The ColumnReducedForm(A,x) command computes a column-reduced form 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 RowReducedForm(A,x) command computes a row-reduced form over such domains.

A column-reduced form is one in which the column leading coefficient matrix has the same column rank as the rank of the matrix of polynomials. A row reduced form has the same properties but with respect to the leading row.

The column-reduced form is obtained by elementary column operations, which include interchanging columns, multiplying a column by a unit, or subtracting a polynomial multiple of one column from another. The row-reduced form uses similar row operations. The method used is a fraction-free algorithm by Beckermann and Labahn.

The optional third argument returns a unimodular matrix of elementary operations having the property that $P=A·U$ in the column-reduced case and $P=U·A$ in the row-reduced case.

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

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

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

$P≔\mathrm{ColumnReducedForm}\left(A\,z\right)$

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

$d≔\mathrm{Degree}\left[\mathrm{column}\right]\left(P\,z\right)$

$d≔\left[1\,1\right]$

$C≔\mathrm{Coeff}\left[\mathrm{column}\right]\left(P\,z\,d\right)$

$C≔\left[\begin{array}{cc}1& 3\\ 0& \mathrm{-1}\end{array}\right]$

$P≔\mathrm{ColumnReducedForm}\left(A\,z\,'U'\right)$

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

$\mathrm{map}\left(\mathrm{expand}\,P-A·U\right)$

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

$\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(C\right)$

$\mathrm{-1}$

$\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(U\right)$

$\frac{1}{2}$

$P≔\mathrm{RowReducedForm}\left(A\,z\right)$

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

$d≔\mathrm{Degree}\left[\mathrm{row}\right]\left(P\,z\right)$

$d≔\left[2\,0\right]$

$C≔\mathrm{Coeff}\left[\mathrm{row}\right]\left(P\,z\,d\right)$

$C≔\left[\begin{array}{cc}1& 3\\ 1& 2\end{array}\right]$

$P≔\mathrm{RowReducedForm}\left(A\,z\,'U'\right)$

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

$\mathrm{map}\left(\mathrm{expand}\,P-U·A\right)$

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

$\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(C\right)$

$\mathrm{-1}$

$\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(U\right)$

$\frac{1}{2}$

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.