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*!

Sponsored ads

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 $.

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)

A normal Hermite-shaped matrix is the triangular scaled matrix $ H $ calculated by the Hermite decomposition (above)

dCode retains ownership of the source code of the script Hermite Normal Form Matrix online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Hermite Normal Form Matrix script for offline use on PC, iPhone or Android, ask for price quote on contact page !

hermite,matrix,reduction,decomposition,normal,form,triangular

Source : https://www.dcode.fr/matrix-hermite

© 2019 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode

Feedback

▲