Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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