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Hermite Normal Form Matrix

Tool for calculating the Hermite normal form (by reducing a matrix to its row echelon form) from a matrix M (with coefficients in Z) the computation yields 2 matrices H and U such that $H = U . M$.

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Hermite Normal Form Matrix -

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# Hermite Normal Form Matrix

## Hermite Form Calculator

### How to calculated the Hermite decomposition?

A matrix $M$ of size $n \times m$ with integer coefficients (natural or relative) has a Hermite decomposition if there exists a triangular matrix $H$ and a unimodular matrix $U$ such that $H = U. M$. Reminder: An upper triangular matrix $H$ is such that $H_ {i, j} = 0$ for $i> j$ and a unimodular matrix is an invertible square matrix with integer coefficients whose determinant is $\pm 1$.

Example: $$M = \begin{bmatrix} 3 & 2 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix} \Rightarrow H = \begin{bmatrix} 0 & -1 & 1 \\ 0 & 1 & 0 \\ -1 & -1 & 3 \end{bmatrix}, U = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$$

There are two forms for the Hermite matrix, an upper triangular matrix such that $H = UM$ (also called Hermite's normal form row style) is a lower triangular matrix such that $H = MU$ ( also called Hermite normal form column style)

dCode uses the LLL algorithm (Lenstra-Lenstra-Lovász) to calculate the Hermite decomposition (the calculation by hand is not recommended)

### What is the Hermite Normal Form?

A normal Hermite-shaped matrix is the triangular scaled matrix $H$ calculated by the Hermite decomposition (above) via reduction to row echelon form of the matrix.

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