Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Over number fields, PARI is used by default, but general-purpose code can be requested. Same vector
space, same bases, different code.:
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: K = A.right_kernel(algorithm=’default’); K
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: A*K.basis_matrix().transpose() == zero_matrix(Q, 2, 2)
True
sage: B = copy(A)
sage: G = A.right_kernel(algorithm=’generic’); G
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: B*G.basis_matrix().transpose() == zero_matrix(Q, 2, 2)
True
sage: K == G
True
For matrices over the integers, several options are possible. The basis can be an LLL-reduced basis or an
echelon basis. The pivot basis isnot available. A heuristic will decide whether to use a p-adic algorithm
from the IML library or an algorithm from the PARI library. Note how specifying the algorithm can mildly
influence the LLL basis.
sage: A = matrix(ZZ, [[0, -1, -1, 2, 9, 4, -4],
... [-1, 1, 0, -2, -7, -1, 6],
... [2, 0, 1, 0, 1, -5, -2],
... [-1, -1, -1, 3, 10, 10, -9],
... [-1, 2, 0, -3, -7, 1, 6]])
sage: A.right_kernel(basis=’echelon’)
Free module of degree 7 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1 5 -8 3 -1 -1 -1]
[ 0 11 -19 5 -2 -3 -3]
sage: B = copy(A)
sage: B.right_kernel(basis=’LLL’)
Free module of degree 7 and rank 2 over Integer Ring
User basis matrix:
[ 2 -1 3 1 0 1 1]
[-5 -3 2 -5 1 -1 -1]
sage: C = copy(A)
sage: C.right_kernel(basis=’pivot’)
Traceback (most recent call last):
...
ValueError: pivot basis only available over a field, not over Integer Ring
sage: D = copy(A)
sage: D.right_kernel(algorithm=’pari’)
Free module of degree 7 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1 5 -8 3 -1 -1 -1]
[ 0 11 -19 5 -2 -3 -3]
sage: E = copy(A)
sage: E.right_kernel(algorithm=’padic’, basis=’LLL’)
238 Chapter 7. Base class for matrices, part 2