X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=a51da3e03fc753e33079fde0debdf6ee12a29b3c;hb=9a6de831e947f663cacf56e409e99ec3aa086c62;hp=933603a40a7be27feb891efa4b53f3f82388f239;hpb=1b6878559ad75aa0064503a962c8c183e13ab91a;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 933603a..a51da3e 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -54,7 +54,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J(1) Traceback (most recent call last): ... - ValueError: not a naturally-represented algebra element + ValueError: not an element of this algebra """ return None @@ -64,7 +64,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): mult_table, prefix='e', category=None, - natural_basis=None, + matrix_basis=None, check_field=True, check_axioms=True): """ @@ -115,7 +115,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): if not all( len(l) == n for l in mult_table ): raise ValueError("multiplication table is not square") - self._natural_basis = natural_basis + self._matrix_basis = matrix_basis if category is None: category = MagmaticAlgebras(field).FiniteDimensional() @@ -149,7 +149,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): def _element_constructor_(self, elt): """ - Construct an element of this algebra from its natural + Construct an element of this algebra from its vector or matrix representation. This gets called only after the parent element _call_ method @@ -177,13 +177,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J(A) Traceback (most recent call last): ... - ArithmeticError: vector is not in free module + ValueError: not an element of this algebra TESTS: Ensure that we can convert any element of the two non-matrix - simple algebras (whose natural representations are their usual - vector representations) back and forth faithfully:: + simple algebras (whose matrix representations are columns) + back and forth faithfully:: sage: set_random_seed() sage: J = HadamardEJA.random_instance() @@ -194,9 +194,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x = J.random_element() sage: J(x.to_vector().column()) == x True - """ - msg = "not a naturally-represented algebra element" + msg = "not an element of this algebra" if elt == 0: # The superclass implementation of random_element() # needs to be able to coerce "0" into the algebra. @@ -208,20 +207,23 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # that the integer 3 belongs to the space of 2-by-2 matrices. raise ValueError(msg) - natural_basis = self.natural_basis() - basis_space = natural_basis[0].matrix_space() - if elt not in basis_space: + if elt not in self.matrix_space(): raise ValueError(msg) # Thanks for nothing! Matrix spaces aren't vector spaces in - # Sage, so we have to figure out its natural-basis coordinates + # Sage, so we have to figure out its matrix-basis coordinates # ourselves. We use the basis space's ring instead of the # element's ring because the basis space might be an algebraic # closure whereas the base ring of the 3-by-3 identity matrix # could be QQ instead of QQbar. - V = VectorSpace(basis_space.base_ring(), elt.nrows()*elt.ncols()) - W = V.span_of_basis( _mat2vec(s) for s in natural_basis ) - coords = W.coordinate_vector(_mat2vec(elt)) + V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols()) + W = V.span_of_basis( _mat2vec(s) for s in self.matrix_basis() ) + + try: + coords = W.coordinate_vector(_mat2vec(elt)) + except ArithmeticError: # vector is not in free module + raise ValueError(msg) + return self.from_vector(coords) def _repr_(self): @@ -510,18 +512,33 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return table(M, header_row=True, header_column=True, frame=True) - def natural_basis(self): + def matrix_basis(self): """ - Return a more-natural representation of this algebra's basis. + Return an (often more natural) representation of this algebras + basis as an ordered tuple of matrices. + + Every finite-dimensional Euclidean Jordan Algebra is a, up to + Jordan isomorphism, a direct sum of five simple + algebras---four of which comprise Hermitian matrices. And the + last type of algebra can of course be thought of as `n`-by-`1` + column matrices (ambiguusly called column vectors) to avoid + special cases. As a result, matrices (and column vectors) are + a natural representation format for Euclidean Jordan algebra + elements. - Every finite-dimensional Euclidean Jordan Algebra is a direct - sum of five simple algebras, four of which comprise Hermitian - matrices. This method returns the original "natural" basis - for our underlying vector space. (Typically, the natural basis - is used to construct the multiplication table in the first place.) + But, when we construct an algebra from a basis of matrices, + those matrix representations are lost in favor of coordinate + vectors *with respect to* that basis. We could eventually + convert back if we tried hard enough, but having the original + representations handy is valuable enough that we simply store + them and return them from this method. - Note that this will always return a matrix. The standard basis - in `R^n` will be returned as `n`-by-`1` column matrices. + Why implement this for non-matrix algebras? Avoiding special + cases for the :class:`BilinearFormEJA` pays with simplicity in + its own right. But mainly, we would like to be able to assume + that elements of a :class:`DirectSumEJA` can be displayed + nicely, without having to have special classes for direct sums + one of whose components was a matrix algebra. SETUP:: @@ -533,7 +550,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = RealSymmetricEJA(2) sage: J.basis() Finite family {0: e0, 1: e1, 2: e2} - sage: J.natural_basis() + sage: J.matrix_basis() ( [1 0] [ 0 0.7071067811865475?] [0 0] [0 0], [0.7071067811865475? 0], [0 1] @@ -544,36 +561,38 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = JordanSpinEJA(2) sage: J.basis() Finite family {0: e0, 1: e1} - sage: J.natural_basis() + sage: J.matrix_basis() ( [1] [0] [0], [1] ) - """ - if self._natural_basis is None: - M = self.natural_basis_space() + if self._matrix_basis is None: + M = self.matrix_space() return tuple( M(b.to_vector()) for b in self.basis() ) else: - return self._natural_basis + return self._matrix_basis - def natural_basis_space(self): + def matrix_space(self): """ - Return the matrix space in which this algebra's natural basis - elements live. + Return the matrix space in which this algebra's elements live, if + we think of them as matrices (including column vectors of the + appropriate size). Generally this will be an `n`-by-`1` column-vector space, except when the algebra is trivial. There it's `n`-by-`n` - (where `n` is zero), to ensure that two elements of the - natural basis space (empty matrices) can be multiplied. + (where `n` is zero), to ensure that two elements of the matrix + space (empty matrices) can be multiplied. + + Matrix algebras override this with something more useful. """ if self.is_trivial(): return MatrixSpace(self.base_ring(), 0) - elif self._natural_basis is None or len(self._natural_basis) == 0: + elif self._matrix_basis is None or len(self._matrix_basis) == 0: return MatrixSpace(self.base_ring(), self.dimension(), 1) else: - return self._natural_basis[0].matrix_space() + return self._matrix_basis[0].matrix_space() @cached_method @@ -700,22 +719,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Vector space of degree 6 and dimension 2... sage: J1 Euclidean Jordan algebra of dimension 3... - sage: J0.one().natural_representation() + sage: J0.one().to_matrix() [0 0 0] [0 0 0] [0 0 1] sage: orig_df = AA.options.display_format sage: AA.options.display_format = 'radical' - sage: J.from_vector(J5.basis()[0]).natural_representation() + sage: J.from_vector(J5.basis()[0]).to_matrix() [ 0 0 1/2*sqrt(2)] [ 0 0 0] [1/2*sqrt(2) 0 0] - sage: J.from_vector(J5.basis()[1]).natural_representation() + sage: J.from_vector(J5.basis()[1]).to_matrix() [ 0 0 0] [ 0 0 1/2*sqrt(2)] [ 0 1/2*sqrt(2) 0] sage: AA.options.display_format = orig_df - sage: J1.one().natural_representation() + sage: J1.one().to_matrix() [1 0 0] [0 1 0] [0 0 0] @@ -970,38 +989,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Ensure that computing the rank actually works, since the ranks of all simple algebras are known and will be cached by default:: - sage: J = HadamardEJA(4) - sage: J.rank.clear_cache() - sage: J.rank() - 4 - - :: - - sage: J = JordanSpinEJA(4) - sage: J.rank.clear_cache() - sage: J.rank() - 2 - - :: - - sage: J = RealSymmetricEJA(3) - sage: J.rank.clear_cache() - sage: J.rank() - 3 - - :: - - sage: J = ComplexHermitianEJA(2) - sage: J.rank.clear_cache() - sage: J.rank() - 2 - - :: + sage: set_random_seed() # long time + sage: J = random_eja() # long time + sage: caches = J.rank() # long time + sage: J.rank.clear_cache() # long time + sage: J.rank() == cached # long time + True - sage: J = QuaternionHermitianEJA(2) - sage: J.rank.clear_cache() - sage: J.rank() - 2 """ return len(self._charpoly_coefficients()) @@ -1101,26 +1095,25 @@ class ConcreteEuclideanJordanAlgebra: TESTS: - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: + Our basis is normalized with respect to the algebra's inner + product, unless we specify otherwise:: sage: set_random_seed() sage: J = ConcreteEuclideanJordanAlgebra.random_instance() sage: all( b.norm() == 1 for b in J.gens() ) True - Since our natural basis is normalized with respect to the natural - inner product, and since we know that this algebra is an EJA, any + Since our basis is orthonormal with respect to the algebra's inner + product, and since we know that this algebra is an EJA, any left-multiplication operator's matrix will be symmetric because - natural->EJA basis representation is an isometry and within the EJA - the operator is self-adjoint by the Jordan axiom:: + natural->EJA basis representation is an isometry and within the + EJA the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() sage: J = ConcreteEuclideanJordanAlgebra.random_instance() sage: x = J.random_element() - sage: x.operator().matrix().is_symmetric() + sage: x.operator().is_self_adjoint() True - """ @staticmethod @@ -1182,7 +1175,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt()) basis = tuple( s.change_ring(field) for s in basis ) self._basis_normalizers = tuple( - ~(self.natural_inner_product(s,s).sqrt()) for s in basis ) + ~(self.matrix_inner_product(s,s).sqrt()) for s in basis ) basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers)) # Now compute the multiplication and inner product tables. @@ -1203,7 +1196,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # HACK: ignore the error here if we don't need the # inner product (as is the case when we construct # a dummy QQ-algebra for fast charpoly coefficients. - ip_table[i][j] = self.natural_inner_product(basis[i], + ip_table[i][j] = self.matrix_inner_product(basis[i], basis[j]) except: pass @@ -1216,7 +1209,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): super(MatrixEuclideanJordanAlgebra, self).__init__(field, mult_table, - natural_basis=basis, + matrix_basis=basis, **kwargs) if algebra_dim == 0: @@ -1239,7 +1232,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # entries in a nice field already. Just compute the thing. return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients() - basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + basis = ( (b/n) for (b,n) in zip(self.matrix_basis(), self._basis_normalizers) ) # Do this over the rationals and convert back at the end. @@ -1299,7 +1292,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): raise NotImplementedError @classmethod - def natural_inner_product(cls,X,Y): + def matrix_inner_product(cls,X,Y): Xu = cls.real_unembed(X) Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() @@ -1378,9 +1371,9 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = RealSymmetricEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1579,9 +1572,9 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @classmethod - def natural_inner_product(cls,X,Y): + def matrix_inner_product(cls,X,Y): """ - Compute a natural inner product in this algebra directly from + Compute a matrix inner product in this algebra directly from its real embedding. SETUP:: @@ -1596,17 +1589,17 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: J = ComplexHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: Xe = x.natural_representation() - sage: Ye = y.natural_representation() + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() sage: X = ComplexHermitianEJA.real_unembed(Xe) sage: Y = ComplexHermitianEJA.real_unembed(Ye) sage: expected = (X*Y).trace().real() - sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual = ComplexHermitianEJA.matrix_inner_product(Xe,Ye) sage: actual == expected True """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 + return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/2 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, @@ -1647,9 +1640,9 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = ComplexHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1874,9 +1867,9 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @classmethod - def natural_inner_product(cls,X,Y): + def matrix_inner_product(cls,X,Y): """ - Compute a natural inner product in this algebra directly from + Compute a matrix inner product in this algebra directly from its real embedding. SETUP:: @@ -1891,17 +1884,17 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: J = QuaternionHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: Xe = x.natural_representation() - sage: Ye = y.natural_representation() + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() sage: X = QuaternionHermitianEJA.real_unembed(Xe) sage: Y = QuaternionHermitianEJA.real_unembed(Ye) sage: expected = (X*Y).trace().coefficient_tuple()[0] - sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual = QuaternionHermitianEJA.matrix_inner_product(Xe,Ye) sage: actual == expected True """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 + return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/4 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, @@ -1942,9 +1935,9 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = QuaternionHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True