Full MatrixSpace of 4 by 4 dense matrices over Rational Field
sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
sage: J.matrix_space()
- Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ Module of 1 by 1 matrices with entries in Quaternion
+ Algebra (-1, -1) with base ring Rational Field over
+ the scalar ring Rational Field
"""
if self.is_trivial():
"""
return (X*Y).trace().real()
-class RealEmbeddedMatrixEJA(MatrixEJA):
- @staticmethod
- def dimension_over_reals():
- r"""
- The dimension of this matrix's base ring over the reals.
-
- The reals are dimension one over themselves, obviously; that's
- just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi`
- have dimension two. Finally, the quaternions have dimension
- four over the reals.
-
- This is used to determine the size of the matrix returned from
- :meth:`real_embed`, among other things.
- """
- raise NotImplementedError
-
- @classmethod
- def real_embed(cls,M):
- """
- Embed the matrix ``M`` into a space of real matrices.
-
- The matrix ``M`` can have entries in any field at the moment:
- the real numbers, complex numbers, or quaternions. And although
- they are not a field, we can probably support octonions at some
- point, too. This function returns a real matrix that "acts like"
- the original with respect to matrix multiplication; i.e.
-
- real_embed(M*N) = real_embed(M)*real_embed(N)
-
- """
- if M.ncols() != M.nrows():
- raise ValueError("the matrix 'M' must be square")
- return M
-
-
- @classmethod
- def real_unembed(cls,M):
- """
- The inverse of :meth:`real_embed`.
- """
- if M.ncols() != M.nrows():
- raise ValueError("the matrix 'M' must be square")
- if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero():
- raise ValueError("the matrix 'M' must be a real embedding")
- return M
-
-
- @classmethod
- def trace_inner_product(cls,X,Y):
- r"""
- Compute the trace inner-product of two real-embeddings.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
- ....: QuaternionHermitianEJA)
-
- EXAMPLES::
-
- sage: set_random_seed()
- sage: J = ComplexHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = J.real_unembed(Xe)
- sage: Y = J.real_unembed(Ye)
- sage: expected = (X*Y).trace().real()
- sage: actual = J.trace_inner_product(Xe,Ye)
- sage: actual == expected
- True
-
- ::
-
- sage: set_random_seed()
- sage: J = QuaternionHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = J.real_unembed(Xe)
- sage: Y = J.real_unembed(Ye)
- sage: expected = (X*Y).trace().coefficient_tuple()[0]
- sage: actual = J.trace_inner_product(Xe,Ye)
- sage: actual == expected
- True
-
- """
- # This does in fact compute the real part of the trace.
- # If we compute the trace of e.g. a complex matrix M,
- # then we do so by adding up its diagonal entries --
- # call them z_1 through z_n. The real embedding of z_1
- # will be a 2-by-2 REAL matrix [a, b; -b, a] whose trace
- # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
- return (X*Y).trace()/cls.dimension_over_reals()
class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
"""
-class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
- # A manual dictionary-cache for the complex_extension() method,
- # since apparently @classmethods can't also be @cached_methods.
- _complex_extension = {}
-
- @classmethod
- def complex_extension(cls,field):
- r"""
- The complex field that we embed/unembed, as an extension
- of the given ``field``.
- """
- if field in cls._complex_extension:
- return cls._complex_extension[field]
-
- # Sage doesn't know how to adjoin the complex "i" (the root of
- # x^2 + 1) to a field in a general way. Here, we just enumerate
- # all of the cases that I have cared to support so far.
- 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
- elif not field.is_exact():
- # RDF or RR
- F = field.complex_field()
- else:
- # Works for QQ and... maybe some other fields.
- R = PolynomialRing(field, 'z')
- z = R.gen()
- F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
-
- cls._complex_extension[field] = F
- return F
-
- @staticmethod
- def dimension_over_reals():
- return 2
-
- @classmethod
- def real_embed(cls,M):
- """
- Embed the n-by-n complex matrix ``M`` into the space of real
- matrices of size 2n-by-2n via the map the sends each entry `z = a +
- bi` to the block matrix ``[[a,b],[-b,a]]``.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
-
- EXAMPLES::
-
- sage: F = QuadraticField(-1, 'I')
- sage: x1 = F(4 - 2*i)
- sage: x2 = F(1 + 2*i)
- sage: x3 = F(-i)
- sage: x4 = F(6)
- sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
- sage: ComplexMatrixEJA.real_embed(M)
- [ 4 -2| 1 2]
- [ 2 4|-2 1]
- [-----+-----]
- [ 0 -1| 6 0]
- [ 1 0| 0 6]
-
- TESTS:
-
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(3)
- sage: F = QuadraticField(-1, 'I')
- sage: X = random_matrix(F, n)
- sage: Y = random_matrix(F, n)
- sage: Xe = ComplexMatrixEJA.real_embed(X)
- sage: Ye = ComplexMatrixEJA.real_embed(Y)
- sage: XYe = ComplexMatrixEJA.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
-
- """
- super().real_embed(M)
- n = M.nrows()
-
- # We don't need any adjoined elements...
