Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 True sage: C == -P True sage: E = A.right_kernel_matrix(algorithm=’default’, basis=’echelon’); E [ 1 0 1 1/2 -1/2] [ 0 1 -1/2 -1/4 -1/4] sage: A*E.transpose() == zero_matrix(QQ, 4, 2) True Since the rationals are a field, we can call the general code available for any field by using the ‘generic’ keyword. sage: A = matrix(QQ, [[1, 0, 1, -3, 1], ... [-5, 1, 0, 7, -3], ... [0, -1, -4, 6, -2], ... [4, -1, 0, -6, 2]]) sage: G = A.right_kernel_matrix(algorithm=’generic’, basis=’echelon’); G [ 1 0 1 1/2 -1/2] [ 0 1 -1/2 -1/4 -1/4] sage: A*G.transpose() == zero_matrix(QQ, 4, 2) True We verify that the rational matrix code is called for both dense and sparse rational matrices, with equal result. sage: A = matrix(QQ, [[1, 0, 1, -3, 1], ... [-5, 1, 0, 7, -3], ... [0, -1, -4, 6, -2], ... [4, -1, 0, -6, 2]], ... sparse=False) sage: B = copy(A).sparse_matrix() sage: set_verbose(1) sage: D = A.right_kernel(); D verbose ... verbose 1 () computing right kernel matrix over the rationals for 4x5 matrix ... verbose 1 () done computing right kernel matrix over the rationals for 4x5 matrix ... Vector space of degree 5 and dimension 2 over Rational Field Basis matrix: [ 1 0 1 1/2 -1/2] [ 0 1 -1/2 -1/4 -1/4] sage: S = B.right_kernel(); S verbose ... verbose 1 () computing right kernel matrix over the rationals for 4x5 matrix ... verbose 1 () done computing right kernel matrix over the rationals for 4x5 matrix ... Vector space of degree 5 and dimension 2 over Rational Field Basis matrix: [ 1 0 1 1/2 -1/2] [ 0 1 -1/2 -1/4 -1/4] sage: set_verbose(0) sage: D == S True Over Number Fields: Kernels are by default computed by PARI, (except for exceptions like the rationals themselves). The raw results from PARI are a pivot basis, so the basis keywords ‘computed’ and ‘pivot’ will return the same 242 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 results. sage: Q = QuadraticField(-7) sage: a = Q.gen(0) sage: A = matrix(Q, [[2, 5-a, 15-a, 16+4*a], ... [2+a, a, -7 + 5*a, -3+3*a]]) sage: C = A.right_kernel_matrix(algorithm=’default’, basis=’computed’); C [ -a -3 1 0] [ -2 -a - 1 0 1] sage: A*C.transpose() == zero_matrix(Q, 2, 2) True sage: P = A.right_kernel_matrix(algorithm=’pari’, basis=’pivot’); P [ -a -3 1 0] [ -2 -a - 1 0 1] sage: A*P.transpose() == zero_matrix(Q, 2, 2) True sage: E = A.right_kernel_matrix(algorithm=’default’, basis=’echelon’); E [ 1 0 7/88*a + 3/88 -3/176*a - 39/176] [ 0 1 -1/88*a - 13/88 13/176*a - 7/176] sage: A*E.transpose() == zero_matrix(Q, 2, 2) True We can bypass using PARI for number fields and use Sage’s general code for matrices over any field. The basis vectors as computed are in pivot format. sage: Q = QuadraticField(-7) sage: a = Q.gen(0) sage: A = matrix(Q, [[2, 5-a, 15-a, 16+4*a],[2+a, a, -7 + 5*a, -3+3*a]]) sage: G = A.right_kernel_matrix(algorithm=’generic’, basis=’computed’); G [ -a -3 1 0] [ -2 -a - 1 0 1] sage: A*G.transpose() == zero_matrix(Q, 2, 2) True We check that number fields are handled by the right routine as part of typical right kernel computation. sage: Q = QuadraticField(-7) sage: a = Q.gen(0) sage: A = matrix(Q, [[2, 5-a, 15-a, 16+4*a],[2+a, a, -7 + 5*a, -3+3*a]]) sage: set_verbose(1) sage: A.right_kernel(algorithm=’default’) verbose ... verbose 1 () computing right kernel matrix over a number field for 2x4 matrix verbose 1 () done computing right kernel matrix over a number field for 2x4 matrix ... Vector space of degree 4 and dimension 2 over Number Field in a with defining polynomial x^2 Basis matrix: [ 1 0 7/88*a + 3/88 -3/176*a - 39/176] [ 0 1 -1/88*a - 13/88 13/176*a - 7/176] sage: set_verbose(0) Over the Finite Field of Order 2: Kernels are computed by the M4RI library using PLUQ matrix decomposition in the _right_kernel_matrix() method. There are no options for the algorithm used. sage: A = matrix(GF(2),[[0, 1, 1, 0, 0, 0], ... [1, 0, 0, 0, 1, 1,], ... [1, 0, 0, 0, 1, 1]]) sage: E = A.right_kernel_matrix(algorithm=’default’, format=’echelon’); E 243
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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