Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
We illustrate each option:
sage: S = A.saturation(p=2)
sage: S = A.saturation(proof=False)
sage: S = A.saturation(max_dets=2)
smith_form()
Returns matrices S, U, and V such that S = U*self*V, and S is in Smith normal form. Thus S is diagonal
with diagonal entries the ordered elementary divisors of S.
Warning: The elementary_divisors function, which returns the diagonal entries of S, is VASTLY
faster than this function.
The elementary divisors are the invariants of the finite abelian group that is the cokernel of this matrix.
They are ordered in reverse by divisibility.
EXAMPLES:
sage: A = MatrixSpace(IntegerRing(), 3)(range(9))
sage: D, U, V = A.smith_form()
sage: D
[1 0 0]
[0 3 0]
[0 0 0]
sage: U
[ 0 1 0]
[ 0 -1 1]
[-1 2 -1]
sage: V
[-1 4 1]
[ 1 -3 -2]
[ 0 0 1]
sage: U*A*V
[1 0 0]
[0 3 0]
[0 0 0]
It also makes sense for nonsquare matrices:
sage: A = Matrix(ZZ,3,2,range(6))
sage: D, U, V = A.smith_form()
sage: D
[1 0]
[0 2]
[0 0]
sage: U
[ 0 1 0]
[ 0 -1 1]
[-1 2 -1]
sage: V
[-1 3]
[ 1 -2]
sage: U * A * V
[1 0]
[0 2]
[0 0]
Empty matrices are handled sensibly (see trac #3068):
334 Chapter 17. Dense matrices over the integer ring