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  in the column-reduced case and  in the row-reduced case.

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.