Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
... d = (M*M).determinant()
... assert d.is_square()
echelon_form(algorithm=’default’, proof=None, include_zero_rows=True, transformation=False,
D=None)
Return the echelon form of this matrix over the integers, also known as the hermit normal form (HNF).
INPUT:
•algorithm – String. The algorithm to use. Valid options are:
–’default’ – Let Sage pick an algorithm (default). Up to 10 rows or columns: pari with flag 0;
Up to 75 rows or columns: pari with flag 1; Larger: use padic algorithm.
–’padic’ - an asymptotically fast p-adic modular algorithm, If your matrix has large coefficients
and is small, you may also want to try this.
–’pari’ - use PARI with flag 1
–’pari0’ - use PARI with flag 0
–’pari4’ - use PARI with flag 4 (use heuristic LLL)
–’ntl’ - use NTL (only works for square matrices of full rank!)
•proof - (default: True); if proof=False certain determinants are computed using a randomized hybrid
p-adic multimodular strategy until it stabilizes twice (instead of up to the Hadamard bound). It is
incredibly unlikely that one would ever get an incorrect result with proof=False.
•include_zero_rows - (default: True) if False, don’t include zero rows
•transformation - if given, also compute transformation matrix; only valid for padic algorithm
•D - (default: None) if given and the algorithm is ‘ntl’, then D must be a multiple of the determinant
and this function will use that fact.
OUTPUT:
The Hermite normal form (=echelon form over Z) of self.
EXAMPLES:
sage: A = MatrixSpace(ZZ,2)([1,2,3,4])
sage: A.echelon_form()
[1 0]
[0 2]
sage: A = MatrixSpace(ZZ,5)(range(25))
sage: A.echelon_form()
[ 5 0 -5 -10 -15]
[ 0 1 2 3 4]
[ 0 0 0 0 0]
[ 0 0 0 0 0]
[ 0 0 0 0 0]
Getting a transformation matrix in the nonsquare case:
sage: A = matrix(ZZ,5,3,[1..15])
sage: H, U = A.hermite_form(transformation=True, include_zero_rows=False)
sage: H
[1 2 3]
[0 3 6]
sage: U
[ 0 0 0 4 -3]
[ 0 0 0 13 -10]
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