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 $.
Hermite Normal Form Matrix - dCode
Tag(s) : Matrix
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Hermite Normal Form Matrix
Hermite Form Calculator
Answers to Questions (FAQ)
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.
Source code
dCode retains ownership of the "Hermite Normal Form Matrix" source code. Except explicit open source licence (indicated Creative Commons / free), the "Hermite Normal Form Matrix" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Hermite Normal Form Matrix" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Hermite Normal Form Matrix" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
The copy-paste of the page "Hermite Normal Form Matrix" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Hermite Normal Form Matrix on dCode.fr [online website], retrieved on 2024-11-07,