- field = M.base_ring().base_ring()
-
- blocks = []
- for z in M.list():
- a = z.real()
- b = z.imag()
- blocks.append(matrix(field, 2, [ [ a, b],
- [-b, a] ]))
-
- return matrix.block(field, n, blocks)
-
-
- @classmethod
- def real_unembed(cls,M):
- """
- The inverse of _embed_complex_matrix().
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
-
- EXAMPLES::
-
- sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [ 9, 10, 11, 12],
- ....: [-10, 9, -12, 11] ])
- sage: ComplexMatrixEJA.real_unembed(A)
- [ 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: M = random_matrix(F, 3)
- sage: Me = ComplexMatrixEJA.real_embed(M)
- sage: ComplexMatrixEJA.real_unembed(Me) == M
- True
-
- """
- super().real_unembed(M)
- n = ZZ(M.nrows())
- d = cls.dimension_over_reals()
- F = cls.complex_extension(M.base_ring())
- i = F.gen()
-
- # Go top-left to bottom-right (reading order), converting every
- # 2-by-2 block we see to a single complex element.
- elements = []
- for k in range(n/d):
- for j in range(n/d):
- submat = M[d*k:d*k+d,d*j:d*j+d]
- if submat[0,0] != submat[1,1]:
- raise ValueError('bad on-diagonal submatrix')
- if submat[0,1] != -submat[1,0]:
- raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0] + submat[0,1]*i
- elements.append(z)
-
- return matrix(F, n/d, elements)
-
-
-class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, ComplexMatrixEJA):
+class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = ComplexHermitianEJA._denormalized_basis(n,ZZ)
- sage: all( M.is_symmetric() for M in B)
+ sage: B = ComplexHermitianEJA._denormalized_basis(n,QQ)
+ sage: all( M.is_hermitian() for M in B)
True
"""
- R = PolynomialRing(ZZ, 'z')
- z = R.gen()
- F = ZZ.extension(z**2 + 1, 'I')
- I = F.gen(1)
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(n, scalars=field)
+ es = A.entry_algebra_gens()
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- Eij = matrix.zero(F,n)
+ basis = []
for i in range(n):
for j in range(i+1):
- # "build" E_ij
- Eij[i,j] = 1
if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
+ E_ii = A.monomial( (i,j,es[0]) )
+ basis.append(E_ii)
else:
- # The second one has a minus because it's conjugated.
- Eij[j,i] = 1 # Eij = Eij + Eij.transpose()
- Sij_real = cls.real_embed(Eij)
- S.append(Sij_real)
- # 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 the entries, we can drop back to the
- # desired real "field" instead of the extension "F".
- return tuple( s.change_ring(field) for s in S )
+ for e in es:
+ E_ij = A.monomial( (i,j,e) )
+ ec = e.conjugate()
+ # If the conjugate has a negative sign in front
+ # of it, (j,i,ec) won't be a monomial!
+ if (j,i,ec) in A.indices():
+ E_ij += A.monomial( (j,i,ec) )
+ else:
+ E_ij -= A.monomial( (j,i,-ec) )
+ basis.append(E_ij)
+ return tuple( basis )
+
+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+ TESTS::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ """
+ return (X*Y).trace().real()
def __init__(self, n, field=AA, **kwargs):
# We know this is a valid EJA, but will double-check
# because the MatrixEJA is not presently a subclass of the
# FDEJA class that defines rank() and one().
self.rank.set_cache(n)
- idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
+ idV = self.matrix_space().one()
self.one.set_cache(self(idV))
@staticmethod
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
-
- # A manual dictionary-cache for the quaternion_extension() method,
- # since apparently @classmethods can't also be @cached_methods.
