Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Note: This row reduction does not use division if the matrix is not over a field (e.g., if the matrix is over
the integers). If you want to calculate the echelon form using division, then use rref(), which assumes
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 reduced row echelon form of self, as an immutable matrix. Note that self is not changed by this
command. Use echelonize() to change self in place.
If the optional parameter transformation=True is specified, the output consists of a pair (E, T ) of
matrices where E is the echelon form of self and T is the transformation matrix.
EXAMPLES:
sage: MS = MatrixSpace(GF(19),2,3)
sage: C = MS.matrix([1,2,3,4,5,6])
sage: C.rank()
2
sage: C.nullity()
0
sage: C.echelon_form()
[ 1 0 18]
[ 0 1 2]
The matrix library used for Z/p-matrices does not return the transformation matrix,
transformation option is ignored:
so the
sage: C.echelon_form(transformation=True)
[ 1 0 18]
[ 0 1 2]
sage: D = matrix(ZZ, 2, 3, [1,2,3,4,5,6])
sage: D.echelon_form(transformation=True)
(
[1 2 3] [ 1 0]
[0 3 6], [ 4 -1]
)
sage: E, T = D.echelon_form(transformation=True)
sage: T*D == E
True
echelonize(algorithm=’default’, cutoff=0, **kwds)
Transform self into a matrix in echelon form over the same base ring as self.
Note: This row reduction does not use division if the matrix is not over a field (e.g., if the matrix is over
the integers). If you want to calculate the echelon form using division, then use rref(), which assumes
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