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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elementary_divisors(algorithm=’pari’)
Return the elementary divisors of self, in order.
Warning: This is MUCH faster than the smith_form function.
The elementary divisors are the invariants of the finite abelian group that is the cokernel of left multiplication
of this matrix. They are ordered in reverse by divisibility.
INPUT:
•self - matrix
•algorithm - (default: ‘pari’)
–’pari’: works robustly, but is slower.
–’linbox’ - use linbox (currently off, broken)
OUTPUT: list of integers
Note: These are the invariants of the cokernel of left multiplication:
sage: M = Matrix([[3,0,1],[0,1,0]])
sage: M
[3 0 1]
[0 1 0]
sage: M.elementary_divisors()
[1, 1]
sage: M.transpose().elementary_divisors()
[1, 1, 0]
EXAMPLES:
sage: matrix(3, range(9)).elementary_divisors()
[1, 3, 0]
sage: matrix(3, range(9)).elementary_divisors(algorithm=’pari’)
[1, 3, 0]
sage: C = MatrixSpace(ZZ,4)([3,4,5,6,7,3,8,10,14,5,6,7,2,2,10,9])
sage: C.elementary_divisors()
[1, 1, 1, 687]
sage: M = matrix(ZZ, 3, [1,5,7, 3,6,9, 0,1,2])
sage: M.elementary_divisors()
[1, 1, 6]
This returns a copy, which is safe to change:
sage: edivs = M.elementary_divisors()
sage: edivs.pop()
6
sage: M.elementary_divisors()
[1, 1, 6]
See Also:
smith_form()
324 Chapter 17. Dense matrices over the integer ring