Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
available, since the integers are not a field.
sage: A = matrix(ZZ, [[8, 0, 7, 1, 3, 4, 6],
... [4, 0, 3, 4, 2, 7, 7],
... [1, 4, 6, 1, 2, 8, 5],
... [0, 3, 1, 2, 3, 6, 2]])
sage: X = A.right_kernel_matrix(algorithm=’default’, basis=’echelon’); X
[ 1 12 3 14 -3 -10 1]
[ 0 35 0 25 -1 -31 17]
[ 0 0 7 12 -3 -1 -8]
sage: A*X.transpose() == zero_matrix(ZZ, 4, 3)
True
sage: X = A.right_kernel_matrix(algorithm=’padic’, basis=’LLL’); X
[ -3 -1 5 7 2 -3 -2]
[ 3 1 2 5 -5 2 -6]
[ -4 -13 2 -7 5 7 -3]
sage: A*X.transpose() == zero_matrix(ZZ, 4, 3)
True
sage: X = A.right_kernel_matrix(algorithm=’pari’, basis=’computed’); X
[ 3 1 -5 -7 -2 3 2]
[ 3 1 2 5 -5 2 -6]
[ 4 13 -2 7 -5 -7 3]
sage: A*X.transpose() == zero_matrix(ZZ, 4, 3)
True
sage: X = A.right_kernel_matrix(algorithm=’padic’, basis=’computed’); X
[ 265 345 -178 17 -297 0 0]
[-242 -314 163 -14 271 -1 0]
[ -36 -47 25 -1 40 0 -1]
sage: A*X.transpose() == zero_matrix(ZZ, 4, 3)
True
We test that the code for integer matrices is called for matrices defined over the integers, both dense and
sparse, with equal result.
sage: A = matrix(ZZ, [[8, 0, 7, 1, 3, 4, 6],
... [4, 0, 3, 4, 2, 7, 7],
... [1, 4, 6, 1, 2, 8, 5],
... [0, 3, 1, 2, 3, 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 integers for 4x7 matrix
verbose 1 () done computing right kernel matrix over the integers for 4x7 matrix
...
Free module of degree 7 and rank 3 over Integer Ring
Echelon basis matrix:
[ 1 12 3 14 -3 -10 1]
[ 0 35 0 25 -1 -31 17]
[ 0 0 7 12 -3 -1 -8]
sage: S = B.right_kernel(); S
verbose ...
verbose 1 () computing right kernel matrix over the integers for 4x7 matrix
verbose 1 () done computing right kernel matrix over the integers for 4x7 matrix
246 Chapter 7. Base class for matrices, part 2