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MatrixPolynomialAlgebra

 ColumnReducedForm
 compute a column-reduced form of a Matrix
 RowReducedForm
 compute a row-reduced form of a Matrix

 Calling Sequence ColumnReducedForm(A, x, U) RowReducedForm(A, x, U)

Parameters

 A - Matrix x - variable name of the polynomial domain U - (optional) name to return unimodular multiplier

Description

 • 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 $P=A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}U$ in the column-reduced case and $P=U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A$ in the row-reduced case.

Examples

 > $\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right):$
 > $A≔⟨⟨{z}^{3}-{z}^{2},{z}^{3}-2{z}^{2}+2z-2⟩|⟨{z}^{3}-2{z}^{2}-1,{z}^{3}-3{z}^{2}+3z-4⟩⟩$
 ${A}{:=}\left[\begin{array}{cc}{{z}}^{{3}}{-}{{z}}^{{2}}& {{z}}^{{3}}{-}{2}{}{{z}}^{{2}}{-}{1}\\ {{z}}^{{3}}{-}{2}{}{{z}}^{{2}}{+}{2}{}{z}{-}{2}& {{z}}^{{3}}{-}{3}{}{{z}}^{{2}}{+}{3}{}{z}{-}{4}\end{array}\right]$ (1)
 > $P≔\mathrm{ColumnReducedForm}\left(A,z\right)$
 ${P}{:=}\left[\begin{array}{cc}{z}& {1}{+}{3}{}{z}\\ {1}& {4}{-}{z}\end{array}\right]$ (2)
 > $d≔\mathrm{Degree}[\mathrm{column}]\left(P,z\right)$
 ${d}{:=}\left[{1}{,}{1}\right]$ (3)
 > $C≔\mathrm{Coeff}[\mathrm{column}]\left(P,z,d\right)$
 ${C}{:=}\left[\begin{array}{rr}{1}& {3}\\ {0}& {-}{1}\end{array}\right]$ (4)
 > $P≔\mathrm{ColumnReducedForm}\left(A,z,'U'\right)$
 ${P}{:=}\left[\begin{array}{cc}{z}& {1}{+}{3}{}{z}\\ {1}& {4}{-}{z}\end{array}\right]$ (5)
 > $\mathrm{map}\left(\mathrm{expand},P-A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}U\right)$
 $\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (6)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(C\right)$
 ${-}{1}$ (7)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(U\right)$
 $\frac{{1}}{{2}}$ (8)
 > $P≔\mathrm{RowReducedForm}\left(A,z\right)$
 ${P}{:=}\left[\begin{array}{cc}{{z}}^{{2}}& {3}{}{{z}}^{{2}}{-}{z}{+}{1}\\ {1}& {2}\end{array}\right]$ (9)
 > $d≔\mathrm{Degree}[\mathrm{row}]\left(P,z\right)$
 ${d}{:=}\left[{2}{,}{0}\right]$ (10)
 > $C≔\mathrm{Coeff}[\mathrm{row}]\left(P,z,d\right)$
 ${C}{:=}\left[\begin{array}{rr}{1}& {3}\\ {1}& {2}\end{array}\right]$ (11)
 > $P≔\mathrm{RowReducedForm}\left(A,z,'U'\right)$
 ${P}{:=}\left[\begin{array}{cc}{{z}}^{{2}}& {3}{}{{z}}^{{2}}{-}{z}{+}{1}\\ {1}& {2}\end{array}\right]$ (12)
 > $\mathrm{map}\left(\mathrm{expand},P-U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\right)$
 $\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (13)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(C\right)$
 ${-}{1}$ (14)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(U\right)$
 $\frac{{1}}{{2}}$ (15)

References

 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.