Compute the Smith decomposition of a 2×3 polynomial matrix:

Check the matrix equation:

Check that u and v are unimodular:

The matrix can have rational coefficients:

Check that the determinants of u and v are constant and verify the required matrix equation:

The matrix can have complex coefficients:

The matrix can have approximate coefficients:

Check that the determinant of v is constant and check the required matrix equation to reasonable precision:

The matrix can have symbolic coefficients:

Check the matrix equation:

Check that v is unimodular; in this case it must be a constant with respect to the polynomial variable x:

When a matrix has rational function entries, PolynomialSmithDecomposition will compute the Smith–McMillan form where {u,v} remain polynomial matrices with constant determinants and elements of s have a denominator:

The second element of the Smith decomposition is a matrix with diagonal elements that are scalar multiples of the Smith normal form diagonal elements given by PolynomialSmithReduce.

Compute the Smith reduction of a 2×2 polynomial matrix:

Compute the decomposition and extract the second part:

The last element in the diagonal of the Smith normal form of a characteristic matrix gives the minimal polynomial:

Compare to MatrixMinimalPolynomial:

The polynomial Smith decomposition can be used to compute the rational canonical form of a matrix; this is the second element in a Frobenius decomposition.

Compute the Smith form of a characteristic matrix and extract the diagonal:

Use this to create companion matrix blocks:

Use SparseArray and Band to reconstruct the companion matrix for m as a diagonal block matrix:

Check the result: