Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
...
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: set_verbose(0)
sage: D == S
True
Over Principal Ideal Domains:
Kernels can be computed using Smith normal form. Only the default algorithm is available, and the ‘pivot’
basis format is not available.
sage: R. = QQ[]
sage: A = matrix(R, [[ 1, y, 1+y^2],
... [y^3, y^2, 2*y^3]])
sage: E = A.right_kernel_matrix(algorithm=’default’, basis=’echelon’); E
[-1 -y 1]
sage: A*E.transpose() == zero_matrix(ZZ, 2, 1)
True
It can be computationally expensive to determine if an integral domain is a principal ideal domain. The
Smith normal form routine can fail for non-PIDs, as in this example.
sage: D. = ZZ[]
sage: A = matrix(D, 2, 2, [[x^2 - x, -x + 5],
... [x^2 - 8, -x + 2]])
sage: A.right_kernel_matrix()
Traceback (most recent call last):
...
ArithmeticError: Ideal Ideal (x^2 - x, x^2 - 8) of Univariate Polynomial Ring in x over Inte
We test that the domain code is called for domains that lack any extra structure.
sage: R. = QQ[]
sage: A = matrix(R, [[ 1, y, 1+y^2],
... [y^3, y^2, 2*y^3]])
sage: set_verbose(1)
sage: A.right_kernel(algorithm=’default’, basis=’echelon’)
verbose ...
verbose 1 () computing right kernel matrix over a domain for 2x3 matrix
verbose 1 () done computing right kernel matrix over a domain for 2x3 matrix
...
Free module of degree 3 and rank 1 over Univariate Polynomial Ring in y over Rational Field
Echelon basis matrix:
[-1 -y 1]
sage: set_verbose(0)
Trivial Cases:
We test two trivial cases. Any possible values for the keywords (algorithm, basis) will return identical
results.
sage: A = matrix(ZZ, 0, 2)
sage: A.right_kernel_matrix()
[1 0]
[0 1]
sage: A = matrix(FiniteField(7), 2, 0)
247