Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: K3
Vector space of degree 2 and dimension 1 over Rational Field
Basis matrix:
[ 1 -1/2]
sage: B = copy(A)
sage: K3 = B.left_kernel(basis=’pivot’)
sage: K3
Vector space of degree 2 and dimension 1 over Rational Field
User basis matrix:
[-2 1]
sage: K3 is K1
False
sage: K3 == K1
True
kernel_on(V, poly=None, check=True)
Return the kernel of self restricted to the invariant subspace V. The result is a vector subspace of V, which
is also a subspace of the ambient space.
INPUT:
•V - vector subspace
•check - (optional) default: True; whether to check that V is invariant under the action of self.
•poly - (optional) default: None; if not None, compute instead the kernel of poly(self) on V.
OUTPUT:
•a subspace
Warning: This function does not check that V is in fact invariant under self if check is False. With
check False this function is much faster.
EXAMPLES:
sage: t = matrix(QQ, 4, [39, -10, 0, -12, 0, 2, 0, -1, 0, 1, -2, 0, 0, 2, 0, -2]); t
[ 39 -10 0 -12]
[ 0 2 0 -1]
[ 0 1 -2 0]
[ 0 2 0 -2]
sage: t.fcp()
(x - 39) * (x + 2) * (x^2 - 2)
sage: s = (t-39)*(t^2-2)
sage: V = s.kernel(); V
Vector space of degree 4 and dimension 3 over Rational Field
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 0 1]
sage: s.restrict(V)
[0 0 0]
[0 0 0]
[0 0 0]
sage: s.kernel_on(V)
Vector space of degree 4 and dimension 3 over Rational Field
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 0 1]
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