Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
A complex subfield of the complex numbers.
sage: C. = CyclotomicField(5)
sage: A = matrix(C, [[ -z^3 - 2*z, -z^3 - 1, 2*z^3 - 2*z^2 + 2*z,
... [ z^3 - 2*z^2 + 1, -z^3 + 2*z^2 - z - 1, -1,
... [-1/2*z^3 - 2*z^2 + z + 1, -z^3 + z - 2, -2*z^3 + 1/2*z^2, 2
sage: G, M = A.gram_schmidt(orthonormal=False)
sage: G
[ -z^3 - 2*z
[ 155/139*z^3 - 161/139*z^2 + 31/139*z + 13/139 -175/139*z
[-10359/19841*z^3 - 36739/39682*z^2 + 24961/39682*z - 11879/39682 -28209/39682*z^3 - 3671/1
sage: M
[ 1
[ 14/139*z^3 + 47/139*z^2 + 145/139*z + 95/139
[ -7/278*z^3 + 199/278*z^2 + 183/139*z + 175/278 -3785/39682*z^3 + 3346/19841*z
sage: M*G - A
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
sage: G*G.conjugate().transpose()
[ 15*z^3 + 15*z^2 + 28
[ 0 463/139*z^3 + 463/139*z^
[ 0
sage: G.row_space() == A.row_space()
True
A slightly edited legacy example.
sage: A = matrix(ZZ, 3, [-1, 2, 5, -11, 1, 1, 1, -1, -3]); A
[ -1 2 5]
[-11 1 1]
[ 1 -1 -3]
sage: G, mu = A.gram_schmidt()
sage: G
[ -1 2 5]
[ -52/5 -1/5 -2]
[ 2/187 36/187 -14/187]
sage: mu
[ 1 0 0]
[ 3/5 1 0]
[ -3/5 -7/187 1]
sage: G.row(0) * G.row(1)
0
sage: G.row(0) * G.row(2)
0
sage: G.row(1) * G.row(2)
0
The relation between mu and A is as follows.
sage: mu*G == A
True
hadamard_bound()
Return an int n such that the absolute value of the determinant of this matrix is at most 10 n .
This is got using both the row norms and the column norms.
This function only makes sense when the base field can be coerced to the real double field RDF or the
MPFR Real Field with 53-bits precision.
178 Chapter 7. Base class for matrices, part 2