Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[0 0], [0 1], [-Infinity, -Infinity]
)
In the following example, self has more rows than columns.
sage: R. = QQ[’t’]
sage: M = matrix([[t,t,t],[0,0,t]], ascend=False)
sage: M.weak_popov_form()
(
[t t t] [1 0]
[0 0 t], [0 1], [1, 1]
)
The next example shows that M must be a matrix with coefficients in a rational function field k(t).
sage: M = matrix([[1,0],[1,1]])
sage: M.weak_popov_form()
Traceback (most recent call last):
...
TypeError: the coefficients of M must lie in a univariate
polynomial ring
NOTES:
•For consistency with LLL and other algorithms in sage, we have opted for row operations; however,
references such as [H] transpose and use column operations.
•There are multiple weak Popov forms of a matrix, so one may want to extend this code to produce a
Popov form (see section 1.2 of [V]). The latter is canonical, but more work to produce.
REFERENCES:
wiedemann(i, t=0)
Application of Wiedemann’s algorithm to the i-th standard basis vector.
INPUT:
•i - an integer
•t - an integer (default: 0) if t is nonzero, use only the first t linear recurrence relations.
IMPLEMENTATION: This is a toy implementation.
EXAMPLES:
sage: t = matrix(QQ, 3, range(9)); t
[0 1 2]
[3 4 5]
[6 7 8]
sage: t.wiedemann(0)
x^2 - 12*x - 18
sage: t.charpoly()
x^3 - 12*x^2 - 18*x
zigzag_form(subdivide=True, transformation=False)
Find a matrix in ZigZag form that is similar to self.
INPUT:
•self - a square matrix with entries from an exact field.
•transformation - default: False - if True return a change-of-basis matrix relating the matrix
and its ZigZag form.
264 Chapter 7. Base class for matrices, part 2