Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: F. = FiniteField(5^2)
sage: A = matrix(F, 3, 4, [[ 1, a, 1+a, a^3+a^5],
... [ a, a^4, a+a^4, a^4+a^8],
... [a^2, a^6, a^2+a^6, a^5+a^10]])
sage: P = A.right_kernel_matrix(algorithm=’default’, basis=’pivot’); P
[ 4 4 1 0]
[ a + 2 3*a + 3 0 1]
sage: A*P.transpose() == zero_matrix(F, 3, 2)
True
sage: E = A.right_kernel_matrix(algorithm=’default’, basis=’echelon’); E
[ 1 0 3*a + 4 2*a + 2]
[ 0 1 2*a 3*a + 3]
sage: A*E.transpose() == zero_matrix(F, 3, 2)
True
This general code can be requested for matrices over any field with the algorithm keyword ‘generic’.
Normally, matrices over the rationals would be handled by specific routines from the IML library. The
default format is an echelon basis, but a pivot basis may be requested, which is identical to the computed
basis.
sage: A = matrix(QQ, 3, 4, [[1,3,-2,4],
... [2,0,2,2],
... [-1,1,-2,0]])
sage: G = A.right_kernel_matrix(algorithm=’generic’); G
[ 1 0 -1/2 -1/2]
[ 0 1 1/2 -1/2]
sage: A*G.transpose() == zero_matrix(QQ, 3, 2)
True
sage: C = A.right_kernel_matrix(algorithm=’generic’, basis=’computed’); C
[-1 1 1 0]
[-1 -1 0 1]
sage: A*C.transpose() == zero_matrix(QQ, 3, 2)
True
We test that the generic code is called for matrices over fields, lacking any more specific routine.
sage: F. = FiniteField(5^2)
sage: A = matrix(F, 3, 4, [[ 1, a, 1+a, a^3+a^5],
... [ a, a^4, a+a^4, a^4+a^8],
... [a^2, a^6, a^2+a^6, a^5+a^10]])
sage: set_verbose(1)
sage: A.right_kernel(algorithm=’default’)
verbose ...
verbose 1 () computing right kernel matrix over an arbitrary field for 3x4 matrix
...
...
Vector space of degree 4 and dimension 2 over Finite Field in a of size 5^2
Basis matrix:
[ 1 0 3*a + 4 2*a + 2]
[ 0 1 2*a 3*a + 3]
sage: set_verbose(0)
Over the Integers:
Either the IML or PARI libraries are used to provide a set of basis vectors. The algorithm keyword can
be used to select either, or when set to ‘default’ a heuristic will choose between the two. Results can be
returned in the ‘compute’ format, straight out of the libraries. Unique to the integers, the basis vectors can
be returned as an LLL basis. Note the similarities and differences in the results. The ‘pivot’ format is not
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