X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=ad4e2c7832f961b4893814551657a7a823ac083d;hb=3940dadefa5ede86fd81917ace7d145e4d3f0da9;hp=618e09cf41d58eec7ade00aca846eb9677f468ae;hpb=21fa036e86711c6c28b6d89af2b1bfe4ceb24b29;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 618e09c..ad4e2c7 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -1,9 +1,53 @@ """ -Euclidean Jordan Algebras. These are formally-real Jordan Algebras; -specifically those where u^2 + v^2 = 0 implies that u = v = 0. They -are used in optimization, and have some additional nice methods beyond -what can be supported in a general Jordan Algebra. - +Representations and constructions for Euclidean Jordan algebras. + +A Euclidean Jordan algebra is a Jordan algebra that has some +additional properties: + + 1. It is finite-dimensional. + 2. Its scalar field is the real numbers. + 3a. An inner product is defined on it, and... + 3b. That inner product is compatible with the Jordan product + in the sense that ` = ` for all elements + `x,y,z` in the algebra. + +Every Euclidean Jordan algebra is formally-real: for any two elements +`x` and `y` in the algebra, `x^{2} + y^{2} = 0` implies that `x = y = +0`. Conversely, every finite-dimensional formally-real Jordan algebra +can be made into a Euclidean Jordan algebra with an appropriate choice +of inner-product. + +Formally-real Jordan algebras were originally studied as a framework +for quantum mechanics. Today, Euclidean Jordan algebras are crucial in +symmetric cone optimization, since every symmetric cone arises as the +cone of squares in some Euclidean Jordan algebra. + +It is known that every Euclidean Jordan algebra decomposes into an +orthogonal direct sum (essentially, a Cartesian product) of simple +algebras, and that moreover, up to Jordan-algebra isomorphism, there +are only five families of simple algebras. We provide constructions +for these simple algebras: + + * :class:`BilinearFormEJA` + * :class:`RealSymmetricEJA` + * :class:`ComplexHermitianEJA` + * :class:`QuaternionHermitianEJA` + +Missing from this list is the algebra of three-by-three octononion +Hermitian matrices, as there is (as of yet) no implementation of the +octonions in SageMath. In addition to these, we provide two other +example constructions, + + * :class:`HadamardEJA` + * :class:`TrivialEJA` + +The Jordan spin algebra is a bilinear form algebra where the bilinear +form is the identity. The Hadamard EJA is simply a Cartesian product +of one-dimensional spin algebras. And last but not least, the trivial +EJA is exactly what you think. Cartesian products of these are also +supported using the usual ``cartesian_product()`` function; as a +result, we support (up to isomorphism) all Euclidean Jordan algebras +that don't involve octonions. SETUP:: @@ -13,7 +57,6 @@ EXAMPLES:: sage: random_eja() Euclidean Jordan algebra of dimension... - """ from itertools import repeat @@ -41,24 +84,50 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): INPUT: - - basis -- a tuple of basis elements in "matrix form," which - must be the same form as the arguments to ``jordan_product`` - and ``inner_product``. In reality, "matrix form" can be either - vectors, matrices, or a Cartesian product (ordered tuple) - of vectors or matrices. All of these would ideally be vector - spaces in sage with no special-casing needed; but in reality - we turn vectors into column-matrices and Cartesian products - `(a,b)` into column matrices `(a,b)^{T}` after converting - `a` and `b` themselves. - - - jordan_product -- function of two elements (in matrix form) - that returns their jordan product in this algebra; this will - be applied to ``basis`` to compute a multiplication table for - the algebra. - - - inner_product -- function of two elements (in matrix form) that - returns their inner product. This will be applied to ``basis`` to - compute an inner-product table (basically a matrix) for this algebra. + - ``basis`` -- a tuple; a tuple of basis elements in "matrix + form," which must be the same form as the arguments to + ``jordan_product`` and ``inner_product``. In reality, "matrix + form" can be either vectors, matrices, or a Cartesian product + (ordered tuple) of vectors or matrices. All of these would + ideally be vector spaces in sage with no special-casing + needed; but in reality we turn vectors into column-matrices + and Cartesian products `(a,b)` into column matrices + `(a,b)^{T}` after converting `a` and `b` themselves. + + - ``jordan_product`` -- a function; afunction of two ``basis`` + elements (in matrix form) that returns their jordan product, + also in matrix form; this will be applied to ``basis`` to + compute a multiplication table for the algebra. + + - ``inner_product`` -- a function; a function of two ``basis`` + elements (in matrix form) that returns their inner + product. This will be applied to ``basis`` to compute an + inner-product table (basically a matrix) for this algebra. + + - ``field`` -- a subfield of the reals (default: ``AA``); the scalar + field for the algebra. + + - ``orthonormalize`` -- boolean (default: ``True``); whether or + not to orthonormalize the basis. Doing so is expensive and + generally rules out using the rationals as your ``field``, but + is required for spectral decompositions. