compute a minimal basis for the nullspace of a matrix of polynomials - Maple Help

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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

withMatrixPolynomialAlgebra:

A:=z5z21,z32z2+2z2,z+1|z32z21,z33z2+3z4,2z3

A:=z5z21z32z21z32z2+2z2z33z2+3z4z+1z3+2

(1)

B:=MinimalBasisleftA,z

B:=z62z5+3z46z3+4z25zz8+3z5z4+2z3+z1z8+3z72z6+z5+3z43z3z2+z2

(2)

mapexpand,B.A

00

(3)

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

B:=MinimalBasisrightA,z

See Also

expand, indets, map, Matrix, MatrixPolynomialAlgebra, MatrixPolynomialAlgebra[MahlerSystem]

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.


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