Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
that the matrix entries are in a field (specifically, the field of fractions of the base ring of the matrix).
INPUT:
•algorithm – string. Which algorithm to use. Choices are
–’default’: Let Sage choose an algorithm (default).
–’classical’: Gauss elimination.
–’strassen’: use a Strassen divide and conquer algorithm (if available)
•cutoff – integer. Only used if the Strassen algorithm is selected.
•transformation – boolean. Whether to also return the transformation matrix. Some matrix
backends do not provide this information, in which case this option is ignored.
OUTPUT:
The matrix self is put into echelon form. Nothing is returned unless the keyword option
transformation=True is specified, in which case the transformation matrix is returned.
EXAMPLES:
sage: a = matrix(QQ,3,range(9)); a
[0 1 2]
[3 4 5]
[6 7 8]
sage: a.echelonize()
sage: a
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
An immutable matrix cannot be transformed into echelon form. Use self.echelon_form() instead:
sage: a = matrix(QQ,3,range(9)); a.set_immutable()
sage: a.echelonize()
Traceback (most recent call last):
...
ValueError: matrix is immutable; please change a copy instead
(i.e., use copy(M) to change a copy of M).
sage: a.echelon_form()
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
Echelon form over the integers is what is also classically often known as Hermite normal form:
sage: a = matrix(ZZ,3,range(9))
sage: a.echelonize(); a
[ 3 0 -3]
[ 0 1 2]
[ 0 0 0]
We compute an echelon form both over a domain and fraction field:
sage: R. = QQ[]
sage: a = matrix(R, 2, [x,y,x,y])
sage: a.echelon_form()
[x y]
[x y]
# not very useful -- why two copies of the same row
154 Chapter 7. Base class for matrices, part 2