We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields. The goal of this paper is to explain how to generalize to a Dedekind domain R many...