# hermite: Hermite Normal Form In numbers: Number-Theoretic Functions

## Description

Hermite normal form over integers (in column-reduced form).

## Usage

 1 hermiteNF(A)

## Arguments

 A integer matrix.

## Details

An mxn-matrix of rank r with integer entries is said to be in Hermite normal form if:

(i) the first r columns are nonzero, the other columns are all zero;
(ii) The first r diagonal elements are nonzero and d[i-1] divides d[i] for i = 2,...,r .
(iii) All entries to the left of nonzero diagonal elements are non-negative
and strictly less than the corresponding diagonal entry.

The lower-triangular Hermite normal form of A is obtained by the following three types of column operations:

(i) exchange two columns
(ii) multiply a column by -1
(iii) Add an integral multiple of a column to another column

U is the unitary matrix such that AU = H, generated by these operations.

## Value

List with two matrices, the Hermite normal form H and the unitary matrix U.

## Note

Another normal form often used in this context is the Smith normal form.

## References

Cohen, H. (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, Vol. 138, Springer-Verlag, Berlin, New York.