from sage.misc.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
-from sage.rings.all import (ZZ, QQ, RR, RLF, CLF,
+from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
Ensure that it says what we think it says::
- sage: JordanSpinEJA(2, field=QQ)
- Euclidean Jordan algebra of dimension 2 over Rational Field
+ sage: JordanSpinEJA(2, field=AA)
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
sage: JordanSpinEJA(3, field=RDF)
Euclidean Jordan algebra of dimension 3 over Real Double Field
Finite family {0: e0, 1: e1, 2: e2}
sage: J.natural_basis()
(
- [1 0] [ 0 1/2*sqrt2] [0 0]
- [0 0], [1/2*sqrt2 0], [0 1]
+ [1 0] [ 0 0.7071067811865475?] [0 0]
+ [0 0], [0.7071067811865475? 0], [0 1]
)
::
return 5
@classmethod
- def random_instance(cls, field=QQ, **kwargs):
+ def random_instance(cls, field=AA, **kwargs):
"""
Return a random instance of this type of algebra.
(r0, r1, r2)
"""
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
V = VectorSpace(field, n)
mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
for i in range(n) ]
return x.to_vector().inner_product(y.to_vector())
-def random_eja(field=QQ, nontrivial=False):
+def random_eja(field=AA, nontrivial=False):
"""
Return a "random" finite-dimensional Euclidean Jordan Algebra.
In theory, our "field" can be any subfield of the reals::
- sage: RealSymmetricEJA(2, AA)
- Euclidean Jordan algebra of dimension 3 over Algebraic Real Field
+ sage: RealSymmetricEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 3 over Real Double Field
sage: RealSymmetricEJA(2, RR)
Euclidean Jordan algebra of dimension 3 over Real Field with
53 bits of precision
return 4 # Dimension 10
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n, field)
super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs)
EXAMPLES::
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: x1 = F(4 - 2*i)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
sage: set_random_seed()
sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
sage: n = ZZ.random_element(n_max)
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
....: [ 9, 10, 11, 12],
....: [-10, 9, -12, 11] ])
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
- [ 2*i + 1 4*i + 3]
- [ 10*i + 9 12*i + 11]
+ [ 2*I + 1 4*I + 3]
+ [ 10*I + 9 12*I + 11]
TESTS:
Unembedding is the inverse of embedding::
sage: set_random_seed()
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: M = random_matrix(F, 3)
sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
field = M.base_ring()
R = PolynomialRing(field, 'z')
z = R.gen()
- F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ if field is AA:
+ # Sage doesn't know how to embed AA into QQbar, i.e. how
+ # to adjoin sqrt(-1) to AA.
+ F = QQbar
+ else:
+ F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
i = F.gen()
# Go top-left to bottom-right (reading order), converting every
sage: Ye = y.natural_representation()
sage: X = ComplexHermitianEJA.real_unembed(Xe)
sage: Y = ComplexHermitianEJA.real_unembed(Ye)
- sage: expected = (X*Y).trace().vector()[0]
+ sage: expected = (X*Y).trace().real()
sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
sage: actual == expected
True
In theory, our "field" can be any subfield of the reals::
- sage: ComplexHermitianEJA(2, AA)
- Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
+ sage: ComplexHermitianEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 4 over Real Double Field
sage: ComplexHermitianEJA(2, RR)
Euclidean Jordan algebra of dimension 4 over Real Field with
53 bits of precision
return ( s.change_ring(field) for s in S )
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs)
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- F = QuadraticField(-1, 'i')
+ F = QuadraticField(-1, 'I')
i = F.gen()
blocks = []
raise ValueError('bad on-diagonal submatrix')
if submat[0,1] != -submat[1,0].conjugate():
raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0].vector()[0] # real part
- z += submat[0,0].vector()[1]*i # imag part
- z += submat[0,1].vector()[0]*j # real part
- z += submat[0,1].vector()[1]*k # imag part
+ z = submat[0,0].real()
+ z += submat[0,0].imag()*i
+ z += submat[0,1].real()*j
+ z += submat[0,1].imag()*k
elements.append(z)
return matrix(Q, n/4, elements)
In theory, our "field" can be any subfield of the reals::
- sage: QuaternionHermitianEJA(2, AA)
- Euclidean Jordan algebra of dimension 6 over Algebraic Real Field
+ sage: QuaternionHermitianEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 6 over Real Double Field
sage: QuaternionHermitianEJA(2, RR)
Euclidean Jordan algebra of dimension 6 over Real Field with
53 bits of precision
return ( s.change_ring(field) for s in S )
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs)
sage: actual == expected
True
"""
- def __init__(self, n, field=QQ, B=None, **kwargs):
+ def __init__(self, n, field=AA, B=None, **kwargs):
if B is None:
self._B = matrix.identity(field, max(0,n-1))
else:
True
"""
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
# This is a special case of the BilinearFormEJA with the identity
# matrix as its bilinear form.
return super(JordanSpinEJA, self).__init__(n, field, **kwargs)
sage: J.one().norm()
0
sage: J.one().subalgebra_generated_by()
- Euclidean Jordan algebra of dimension 0 over Rational Field
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
sage: J.rank()
0
"""
- def __init__(self, field=QQ, **kwargs):
+ def __init__(self, field=AA, **kwargs):
mult_table = []
fdeja = super(TrivialEJA, self)
# The rank is zero using my definition, namely the dimension of the
We should always get back an element of the algebra::
sage: set_random_seed()
- sage: p = PolynomialRing(QQ, 't').random_element()
+ sage: p = PolynomialRing(AA, 't').random_element()
sage: J = random_eja()
sage: x = J.random_element()
sage: x.apply_univariate_polynomial(p) in J
The spectral decomposition of a non-regular element should always
contain at least one non-minimal idempotent::
- sage: J = RealSymmetricEJA(3, AA)
+ sage: J = RealSymmetricEJA(3)
sage: x = sum(J.gens())
sage: x.is_regular()
False
On the other hand, the spectral decomposition of a regular
element should always be in terms of minimal idempotents::
- sage: J = JordanSpinEJA(4, AA)
+ sage: J = JordanSpinEJA(4)
sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
sage: x.is_regular()
True
sage: set_random_seed()
sage: n_max = RealSymmetricEJA._max_test_case_size()
sage: n = ZZ.random_element(1, n_max)
- sage: J1 = RealSymmetricEJA(n,QQ)
- sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
- sage: X = random_matrix(QQ,n)
+ sage: J1 = RealSymmetricEJA(n)
+ sage: J2 = RealSymmetricEJA(n,normalize_basis=False)
+ sage: X = random_matrix(AA,n)
sage: X = X*X.transpose()
sage: x1 = J1(X)
sage: x2 = J2(X)
sage: J = HadamardEJA(2)
sage: x = sum(J.gens())
sage: x.norm()
- sqrt(2)
+ 1.414213562373095?
::
sage: n = x_vec.degree()
sage: x0 = x_vec[0]
sage: x_bar = x_vec[1:]
- sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
+ sage: A = matrix(AA, 1, [x_vec.inner_product(x_vec)])
sage: B = 2*x0*x_bar.row()
sage: C = 2*x0*x_bar.column()
- sage: D = matrix.identity(QQ, n-1)
+ sage: D = matrix.identity(AA, n-1)
sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
sage: D = D + 2*x_bar.tensor_product(x_bar)
sage: Q = matrix.block(2,2,[A,B,C,D])
The spectral decomposition of the identity is ``1`` times itself,
and the spectral decomposition of zero is ``0`` times the identity::
- sage: J = RealSymmetricEJA(3,AA)
+ sage: J = RealSymmetricEJA(3)
sage: J.one()
e0 + e2 + e5
sage: J.one().spectral_decomposition()
TESTS::
- sage: J = RealSymmetricEJA(4,AA)
+ sage: J = RealSymmetricEJA(4)
sage: x = sum(J.gens())
sage: sd = x.spectral_decomposition()
sage: l0 = sd[0][0]
sage: J = HadamardEJA(2)
sage: x = sum(J.gens())
sage: x.trace_norm()
- sqrt(2)
+ 1.414213562373095?
::
sage: J = JordanSpinEJA(4)
sage: x = sum(J.gens())
sage: x.trace_norm()
- 2*sqrt(2)
+ 2.828427124746190?
"""
return self.trace_inner_product(self).sqrt()
projects / sage.d.git / commitdiff