X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=58d5ce03a495ddf2eb7a9c4cba0042145be0c037;hb=bc418347d7267c8eaee6f3989db000b2f3b77165;hp=eca279f68156ef1cd8e407f2f8ad25661af1e517;hpb=3f7be6593b9eb870406dbb511feaf85c58c75121;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index eca279f..58d5ce0 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -750,23 +750,57 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): 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() @@ -774,6 +808,27 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): 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:: @@ -785,6 +840,15 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): 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. @@ -1201,6 +1265,7 @@ class RationalBasisEJA(FiniteDimensionalEJA): if not all( all(b_i in QQ for b_i in b.list()) for b in basis ): raise TypeError("basis not rational") + self._rational_algebra = None if field is not QQ: # There's no point in constructing the extra algebra if this # one is already rational. @@ -1602,6 +1667,38 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA): class ComplexMatrixEJA(MatrixEJA): + # 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 @@ -1656,9 +1753,10 @@ class ComplexMatrixEJA(MatrixEJA): 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) @@ -1697,26 +1795,7 @@ class ComplexMatrixEJA(MatrixEJA): super(ComplexMatrixEJA,cls).real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() - - # If "M" was normalized, its base ring might have roots - # adjoined and they can stick around after unembedding. - field = M.base_ring() - R = PolynomialRing(field, 'z') - z = R.gen() - - # 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. - F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) + F = cls.complex_extension(M.base_ring()) i = F.gen() # Go top-left to bottom-right (reading order), converting every @@ -1812,7 +1891,6 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): 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 @@ -1830,18 +1908,27 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): # * 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". @@ -1877,6 +1964,25 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): return cls(n, **kwargs) class QuaternionMatrixEJA(MatrixEJA): + + # 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 @@ -1981,8 +2087,7 @@ class QuaternionMatrixEJA(MatrixEJA): # Use the base ring of the matrix to ensure that its entries can be # multiplied by elements of the quaternion algebra. - field = M.base_ring() - Q = QuaternionAlgebra(field,-1,-1) + Q = cls.quaternion_extension(M.base_ring()) i,j,k = Q.gens() # Go top-left to bottom-right (reading order), converting every @@ -2097,23 +2202,39 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): # * 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".