Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
The routine expects matrices over the rational function field, but other examples below show how one can
provide matrices over the ring of polynomials (whose quotient field is the rational function field).
sage: R. = GF(3)[’t’]
sage: K = FractionField(R)
sage: M = matrix([[(t-1)^2/t],[(t-1)]])
sage: M.weak_popov_form()
(
[ 0] [ t 2*t + 1]
[(2*t + 1)/t], [ 1 2], [-Infinity, 0]
)
If self is an nx1 matrix with at least one non-zero entry, W has a single non-zero entry and that entry is a
scalar multiple of the greatest-common-divisor of the entries of self.
sage: M1 = matrix([[t*(t-1)*(t+1)],[t*(t-2)*(t+2)],[t]])
sage: output1 = M1.weak_popov_form()
sage: output1
(
[0] [ 1 0 2*t^2 + 1]
[0] [ 0 1 2*t^2 + 1]
[t], [ 0 0 1], [-Infinity, -Infinity, 1]
)
We check that the output is the same for a matrix M if its entries are rational functions intead of polynomials.
We also check that the type of the output follows the documentation. See #9063
sage: M2 = M1.change_ring(K)
sage: output2 = M2.weak_popov_form()
sage: output1 == output2
True
sage: output1[0].base_ring() is K
True
sage: output2[0].base_ring() is K
True
sage: output1[1].base_ring() is R
True
sage: output2[1].base_ring() is R
True
The following is the first half of example 5 in [H] except that we have transposed self; [H] uses column
operations and we use row.
sage: R. = QQ[’t’]
sage: M = matrix([[t^3 - t,t^2 - 2],[0,t]]).transpose()
sage: M.weak_popov_form()
(
[ t -t^2] [ 1 -t]
[t^2 - 2 t], [ 0 1], [2, 2]
)
The next example demonstrates what happens when self is a zero matrix.
sage: R. = GF(5)[’t’]
sage: K = FractionField(R)
sage: M = matrix([[K(0),K(0)],[K(0),K(0)]])
sage: M.weak_popov_form()
(
[0 0] [1 0]
263