Built-in Mathematica Symbol

HermiteDecomposition

HermiteDecomposition[m]
gives the Hermite normal form decomposition of an integer matrix

.

The result is given in the form where is a unimodular matrix, is an upper-triangular matrix, and .

Decompose m into a unimodular matrix u and an upper-triangular matrix r:

The determinant of u is a unit:

Decompose m into a unimodular matrix u and an upper-triangular matrix r:

The determinant of u is a unit:

A singular matrix m:

The number of nonzero rows in r is equal to the rank of m:

Rectangular matrices m:

A rational matrix m:

u is a unimodular integer matrix; r is an upper-triangular rational matrix:

A Gaussian integer matrix m:

u is a unimodular Gaussian integer matrix; r is an upper-triangular Gaussian integer matrix:

A Gaussian rational matrix m:

u is a unimodular Gaussian integer matrix; r is an upper-triangular Gaussian rational matrix:

Solve a linear Diophantine equation A.x

b:

Form a homogenized system:

Rows of r are integer combinations of rows of m:

The second row of r gives a solution of A.x

b:

The last two rows of r give a basis of solutions to the homogenous equation A.x

0:

Reduce uses HermiteDecomposition to solve linear Diophantine equations:

HermiteDecomposition gives a unimodular matrix and an upper-triangular matrix:

The matrices satisfy the equation u.m

r:

Det of a unimodular matrix is a unit:

Inverse of a unimodular integer matrix is an integer matrix:

Elements of r form an upper-triangular basis of the lattice generated by the rows of m:

The inverse of u provides an explicit representation of rows of m in terms of the basis:

LatticeReduce gives a basis consisting of shorter vectors: