Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
... [4*a^2 + 3*a + 4, 4*a^2 + 2, 3*a, 2*a^2 + 4*a + 2],
... [ 3*a^2 + a, 3*a, 3*a^2 + 2, 3*a^2 + 2*a + 3],
... [2*a^2 + 2*a + 1, 2*a^2 + 4*a + 2, 3*a^2 + 2*a + 3, 3*a^2 + 2*a + 4]])
sage: A.is_symmetric()
True
sage: L, d = A.indefinite_factorization()
sage: D = diagonal_matrix(d)
sage: L
[ 1 0 0 0]
[4*a^2 + 4*a + 3 1 0 0]
[ 3 4*a^2 + a + 2 1 0]
[ 4*a^2 + 4 2*a^2 + 3*a + 3 2*a^2 + 3*a + 1 1]
sage: D
[ a^2 + 2*a 0 0 0]
[ 0 2*a^2 + 2*a + 4 0 0]
[ 0 0 3*a^2 + 4*a + 3 0]
[ 0 0 0 a^2 + 3*a]
sage: A == L*D*L.transpose()
True
AUTHOR:
•Rob Beezer (2012-05-24)
integer_kernel(ring=’ZZ’)
Return the kernel of this matrix over the given ring (which should be either the base ring, or a PID whose
fraction field is the base ring).
Assume that the base field of this matrix has a numerator and denominator functions for its elements, e.g.,
it is the rational numbers or a fraction field. This function computes a basis over the integers for the kernel
of self.
If the matrix is not coercible into QQ, then the PID itself should be given as a second argument, as in the
third example below.
EXAMPLES:
sage: A = MatrixSpace(QQ, 4)(range(16))
sage: A.integer_kernel()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1 0 -3 2]
[ 0 1 -2 1]
The integer kernel even makes sense for matrices with fractional entries:
sage: A = MatrixSpace(QQ, 2)([’1/2’,0, 0, 0])
sage: A.integer_kernel()
Free module of degree 2 and rank 1 over Integer Ring
Echelon basis matrix:
[0 1]
An example over a bigger ring:
sage: L. = NumberField(x^2 - x + 2)
sage: OL = L.ring_of_integers()
sage: A = matrix(L, 2, [1, w/2])
sage: A.integer_kernel(OL)
Free module of degree 2 and rank 1 over Maximal Order in Number Field in w with defining pol
Echelon basis matrix:
[ -1 -w + 1]
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