Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 1 1/15*a1 + 2/5 2/15*a1 - 1/5], 1)
]
sage: e, v, n = es[0]; v = v.basis()[0]
sage: delta = e*v - v*A
sage: abs(abs(delta)) < 1e-10
True
The same computation, but with implicit base change to a field.
sage: A = matrix(ZZ,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: A.eigenspaces_left(format=’galois’)
[
(0, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[ 1 -2 1]),
(a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi
User basis matrix:
[ 1 1/15*a1 + 2/5 2/15*a1 - 1/5])
]
We compute the left eigenspaces of the matrix of the Hecke operator T 2 on level 43 modular symbols, both
with all eigenvalues (the default) and with one subspace per factor.
sage: A = ModularSymbols(43).T(2).matrix(); A
[ 3 0 0 0 0 0 -1]
[ 0 -2 1 0 0 0 0]
[ 0 -1 1 1 0 -1 0]
[ 0 -1 0 -1 2 -1 1]
[ 0 -1 0 1 1 -1 1]
[ 0 0 -2 0 2 -2 1]
[ 0 0 -1 0 1 0 -1]
sage: A.base_ring()
Rational Field
sage: f = A.charpoly(); f
x^7 + x^6 - 12*x^5 - 16*x^4 + 36*x^3 + 52*x^2 - 32*x - 48
sage: factor(f)
(x - 3) * (x + 2)^2 * (x^2 - 2)^2
sage: A.eigenspaces_left(algebraic_multiplicity=True)
[
(3, Vector space of degree 7 and dimension 1 over Rational Field
User basis matrix:
[ 1 0 1/7 0 -1/7 0 -2/7], 1),
(-2, Vector space of degree 7 and dimension 2 over Rational Field
User basis matrix:
[ 0 1 0 1 -1 1 -1]
[ 0 0 1 0 -1 2 -1], 2),
(-1.414213562373095, Vector space of degree 7 and dimension 2 over Algebraic Field
User basis matrix:
[ 0 1 0 -1 0.414213562
[ 0 0 1 0
(1.414213562373095, Vector space of degree 7 and dimension 2 over Algebraic Field
User basis matrix:
[ 0 1 0 -1 -2.414
[ 0 0 1 0
]
sage: A.eigenspaces_left(format=’galois’, algebraic_multiplicity=True)
207