Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
(a2, Vector space of degree 7 and dimension 2 over Number Field in a2 with defining polynomi
[ 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)
[
(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),
User basis matrix:
[ 0 1 0 -1 -a2 - 1 1 -1]
[ 0 0 1 0 -1 0 -a2 + 1], 2)
]
Next we compute the left eigenspaces over the finite field of order 11.
sage: A = ModularSymbols(43, base_ring=GF(11), sign=1).T(2).matrix(); A
[ 3 9 0 0]
[ 0 9 0 1]
[ 0 10 9 2]
[ 0 9 0 2]
sage: A.base_ring()
Finite Field of size 11
sage: A.charpoly()
x^4 + 10*x^3 + 3*x^2 + 2*x + 1
sage: A.eigenspaces_left(format=’galois’, var = ’beta’)
[
(9, Vector space of degree 4 and dimension 1 over Finite Field of size 11
User basis matrix:
[0 0 1 5]),
(3, Vector space of degree 4 and dimension 1 over Finite Field of size 11
User basis matrix:
[1 6 0 6]),
(beta2, Vector space of degree 4 and dimension 1 over Univariate Quotient Polynomial Ring in
User basis matrix:
[ 0 1 0 5*beta2 + 10])
]
This method is only applicable to exact matrices. The “eigenmatrix” routines for matrices with doubleprecision
floating-point entries (RDF, CDF) are the best alternative. (Since some platforms return eigenvectors
that are the negatives of those given here, this one example is not tested here.) There are also
“eigenmatrix” routines for matrices with symbolic entries.
sage: A = matrix(QQ, 3, 3, range(9))
sage: A.change_ring(RR).eigenspaces_left()
Traceback (most recent call last):
...
160 Chapter 7. Base class for matrices, part 2