The MinimalBasis(A,x) and MinimalBasis[right](A,x) commands compute a minimal basis for the right nullspace of an m x n rectangular matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q, that is, univariate polynomials in x with coefficients in Q(a1,...,an).

The MinimalBasis[left](A,x) command computes a minimal basis for the left nullspace.

The computed minimal basis is returned as a matrix of polynomials.  A minimal basis for the right nullspace is specified by the columns of the matrix, whereas a minimal basis for the left nullspace is specified by the rows of the matrix. If the nullspace is trivial then the result returned is NULL.

The right minimal indices of A are specified by the column degrees of the returned matrix.  The left minimal indices of A are specified by the row degrees of the returned matrix.

$\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right)\:$

$A≔⟨⟨z^5-z^2-1\,z^3-2z^2+2z-2\,z+1⟩|⟨z^3-2z^2-1\,z^3-3z^2+3z-4\,2-z^3⟩⟩$

$A≔\left[\begin{array}{cc}{z}^5-z^2-1& z^3-2z^2-1\\ z^3-2z^2+2z-2& z^3-3z^2+3z-4\\ z+1& -z^3+2\end{array}\right]$

$B≔\mathrm{MinimalBasis}\left[\mathrm{left}\right]\left(A\,z\right)$

$B≔\left[\begin{array}{ccc}{z}^6-2z^5+3z^4-6z^3+4z^2-5z& -z^8+3z^5-z^4+2z^3+z-1& -z^8+3z^7-2z^6+z^5+3z^4-3z^3-z^2+z-2\end{array}\right]$

$\mathrm{map}\left(\mathrm{expand}\,B·A\right)$

$\left[\begin{array}{cc}0& 0\end{array}\right]$

$B≔\mathrm{MinimalBasis}\left[\mathrm{right}\right]\left(A\,z\right)$

Beckermann, B. and Labahn, G. "Fraction-free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications, Vol. 22 No. 1, (2000): 114-144.