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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We test that an exception is raised when the block is out of bounds:
sage: matrix([1]).set_block(0,1,matrix([1]))
Traceback (most recent call last):
...
IndexError: matrix window index out of range
smith_form()
If self is a matrix over a principal ideal domain R, return matrices D, U, V over R such that D = U * self *
V, U and V have unit determinant, and D is diagonal with diagonal entries the ordered elementary divisors
of self, ordered so that D i | D i+1 . Note that U and V are not uniquely defined in general, and D is defined
only up to units.
INPUT:
•self - a matrix over an integral domain. If the base ring is not a PID, the routine might work, or else
it will fail having found an example of a non-principal ideal. Note that we do not call any methods
to check whether or not the base ring is a PID, since this might be quite expensive (e.g. for rings of
integers of number fields of large degree).
ALGORITHM: Lifted wholesale from http://en.wikipedia.org/wiki/Smith_normal_form
See Also:
elementary_divisors()
AUTHORS:
•David Loeffler (2008-12-05)
EXAMPLES:
An example over the ring of integers of a number field (of class number 1):
sage: OE = NumberField(x^2 - x + 2,’w’).ring_of_integers()
sage: w = OE.ring_generators()[0]
sage: m = Matrix([ [1, w],[w,7]])
sage: d, u, v = m.smith_form()
sage: (d, u, v)
(
[ 1 0] [ 1 0] [ 1 -w]
[ 0 -w + 9], [-w 1], [ 0 1]
)
sage: u * m * v == d
True
sage: u.base_ring() == v.base_ring() == d.base_ring() == OE
True
sage: u.det().is_unit() and v.det().is_unit()
True
An example over the polynomial ring QQ[x]:
sage: R. = QQ[]; m=x*matrix(R,2,2,1) - matrix(R, 2,2,[3,-4,1,-1]); m.smith_form()
(
[ 1 0] [ 0 -1] [ 1 x + 1]
[ 0 x^2 - 2*x + 1], [ 1 x - 3], [ 0 1]
)
An example over a field:
252 Chapter 7. Base class for matrices, part 2