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 online "Hermite Normal Form Matrix" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Hermite Normal Form Matrix" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, 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, copy-paste, or API access for "Hermite Normal Form Matrix" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.
Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
Questions / Comments
Thanks to your feedback and relevant comments, dCode has developed the best 'Hermite Normal Form Matrix' tool, so feel free to write! Thank you!