sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
- EXAMPLES::
+ EXAMPLES:
+
+ We can compute unit element in the Hadamard EJA::
+
+ sage: J = HadamardEJA(5)
+ sage: J.one()
+ e0 + e1 + e2 + e3 + e4
+
+ The unit element in the Hadamard EJA is inherited in the
+ subalgebras generated by its elements::
sage: J = HadamardEJA(5)
sage: J.one()
e0 + e1 + e2 + e3 + e4
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.one()
+ f0
+ sage: A.one().superalgebra_element()
+ e0 + e1 + e2 + e3 + e4
TESTS:
- The identity element acts like the identity::
+ The identity element acts like the identity, regardless of
+ whether or not we orthonormalize::
sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
True
+ sage: A = x.subalgebra_generated_by()
+ sage: y = A.random_element()
+ sage: A.one()*y == y and y*A.one() == y
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: x = J.random_element()
+ sage: J.one()*x == x and x*J.one() == x
+ True
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: y = A.random_element()
+ sage: A.one()*y == y and y*A.one() == y
+ True
- The matrix of the unit element's operator is the identity::
+ The matrix of the unit element's operator is the identity,
+ regardless of the base field and whether or not we
+ orthonormalize::
sage: set_random_seed()
sage: J = random_eja()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
sage: actual == expected
True
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by()
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: actual = J.one().operator().matrix()
+ sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ sage: actual == expected
+ True
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
Ensure that the cached unit element (often precomputed by
hand) agrees with the computed one::
sage: J.one() == cached
True
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: cached = J.one()
+ sage: J.one.clear_cache()
+ sage: J.one() == cached
+ True
+
"""
# We can brute-force compute the matrices of the operators
# that correspond to the basis elements of this algebra.
blocks = []
for z in M.list():
- a = z.list()[0] # real part, I guess
- b = z.list()[1] # imag part, I guess
- blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
+ a = z.real()
+ b = z.imag()
+ blocks.append(matrix(field, 2, [ [ a, b],
+ [-b, a] ]))
return matrix.block(field, n, blocks)
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: field = QuadraticField(2, 'sqrt2')
sage: B = ComplexHermitianEJA._denormalized_basis(n)
sage: all( M.is_symmetric() for M in B)
True
# * The diagonal will (as a result) be real.
#
S = []
+ Eij = matrix.zero(F,n)
for i in range(n):
for j in range(i+1):
- Eij = matrix(F, n, lambda k,l: k==i and l==j)
+ # "build" E_ij
+ Eij[i,j] = 1
if i == j:
Sij = cls.real_embed(Eij)
S.append(Sij)
else:
# The second one has a minus because it's conjugated.
- Sij_real = cls.real_embed(Eij + Eij.transpose())
+ Eij[j,i] = 1 # Eij = Eij + Eij.transpose()
+ Sij_real = cls.real_embed(Eij)
S.append(Sij_real)
- Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose())
+ # Eij = I*Eij - I*Eij.transpose()
+ Eij[i,j] = I
+ Eij[j,i] = -I
+ Sij_imag = cls.real_embed(Eij)
S.append(Sij_imag)
+ Eij[j,i] = 0
+ # "erase" E_ij
+ Eij[i,j] = 0
# Since we embedded these, we can drop back to the "field" that we
# started with instead of the complex extension "F".
# * The diagonal will (as a result) be real.
#
S = []
+ Eij = matrix.zero(Q,n)
for i in range(n):
for j in range(i+1):
- Eij = matrix(Q, n, lambda k,l: k==i and l==j)
+ # "build" E_ij
+ Eij[i,j] = 1
if i == j:
Sij = cls.real_embed(Eij)
S.append(Sij)
else:
# The second, third, and fourth ones have a minus
# because they're conjugated.
- Sij_real = cls.real_embed(Eij + Eij.transpose())
+ # Eij = Eij + Eij.transpose()
+ Eij[j,i] = 1
+ Sij_real = cls.real_embed(Eij)
S.append(Sij_real)
- Sij_I = cls.real_embed(I*Eij - I*Eij.transpose())
+ # Eij = I*(Eij - Eij.transpose())
+ Eij[i,j] = I
+ Eij[j,i] = -I
+ Sij_I = cls.real_embed(Eij)
S.append(Sij_I)
- Sij_J = cls.real_embed(J*Eij - J*Eij.transpose())
+ # Eij = J*(Eij - Eij.transpose())
+ Eij[i,j] = J
+ Eij[j,i] = -J
+ Sij_J = cls.real_embed(Eij)
S.append(Sij_J)
- Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
+ # Eij = K*(Eij - Eij.transpose())
+ Eij[i,j] = K
+ Eij[j,i] = -K
+ Sij_K = cls.real_embed(Eij)
S.append(Sij_K)
+ Eij[j,i] = 0
+ # "erase" E_ij
+ Eij[i,j] = 0
# Since we embedded these, we can drop back to the "field" that we
# started with instead of the quaternion algebra "Q".
projects / sage.d.git / blobdiff