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MatrixPolynomialAlgebra

 MinimalBasis
 compute a minimal basis for the nullspace of a matrix of polynomials

 Calling Sequence MinimalBasis(A, x) MinimalBasis[right](A, x) MinimalBasis[left](A, x)

Parameters

 A - Matrix x - variable name of the polynomial domain

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right):$
 > $A≔⟨⟨{z}^{5}-{z}^{2}-1,{z}^{3}-2{z}^{2}+2z-2,z+1⟩|⟨{z}^{3}-2{z}^{2}-1,{z}^{3}-3{z}^{2}+3z-4,2-{z}^{3}⟩⟩$
 ${A}{≔}\left[\begin{array}{cc}{{z}}^{{5}}{-}{{z}}^{{2}}{-}{1}& {{z}}^{{3}}{-}{2}{}{{z}}^{{2}}{-}{1}\\ {{z}}^{{3}}{-}{2}{}{{z}}^{{2}}{+}{2}{}{z}{-}{2}& {{z}}^{{3}}{-}{3}{}{{z}}^{{2}}{+}{3}{}{z}{-}{4}\\ {z}{+}{1}& {-}{{z}}^{{3}}{+}{2}\end{array}\right]$ (1)
 > $B≔\mathrm{MinimalBasis}[\mathrm{left}]\left(A,z\right)$
 ${B}{≔}\left[\begin{array}{ccc}{{z}}^{{6}}{-}{2}{}{{z}}^{{5}}{+}{3}{}{{z}}^{{4}}{-}{6}{}{{z}}^{{3}}{+}{4}{}{{z}}^{{2}}{-}{5}{}{z}& {-}{{z}}^{{8}}{+}{3}{}{{z}}^{{5}}{-}{{z}}^{{4}}{+}{2}{}{{z}}^{{3}}{+}{z}{-}{1}& {-}{{z}}^{{8}}{+}{3}{}{{z}}^{{7}}{-}{2}{}{{z}}^{{6}}{+}{{z}}^{{5}}{+}{3}{}{{z}}^{{4}}{-}{3}{}{{z}}^{{3}}{-}{{z}}^{{2}}{+}{z}{-}{2}\end{array}\right]$ (2)
 > $\mathrm{map}\left(\mathrm{expand},B\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\right)$
 $\left[\begin{array}{rr}{0}& {0}\end{array}\right]$ (3)

The next example returns NULL, so the right nullspace is {0}.

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

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.