Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
-6/5*x^2*y^2 - 3*x*y^3 + 6/5*x^2*y + 11/12*x*y^2 - 18*y^2 - 3/4*y
sage: C.change_ring(R.change_ring(QQbar)).det()
(-6/5)*x^2*y^2 + (-3)*x*y^3 + 6/5*x^2*y + 11/12*x*y^2 + (-18)*y^2 + (-3/4)*y
Finally, we check whether the Singular interface is working:
sage: R. = RR[]
sage: C = random_matrix(R, 2, 2, terms=2)
sage: C
[0.368965517352886*y^2 + 0.425700773972636*x -0.800362171389760*y^2 - 0.807635502485287]
[ 0.706173539423122*y^2 - 0.915986060298440 0.897165181570476*y + 0.107903328188376]
sage: C.determinant()
0.565194587390682*y^4 + 0.331023015369146*y^3 + 0.381923912175852*x*y - 0.122977163520282*y^
ALGORITHM: Calls Singular, libSingular or native implementation.
TESTS:
sage: R = PolynomialRing(QQ, 9, ’x’)
sage: matrix(R, 0, 0).det()
1
sage: R. = QQ[]
sage: m = matrix([[y,y,y,y]] * 4) # larger than 3x3
sage: m.charpoly() # put charpoly in the cache
x^4 - 4*y*x^3
sage: m.det()
0
echelon_form(algorithm=’row_reduction’, **kwds)
Return an echelon form of self using chosen algorithm.
By default only a usual row reduction with no divisions or column swaps is returned.
If Gauss-Bareiss algorithm is chosen,
swapped_columns().
INPUT:
column swaps are recorded and can be retrieved via
•algorithm – string, which algorithm to use (default: ‘row_reduction’). Valid options are:
OUTPUT:
–’row_reduction’ (default) – reduce as far as possible, only divide by constant entries
–’frac’ – reduced echelon form over fraction field
–’bareiss’ – fraction free Gauss-Bareiss algorithm with column swaps
The row echelon form of A depending on the chosen algorithm, as an immutable matrix. Note that self
is not changed by this command. Use A.echelonize()‘ to change A in place.
EXAMPLES:
sage: P. = PolynomialRing(GF(127), 2)
sage: A = matrix(P, 2, 2, [1, x, 1, y])
sage: A
[1 x]
[1 y]
sage: A.echelon_form()
[ 1 x]
[ 0 -x + y]
400 Chapter 22. Dense matrices over multivariate polynomials over fields