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