Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm=’pari0’)
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm=’pari4’)
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm=’padic’)
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm=’default’)
[1 2 3]
[0 3 6]
[0 0 0]
The ‘ntl’ algorithm doesn’t work on matrices that do not have full rank.:
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm=’ntl’)
Traceback (most recent call last):
...
ValueError: ntl only computes HNF for square matrices of full rank.
sage: matrix(ZZ,3,[0] +[2..9]).hermite_form(algorithm=’ntl’)
[1 0 0]
[0 1 0]
[0 0 3]
TESTS:
This example illustrated trac 2398:
sage: a = matrix([(0, 0, 3), (0, -2, 2), (0, 1, 2), (0, -2, 5)])
sage: a.hermite_form()
[0 1 2]
[0 0 3]
[0 0 0]
[0 0 0]
Check that #12280 is fixed:
sage: m = matrix([(-2, 1, 9, 2, -8, 1, -3, -1, -4, -1),
... (5, -2, 0, 1, 0, 4, -1, 1, -2, 0),
... (-11, 3, 1, 0, -3, -2, -1, -11, 2, -2),
... (-1, 1, -1, -2, 1, -1, -1, -1, -1, 7),
... (-2, -1, -1, 1, 1, -2, 1, 0, 2, -4)]).stack(
... 200 * identity_matrix(ZZ, 10))
sage: matrix(ZZ,m).hermite_form(algorithm=’pari’, include_zero_rows=False)
[ 1 0 2 0 13 5 1 166 72 69]
[ 0 1 1 0 20 4 15 195 65 190]
[ 0 0 4 0 24 5 23 22 51 123]
[ 0 0 0 1 23 7 20 105 60 151]
[ 0 0 0 0 40 4 0 80 36 68]
[ 0 0 0 0 0 10 0 100 190 170]
[ 0 0 0 0 0 0 25 0 100 150]
[ 0 0 0 0 0 0 0 200 0 0]
[ 0 0 0 0 0 0 0 0 200 0]
[ 0 0 0 0 0 0 0 0 0 200]
328 Chapter 17. Dense matrices over the integer ring