Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
EXAMPLES:
sage: a = matrix(ZZ, 3, [1,2,5,7,-3,4,2,1,123])
sage: a.hadamard_bound()
4
sage: a.det()
-2014
sage: 10^4
10000
In this example the Hadamard bound has to be computed (automatically) using MPFR instead of doubles,
since doubles overflow:
sage: a = matrix(ZZ, 2, [2^10000,3^10000,2^50,3^19292])
sage: a.hadamard_bound()
12215
sage: len(str(a.det()))
12215
hermite_form(include_zero_rows=True, transformation=False)
Return the Hermite form of self, if it is defined.
INPUT:
•include_zero_rows – bool (default: True); if False the zero rows in the output matrix are deleted.
•transformation – bool (default: False) a matrix U such that U*self == H.
OUTPUT:
•matrix H
•(optional) transformation matrix U such that U*self == H, possibly with zero rows deleted...
EXAMPLES:
sage: M = FunctionField(GF(7),’x’).maximal_order()
sage: K. = FunctionField(GF(7)); M = K.maximal_order()
sage: A = matrix(M, 2, 3, [x, 1, 2*x, x, 1+x, 2])
sage: A.hermite_form()
[ x 1 2*x]
[ 0 x 5*x + 2]
sage: A.hermite_form(transformation=True)
(
[ x 1 2*x] [1 0]
[ 0 x 5*x + 2], [6 1]
)
sage: A = matrix(M, 2, 3, [x, 1, 2*x, 2*x, 2, 4*x])
sage: A.hermite_form(transformation=True, include_zero_rows=False)
([ x 1 2*x], [1 0])
sage: H, U = A.hermite_form(transformation=True, include_zero_rows=True); H, U
(
[ x 1 2*x] [1 0]
[ 0 0 0], [5 1]
)
sage: U*A == H
True
sage: H, U = A.hermite_form(transformation=True, include_zero_rows=False)
sage: U*A
[ x 1 2*x]
sage: U*A == H
True
179