Bricklaying and the Hermite Normal Form

Bricklaying and the 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...

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.

A Linear Space Algorithm for Computing the Hermite Normal Form

Computing the Hermite Normal Form of an n n integer matrix using the best urrent algorithms typi ally requires O(n3 logM) spa e, whereM is a bound on the entries of the input matrix. Although polynomial in the input size (whi h is O(n2 logM)), this spa e blow-up an easily be ome a serious issue in pra ti e when working on big integer matri es. In this paper we present a new algorithm for omputi...

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 ...