- _quaternion_extension = {}
-
- @classmethod
- def quaternion_extension(cls,field):
- r"""
- The quaternion field that we embed/unembed, as an extension
- of the given ``field``.
- """
- if field in cls._quaternion_extension:
- return cls._quaternion_extension[field]
-
- Q = QuaternionAlgebra(field,-1,-1)
-
- cls._quaternion_extension[field] = Q
- return Q
-
- @staticmethod
- def dimension_over_reals():
- return 4
-
- @classmethod
- def real_embed(cls,M):
- """
- Embed the n-by-n quaternion matrix ``M`` into the space of real
- matrices of size 4n-by-4n by first sending each quaternion entry `z
- = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
- c+di],[-c + di, a-bi]]`, and then embedding those into a real
- matrix.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
-
- EXAMPLES::
-
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: i,j,k = Q.gens()
- sage: x = 1 + 2*i + 3*j + 4*k
- sage: M = matrix(Q, 1, [[x]])
- sage: QuaternionMatrixEJA.real_embed(M)
- [ 1 2 3 4]
- [-2 1 -4 3]
- [-3 4 1 -2]
- [-4 -3 2 1]
-
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(2)
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: X = random_matrix(Q, n)
- sage: Y = random_matrix(Q, n)
- sage: Xe = QuaternionMatrixEJA.real_embed(X)
- sage: Ye = QuaternionMatrixEJA.real_embed(Y)
- sage: XYe = QuaternionMatrixEJA.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
-
- """
- super().real_embed(M)
- quaternions = M.base_ring()
- n = M.nrows()
-
- F = QuadraticField(-1, 'I')
- i = F.gen()
-
- blocks = []
- for z in M.list():
- t = z.coefficient_tuple()
- a = t[0]
- b = t[1]
- c = t[2]
- d = t[3]
- cplxM = matrix(F, 2, [[ a + b*i, c + d*i],
- [-c + d*i, a - b*i]])
- realM = ComplexMatrixEJA.real_embed(cplxM)
- blocks.append(realM)
-
- # We should have real entries by now, so use the realest field
- # we've got for the return value.
- return matrix.block(quaternions.base_ring(), n, blocks)
-
-
- @classmethod
- def real_unembed(cls,M):
- """
- The inverse of _embed_quaternion_matrix().
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
-
- EXAMPLES::
-
- sage: M = matrix(QQ, [[ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [-3, 4, 1, -2],
- ....: [-4, -3, 2, 1]])
- sage: QuaternionMatrixEJA.real_unembed(M)
- [1 + 2*i + 3*j + 4*k]
-
- TESTS:
-
- Unembedding is the inverse of embedding::
-
- sage: set_random_seed()
- sage: Q = QuaternionAlgebra(QQ, -1, -1)
- sage: M = random_matrix(Q, 3)
- sage: Me = QuaternionMatrixEJA.real_embed(M)
- sage: QuaternionMatrixEJA.real_unembed(Me) == M
- True
-
- """
- super().real_unembed(M)
- n = ZZ(M.nrows())
- d = cls.dimension_over_reals()
-
- # Use the base ring of the matrix to ensure that its entries can be
- # multiplied by elements of the quaternion algebra.
- Q = cls.quaternion_extension(M.base_ring())
- i,j,k = Q.gens()
-
- # Go top-left to bottom-right (reading order), converting every
- # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
- # quaternion block.
- elements = []
- for l in range(n/d):
- for m in range(n/d):
- submat = ComplexMatrixEJA.real_unembed(
- M[d*l:d*l+d,d*m:d*m+d] )
- if submat[0,0] != submat[1,1].conjugate():
- raise ValueError('bad on-diagonal submatrix')
- if submat[0,1] != -submat[1,0].conjugate():
- raise ValueError('bad off-diagonal submatrix')
- 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/d, elements)
-
-
-class QuaternionHermitianEJA(RationalBasisEJA,
- ConcreteEJA,
- QuaternionMatrixEJA):
+class QuaternionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = QuaternionHermitianEJA._denormalized_basis(n,ZZ)
- sage: all( M.is_symmetric() for M in B )
+ sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
+ sage: all( M.is_hermitian() for M in B )
True
"""
- Q = QuaternionAlgebra(QQ,-1,-1)
- I,J,K = Q.gens()
+ from mjo.hurwitz import QuaternionMatrixAlgebra
+ A = QuaternionMatrixAlgebra(n, scalars=field)
+ es = A.entry_algebra_gens()
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- Eij = matrix.zero(Q,n)
+ basis = []
for i in range(n):
for j in range(i+1):
- # "build" E_ij
- Eij[i,j] = 1
if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
+ E_ii = A.monomial( (i,j,es[0]) )
+ basis.append(E_ii)
else:
- # The second, third, and fourth ones have a minus
- # because they're conjugated.
