Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
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ZeroDivisionError: The modulus cannot be zero
saturation(p=0, proof=None, max_dets=5)
Return a saturation matrix of self, which is a matrix whose rows span the saturation of the row span of self.
This is not unique.
The saturation of a Z module M embedded in Z n is the a module S that contains M with finite index such
that Z n /S is torsion free. This function takes the row span M of self, and finds another matrix of full rank
with row span the saturation of M.
INPUT:
•p - (default: 0); if nonzero given, saturate only at the prime p, i.e., return a matrix whose row span is
a Z-module S that contains self and such that the index of S in its saturation is coprime to p. If p is
None, return full saturation of self.
•proof - (default: use proof.linear_algebra()); if False, the determinant calculations are done with
proof=False.
•max_dets - (default: 5); technical parameter - max number of determinant to compute when bounding
prime divisor of self in its saturation.
OUTPUT:
•matrix - a matrix over ZZ
Note: The result is not cached.
ALGORITHM: 1. Replace input by a matrix of full rank got from a subset of the rows. 2. Divide out
any common factors from rows. 3. Check max_dets random dets of submatrices to see if their GCD (with
p) is 1 - if so matrix is saturated and we’re done. 4. Finally, use that if A is a matrix of full rank, then
hnf(transpose(A)) −1 ∗ A is a saturation of A.
EXAMPLES:
sage: A = matrix(ZZ, 3, 5, [-51, -1509, -71, -109, -593, -19, -341, 4, 86, 98, 0, -246, -11,
sage: A.echelon_form()
[ 1 5 2262 20364 56576]
[ 0 6 35653 320873 891313]
[ 0 0 42993 386937 1074825]
sage: S = A.saturation(); S
[ -51 -1509 -71 -109 -593]
[ -19 -341 4 86 98]
[ 35 994 43 51 347]
Notice that the saturation spans a different module than A.
sage: S.echelon_form()
[ 1 2 0 8 32]
[ 0 3 0 -2 -6]
[ 0 0 1 9 25]
sage: V = A.row_space(); W = S.row_space()
sage: V.is_submodule(W)
True
sage: V.index_in(W)
85986
sage: V.index_in_saturation()
85986
333