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MatrixPolynomialAlgebra[MinimalBasis] - compute a minimal basis for the nullspace of a matrix of polynomials
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Calling Sequence
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MinimalBasis(A, x)
MinimalBasis[right](A, x)
MinimalBasis[left](A, x)
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Parameters
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A
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Matrix
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x
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variable name of the polynomial domain
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Description
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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).
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The MinimalBasis[left](A,x) command computes a minimal basis for the left nullspace.
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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.
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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.
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Examples
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![A := Matrix([[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, 2-z^3]])](/support/helpjp/helpview.aspx?si=7301/file02412/math102.png)
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](/support/helpjp/helpview.aspx?si=7301/file02412/math106.png)
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![B := Matrix([[-2*z^5+3*z^4-6*z^3+4*z^2-5*z+z^6, -1+3*z^5-z^4+2*z^3+z-z^8, -2-z^8+3*z^7-2*z^6+z^5+3*z^4-3*z^3-z^2+z]])](/support/helpjp/helpview.aspx?si=7301/file02412/math109.png)
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![Matrix([[0, 0]])](/support/helpjp/helpview.aspx?si=7301/file02412/math116.png)
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The next example returns NULL, so the right nullspace is {0}.
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](/support/helpjp/helpview.aspx?si=7301/file02412/math122.png)
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References
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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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