- # Eij = Eij + Eij.transpose()
- Eij[j,i] = 1
- Sij_real = cls.real_embed(Eij)
- S.append(Sij_real)
- # Eij = I*(Eij - Eij.transpose())
- Eij[i,j] = I
- Eij[j,i] = -I
- Sij_I = cls.real_embed(Eij)
- S.append(Sij_I)
- # Eij = J*(Eij - Eij.transpose())
- Eij[i,j] = J
- Eij[j,i] = -J
- Sij_J = cls.real_embed(Eij)
- S.append(Sij_J)
- # 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 the entries, we can drop back to the
- # desired real "field" instead of the quaternion algebra "Q".
- return tuple( s.change_ring(field) for s in S )
+ for e in es:
+ E_ij = A.monomial( (i,j,e) )
+ ec = e.conjugate()
+ # If the conjugate has a negative sign in front
+ # of it, (j,i,ec) won't be a monomial!
+ if (j,i,ec) in A.indices():
+ E_ij += A.monomial( (j,i,ec) )
+ else:
+ E_ij -= A.monomial( (j,i,-ec) )
+ basis.append(E_ij)
+
+ return tuple( basis )
+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ Overload the superclass method because the quaternions are weird
+ and we need to use ``coefficient_tuple()`` to get the realpart.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
+
+ TESTS::
+
+ sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ """
+ return (X*Y).trace().coefficient_tuple()[0]
+
def __init__(self, n, field=AA, **kwargs):
# We know this is a valid EJA, but will double-check
# if the user passes check_axioms=True.
# because the MatrixEJA is not presently a subclass of the
# FDEJA class that defines rank() and one().
self.rank.set_cache(n)
- idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
+ idV = self.matrix_space().one()
self.one.set_cache(self(idV))
27
"""
- from mjo.octonions import OctonionMatrixAlgebra
- MS = OctonionMatrixAlgebra(n, scalars=field)
- es = MS.entry_algebra().gens()
+ from mjo.hurwitz import OctonionMatrixAlgebra
+ A = OctonionMatrixAlgebra(n, scalars=field)
+ es = A.entry_algebra_gens()
basis = []
for i in range(n):
for j in range(i+1):
if i == j:
- E_ii = MS.monomial( (i,j,es[0]) )
+ E_ii = A.monomial( (i,j,es[0]) )
basis.append(E_ii)
else:
for e in es:
- E_ij = MS.monomial( (i,j,e) )
+ E_ij = A.monomial( (i,j,e) )
ec = e.conjugate()
# If the conjugate has a negative sign in front
# of it, (j,i,ec) won't be a monomial!
- if (j,i,ec) in MS.indices():
- E_ij += MS.monomial( (j,i,ec) )
+ if (j,i,ec) in A.indices():
+ E_ij += A.monomial( (j,i,ec) )
else:
- E_ij -= MS.monomial( (j,i,-ec) )
+ E_ij -= A.monomial( (j,i,-ec) )
basis.append(E_ij)
return tuple( basis )
-2
"""
- return (X*Y).trace().real().coefficient(0)
+ return (X*Y).trace().coefficient(0)
class AlbertEJA(OctonionHermitianEJA):
class HadamardEJA(RationalBasisEJA, ConcreteEJA):
"""
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
+ Return the Euclidean Jordan algebra on `R^n` with the Hadamard
+ (pointwise real-number multiplication) Jordan product and the
+ usual inner-product.
- Note: this is nothing more than the Cartesian product of ``n``
- copies of the spin algebra. Once Cartesian product algebras
- are implemented, this can go.
+ This is nothing more than the Cartesian product of ``n`` copies of
+ the one-dimensional Jordan spin algebra, and is the most common
+ example of a non-simple Euclidean Jordan algebra.
SETUP::
sage: HadamardEJA(3, prefix='r').gens()
(r0, r1, r2)
-
"""
def __init__(self, n, field=AA, **kwargs):
if n == 0:
projects / sage.d.git / blobdiff