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS: + + We should compute that an element subalgebra is associative even + if we circumvent the element method:: + + sage: set_random_seed() + sage: J = random_eja(field=QQ,orthonormalize=False) + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: basis = tuple(b.superalgebra_element() for b in A.basis()) + sage: J.subalgebra(basis, orthonormalize=False).is_associative() + True """ Element = FiniteDimensionalEJAElement @@ -69,7 +138,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): inner_product, field=AA, orthonormalize=True, - associative=False, + associative=None, cartesian_product=False, check_field=True, check_axioms=True, @@ -126,7 +195,20 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() + category = category.WithBasis().Unital().Commutative() + + if associative is None: + # We should figure it out. As with check_axioms, we have to do + # this without the help of the _jordan_product_is_associative() + # method because we need to know the category before we + # initialize the algebra. + associative = all( jordan_product(jordan_product(bi,bj),bk) + == + jordan_product(bi,jordan_product(bj,bk)) + for bi in basis + for bj in basis + for bk in basis) + if associative: # Element subalgebras can take advantage of this. category = category.Associative() @@ -257,6 +339,35 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): def product_on_basis(self, i, j): + r""" + Returns the Jordan product of the `i` and `j`th basis elements. + + This completely defines the Jordan product on the algebra, and + is used direclty by our superclass machinery to implement + :meth:`product`. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: n = J.dimension() + sage: ei = J.zero() + sage: ej = J.zero() + sage: ei_ej = J.zero()*J.zero() + sage: if n > 0: + ....: i = ZZ.random_element(n) + ....: j = ZZ.random_element(n) + ....: ei = J.gens()[i] + ....: ej = J.gens()[j] + ....: ei_ej = J.product_on_basis(i,j) + sage: ei*ej == ei_ej + True + + """ # We only stored the lower-triangular portion of the # multiplication table. if j <= i: @@ -341,6 +452,18 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): """ return "Associative" in self.category().axioms() + def _is_commutative(self): + r""" + Whether or not this algebra's multiplication table is commutative. + + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. + """ + return all( self.product_on_basis(i,j) == self.product_on_basis(i,j) + for i in range(self.dimension()) + for j in range(self.dimension()) ) + def _is_jordanian(self): r""" Whether or not this algebra's multiplication table respects the @@ -348,7 +471,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): We only check one arrangement of `x` and `y`, so for a ``True`` result to be truly true, you should also check - :meth:`is_commutative`. This method should of course always + :meth:`_is_commutative`. This method should of course always return ``True``, unless this algebra was constructed with ``check_axioms=False`` and passed an invalid multiplication table. """ @@ -358,6 +481,92 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): for i in range(self.dimension()) for j in range(self.dimension()) ) + def _jordan_product_is_associative(self): + r""" + Return whether or not this algebra's Jordan product is + associative; that is, whether or not `x*(y*z) = (x*y)*z` + for all `x,y,x`. + + This method should agree with :meth:`is_associative` unless + you lied about the value of the ``associative`` parameter + when you constructed the algebra. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA) + + EXAMPLES:: + + sage: J = RealSymmetricEJA(4, orthonormalize=False) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A._jordan_product_is_associative() + True + + :: + + sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A._jordan_product_is_associative() + True + + :: + + sage: J = QuaternionHermitianEJA(2) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A._jordan_product_is_associative() + True + + TESTS: + + The values we've presupplied to the constructors agree with + the computation:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.is_associative() == J._jordan_product_is_associative() + True + + """ + R = self.base_ring() + + # Used to check whether or not something is zero. + epsilon = R.zero() + if not R.is_exact(): + # I don't know of any examples that make this magnitude + # necessary because I don't know how to make an + # associative algebra when the element subalgebra + # construction is unreliable (as it is over RDF; we can't + # find the degree of an element because we can't compute + # the rank of a matrix). But even multiplication of floats + # is non-associative, so *some* epsilon is needed... let's + # just take the one from _inner_product_is_associative? + epsilon = 1e-15 + + for i in range(self.dimension()): + for j in range(self.dimension()): + for k in range(self.dimension()): + x = self.gens()[i] + y = self.gens()[j] + z = self.gens()[k] + diff = (x*y)*z - x*(y*z) + + if diff.norm() > epsilon: + return False + + return True + def _inner_product_is_associative(self): r""" Return whether or not this algebra's inner product `B` is @@ -367,11 +576,14 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): this algebra was constructed with ``check_axioms=False`` and passed an invalid Jordan or inner-product. """ + R = self.base_ring() - # Used to check whether or not something is zero