HermiteDecomposition

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 :

Form a homogenized system:

Rows of r are integer combinations of rows of m:

The second row of r gives a solution of :

The last two rows of r give a basis of solutions to the homogeneous equation :

Reduce uses HermiteDecomposition to solve linear Diophantine equations:

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

The matrices satisfy the equation :

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: