Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
The reduced row echelon form over the fraction field is as follows:
sage: A.echelon_form(’frac’) # over fraction field
[1 0]
[0 1]
Alternatively, the Gauss-Bareiss algorithm may be chosen:
sage: E = A.echelon_form(’bareiss’); E
[ 1 y]
[ 0 x - y]
After the application of the Gauss-Bareiss algorithm the swapped columns may inspected:
sage: E.swapped_columns(), E.pivots()
((0, 1), (0, 1))
sage: A.swapped_columns(), A.pivots()
(None, (0, 1))
Another approach is to row reduce as far as possible:
sage: A.echelon_form(’row_reduction’)
[ 1 x]
[ 0 -x + y]
echelonize(algorithm=’row_reduction’, **kwds)
Transform self into a matrix in echelon form over the same base ring as self.
If Gauss-Bareiss algorithm is chosen,
swapped_columns().
INPUT:
•algorithm – string, which algorithm to use. Valid options are:
EXAMPLES:
column swaps are recorded and can be retrieved via
–’row_reduction’ – reduce as far as possible, only divide by constant entries
–’bareiss’ – fraction free Gauss-Bareiss algorithm with column swaps
sage: P. = PolynomialRing(QQ, 2)
sage: A = matrix(P, 2, 2, [1/2, x, 1, 3/4*y+1])
sage: A
[ 1/2 x]
[ 1 3/4*y + 1]
sage: B = copy(A)
sage: B.echelonize(’bareiss’); B
[ 1 3/4*y + 1]
[ 0 x - 3/8*y - 1/2]
sage: B = copy(A)
sage: B.echelonize(’row_reduction’); B
[ 1 2*x]
[ 0 -2*x + 3/4*y + 1]
sage: P. = PolynomialRing(QQ, 2)
sage: A = matrix(P,2,3,[2,x,0,3,y,1]); A
[2 x 0]
[3 y 1]
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