in an inexact - # ring. This number is sufficient to allow the construction of - # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. - epsilon = 1e-16 + # Used to check whether or not something is zero. + epsilon = R.zero() + if not R.is_exact(): + # This choice is sufficient to allow the construction of + # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. + epsilon = 1e-15 for i in range(self.dimension()): for j in range(self.dimension()): @@ -381,12 +593,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): z = self.gens()[k] diff = (x*y).inner_product(z) - x.inner_product(y*z) - if self.base_ring().is_exact(): - if diff != 0: - return False - else: - if diff.abs() > epsilon: - return False + if diff.abs() > epsilon: + return False return True @@ -400,7 +608,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, ....: HadamardEJA, ....: RealSymmetricEJA) @@ -433,27 +642,31 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): TESTS: - Ensure that we can convert any element of the two non-matrix - simple algebras (whose matrix representations are columns) - back and forth faithfully:: + Ensure that we can convert any element back and forth + faithfully between its matrix and algebra representations:: sage: set_random_seed() - sage: J = HadamardEJA.random_instance() - sage: x = J.random_element() - sage: J(x.to_vector().column()) == x - True - sage: J = JordanSpinEJA.random_instance() + sage: J = random_eja() sage: x = J.random_element() - sage: J(x.to_vector().column()) == x + sage: J(x.to_matrix()) == x True + We cannot coerce elements between algebras just because their + matrix representations are compatible:: + + sage: J1 = HadamardEJA(3) + sage: J2 = JordanSpinEJA(3) + sage: J2(J1.one()) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + sage: J1(J2.zero()) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra """ 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. - return self.zero() - elif elt in self.base_ring(): + if elt in self.base_ring(): # Ensure that no base ring -> algebra coercion is performed # by this method. There's some stupidity in sage that would # otherwise propagate to this method; for example, sage thinks @@ -461,9 +674,11 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): raise ValueError(msg) try: + # Try to convert a vector into a column-matrix... elt = elt.column() except (AttributeError, TypeError): - # Try to convert a vector into a column-matrix + # and ignore failure, because we weren't really expecting + # a vector as an argument anyway. pass if elt not in self.matrix_space(): @@ -477,7 +692,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # could be QQ instead of QQbar. # # And, we also have to handle Cartesian product bases (when - # the matric basis consists of tuples) here. The "good news" + # the matrix basis consists of tuples) here. The "good news" # is that we're already converting everything to long vectors, # and that strategy works for tuples as well. # @@ -787,12 +1002,49 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): we think of them as matrices (including column vectors of the appropriate size). - Generally this will be an `n`-by-`1` column-vector space, + "By default" this will be an `n`-by-`1` column-matrix space, except when the algebra is trivial. There it's `n`-by-`n` (where `n` is zero), to ensure that two elements of the matrix - space (empty matrices) can be multiplied. + space (empty matrices) can be multiplied. For algebras of + matrices, this returns the space in which their + real embeddings live. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: JordanSpinEJA, + ....: QuaternionHermitianEJA, + ....: TrivialEJA) + + EXAMPLES: + + By default, the matrix representation is just a column-matrix + equivalent to the vector representation:: + + sage: J = JordanSpinEJA(3) + sage: J.matrix_space() + Full MatrixSpace of 3 by 1 dense matrices over Algebraic + Real Field + + The matrix representation in the trivial algebra is + zero-by-zero instead of the usual `n`-by-one:: + + sage: J = TrivialEJA() + sage: J.matrix_space() + Full MatrixSpace of 0 by 0 dense matrices over Algebraic + Real Field + + The matrix space for complex/quaternion Hermitian matrix EJA + is the space in which their real-embeddings live, not the + original complex/quaternion matrix space:: + + sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) + sage: J.matrix_space() + 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 - Matrix algebras override this with something more useful. """ if self.is_trivial(): return MatrixSpace(self.base_ring(), 0) @@ -1350,6 +1602,13 @@ 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") + super().__init__(basis, + jordan_product, + inner_product, + field=field, + check_field=check_field, + **kwargs) + self._rational_algebra = None if field is not QQ: # There's no point in constructing the extra algebra if this @@ -1363,17 +1622,11 @@ class RationalBasisEJA(FiniteDimensionalEJA): jordan_product, inner_product, field=QQ, + associative=self.is_associative(), orthonormalize=False, check_field=False, check_axioms=False) - super().__init__(basis, - jordan_product, - inner_product, - field=field, - check_field=check_field, - **kwargs) - @cached_method def _charpoly_coefficients(self): r""" @@ -1646,9 +1899,9 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA): In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, field=RDF) + sage: RealSymmetricEJA(2, field=RDF, check_axioms=True) Euclidean Jordan algebra of dimension 3 over Real Double Field - sage: RealSymmetricEJA(2, field=RR) + sage: RealSymmetricEJA(2, field=RR, check_axioms=True) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1737,10 +1990,15 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA): # if the user passes check_axioms=True. if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n), - self.jordan_product, - self.trace_inner_product, - **kwargs) + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + associative=associative, + **kwargs) # TODO: this could be factored out somehow, but is left here # because the MatrixEJA is not presently a subclass of the @@ -1830,7 +2088,7 @@ class ComplexMatrixEJA(MatrixEJA): True """ - super(ComplexMatrixEJA,cls).real_embed(M) + super().real_embed(M) n = M.nrows() # We don't need any adjoined elements... @@ -1877,7 +2135,7 @@ class ComplexMatrixEJA(MatrixEJA): True """ - super(ComplexMatrixEJA,cls).real_unembed(M) + super().real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() F = cls.complex_extension(M.base_ring()) @@ -1914,9 +2172,9 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, field=RDF) + sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True) Euclidean Jordan algebra of dimension 4 over Real Double Field - sage: ComplexHermitianEJA(2, field=RR) + sage: ComplexHermitianEJA(2, field=RR, check_axioms=True) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -2025,10 +2283,15 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): # if the user passes check_axioms=True. if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n), - self.jordan_product, - self.trace_inner_product, - **kwargs) + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + associative=associative, + **kwargs) # TODO: this could be factored out somehow, but is left here # because the MatrixEJA is not presently a subclass of the # FDEJA class that defines rank() and one(). @@ -2111,7 +2374,7 @@ class QuaternionMatrixEJA(MatrixEJA): True """ - super(QuaternionMatrixEJA,cls).real_embed(M) + super().real_embed(M) quaternions = M.base_ring() n = M.nrows() @@ -2166,7 +2429,7 @@ class QuaternionMatrixEJA(MatrixEJA): True """ - super(QuaternionMatrixEJA,cls).real_unembed(M) + super().real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() @@ -2211,9 +2474,9 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, field=RDF) + sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True) Euclidean Jordan algebra of dimension 6 over Real Double Field - sage: QuaternionHermitianEJA(2, field=RR) + sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -2331,10 +2594,16 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): # if the user passes check_axioms=True. if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n), - self.jordan_product, - self.trace_inner_product, - **kwargs) + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + associative=associative, + **kwargs) + # TODO: this could be factored out somehow, but is left here # because the MatrixEJA is not presently a subclass of the # FDEJA class that defines rank() and one(). @@ -2560,10 +2829,17 @@ class BilinearFormEJA(ConcreteEJA): n = B.nrows() column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() ) - super(BilinearFormEJA, self).__init__(column_basis, - jordan_product, - inner_product, - **kwargs) + + # TODO: I haven't actually checked this, but it seems legit. + associative = False + if n <= 2: + associative = True + + super().__init__(column_basis, + jordan_product, + inner_product, + associative=associative, + **kwargs) # The rank of this algebra is two, unless we're in a # one-dimensional ambient space (because the rank is bounded @@ -2668,7 +2944,7 @@ class JordanSpinEJA(BilinearFormEJA): # But also don't pass check_field=False here, because the user # can pass in a field! - super(JordanSpinEJA, self).__init__(B, **kwargs) + super().__init__(B, **kwargs) @staticmethod def _max_random_instance_size(): @@ -2726,10 +3002,12 @@ class TrivialEJA(ConcreteEJA): if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(TrivialEJA, self).__init__(basis, - jordan_product, - inner_product, - **kwargs) + super().__init__(basis, + jordan_product, + inner_product, + associative=True, + **kwargs) + # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. self.rank.set_cache(0) @@ -3132,4 +3410,42 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA +class RationalBasisCartesianProductEJA(CartesianProductEJA, + RationalBasisEJA): + r""" + A separate class for products of algebras for which we know a + rational basis. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + This gives us fast characteristic polynomial computations in + product algebras, too:: + + + sage: J1 = JordanSpinEJA(2) + sage: J2 = RealSymmetricEJA(3) + sage: J = cartesian_product([J1,J2]) + sage: J.characteristic_polynomial_of().degree() + 5 + sage: J.rank() + 5 + + """ + def __init__(self, algebras, **kwargs): + CartesianProductEJA.__init__(self, algebras, **kwargs) + + self._rational_algebra = None + if self.vector_space().base_field() is not QQ: + self._rational_algebra = cartesian_product([ + r._rational_algebra for r in algebras + ]) + + +RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA + random_eja = ConcreteEJA.random_instance