Compute the Hermite reduction of a 2x2 matrix:

Compute the Hermite reduction of a 2x3 matrix of low-degree polynomials:

The matrix can have rational coefficients:

The matrix can have complex coefficients:

The matrix can have approximate coefficients:

The matrix can have symbolic coefficients:

Create an underdetermined system using a 6×8 matrix of random quartic polynomials:

A function to find polynomial solutions to linear systems with polynomial coefficients:

Solve this system:

Check the solution:

One can get a solution faster using LinearSolve:

But several components are rational functions rather than polynomials:

One can use PolynomialHermiteReduce to compute left-extended GCDs of two polynomial matrices.

Construct a pair of matrices with a known nontrivial left common divisor:

Join these transposed and compute the Hermite reduction:

The GCD matrix is obtained from h:

Create a solver over the univariate polynomial ring:

Use it to recover the GCD cofactors:

Check these:

While a matrix GCD is not unique, any one is related to another by a unimodular transformation.

Construct the unimodular matrix transforming gcd to the common multiplier m3:

Check that is unimodular and transforms gcd to m3:

Compute the Hermite normal form of a matrix:

The Hermite normal form can also be obtained from the second matrix returned by PolynomialHermiteDecomposition:

The Hermite form of a matrix is unique only up to constant multiples of the rows:

Only consider second element of second rows to avoid a division by zero:

Create a pair of polynomials with a known common divisor (which is the GCD with high probability):

Compute the GCD of this pair:

Compute the Hermite reduction:

The first row of h will recover the GCD up to a unit factor:

PolynomialHermiteReduce works over the Euclidean ring of univariate polynomials:

The entries above any pivot are always lower in degree than that pivot:

In contrast, HermiteReduce works over the Euclidean ring of the integers:

Now the entries above any pivot are always smaller in absolute value than that pivot:

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: