Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: E = A.echelon_form(’bareiss’); E
[1 3 y]
[0 2 x]
sage: E.swapped_columns()
(2, 0, 1)
sage: A.pivots()
(0, 1, 2)
pivots()
Return the pivot column positions of this matrix as a list of integers.
This returns a list, of the position of the first nonzero entry in each row of the echelon form.
OUTPUT:
A list of Python ints.
EXAMPLES:
sage: matrix([PolynomialRing(GF(2), 2, ’x’).gen()]).pivots()
(0,)
sage: K = QQ[’x,y’]
sage: x, y = K.gens()
sage: m = matrix(K, [(-x, 1, y, x - y), (-x*y, y, y^2 - 1, x*y - y^2 + x), (-x*y + x, y - 1,
sage: m.pivots()
(0, 2)
sage: m.rank()
2
swapped_columns()
Return which columns were swapped during the Gauss-Bareiss reduction
OUTPUT:
Return a tuple representing the column swaps during the last application of the Gauss-Bareiss algorithm
(see echelon_form() for details).
The tuple as length equal to the rank of self and the value at the i-th position indicates the source column
which was put as the i-th column.
If no Gauss-Bareiss reduction was performed yet, None is returned.
EXAMPLES:
sage: R. = QQ[]
sage: C = random_matrix(R, 2, 2, terms=2)
sage: C.swapped_columns()
sage: E = C.echelon_form(’bareiss’)
sage: E.swapped_columns()
(0, 1)
402 Chapter 22. Dense matrices over multivariate polynomials over fields