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

The determinant of u has absolute value 1:

Confirm the decomposition:

HermiteDecomposition can be used to determine if two integer lattices are equivalent. If the generators of the lattices are put in row matrices, the lattices are equal if and only if the matrices of the Hermite decompositions of the corresponding matrices are the same. Consider the lattices generated by , , and , , :

As the matrices of and are equal, the two sets of generators create the same lattice:

Confirm the result using LatticeReduce, which gives the same reduction up to overall sign:

Solve a linear Diophantine equation :

Form a homogenized system:

Rows of are integer combinations of rows of :

The second row of gives a solution of :

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

Reduce uses HermiteDecomposition to solve linear Diophantine equations:

The Smith decomposition can often be found by two iterations of the Hermite decomposition:

The first use of HermiteDecomposition gives a unimodular matrix and an upper triangular matrix:

A second application, to , gives another unimodular matrix and a diagonal matrix:

Compare , and with the result given by SmithDecomposition:

HermiteDecomposition gives a unimodular matrix and an upper triangular matrix:

The matrices satisfy the equation u.m==r:

Det of a unimodular matrix has absolute value 1:

Inverse of a unimodular integer matrix is an integer matrix:

For an matrix , the matrix of HermiteDecomposition[a] has the same dimensions:

The matrix as a square matrix:

The matrix always has integer values, irrespective of whether the input is integer or rational:

This extends to the real and imaginary parts of the result if the input matrix is complex:

The matrix has integer values if the input matrix is integer:

If the input matrix has noninteger rational elements, so will the matrix:

For a real-valued input, the entries of the matrix are non-negative:

For complex-valued input, the imaginary parts of the entries are non-negative:

The real parts can be negative:

Both HermiteDecomposition and QRDecomposition give unimodular and triangular matrices:

For HermiteDecomposition the decomposition is and the matrices returned are integer:

For QRDecomposition the decomposition is and matrices returned can have any values:

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

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

LatticeReduce gives a basis consisting of shorter vectors:

The second matrix in the Hermite decomposition is the Hermite normal form given by HermiteReduce:

HermiteDecomposition works over the Euclidean ring of the integers:

The entries above each pivot are always smaller in absolute value than that pivot:

The determinant of u must be 1 or , as these are the only invertible integers:

In contrast, PolynomialHermiteDecomposition works over the Euclidean ring of univariate polynomials:

Now the entries above each pivot are always lower in degree than that pivot:

The determinant of u2 must be a nonzero number, as these are exactly the invertible polynomials:

Note that integers and polynomials are different as rings, and in particular one does not recover anything resembling the integer Hermite decomposition by making the polynomial variable zero: