X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=5bb4cee119e31cde1fd17accc7cc6e0bdee7f041;hb=fcdcc7a37d4b5ee83d5ff0bc98fe2b63c61a7f51;hp=1302ca18bf9988f57048cdd9035e79212aa7ca3d;hpb=858aa3653fd2e4ae8573f472cf3f0d698072c185;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 1302ca1..5bb4cee 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 @@ -33,7 +76,7 @@ from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, QuadraticField) from mjo.eja.eja_element import FiniteDimensionalEJAElement from mjo.eja.eja_operator import FiniteDimensionalEJAOperator -from mjo.eja.eja_utils import _mat2vec +from mjo.eja.eja_utils import _all2list, _mat2vec class FiniteDimensionalEJA(CombinatorialFreeModule): r""" @@ -41,16 +84,50 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): INPUT: - - basis -- a tuple of basis elements in their matrix form. + - ``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. - - 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. + SETUP:: - - 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. + 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 @@ -61,11 +138,13 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): inner_product, field=AA, orthonormalize=True, - associative=False, + associative=None, + cartesian_product=False, check_field=True, check_axioms=True, - prefix='e', - category=None): + prefix='e'): + + n = len(basis) if check_field: if not field.is_subring(RR): @@ -74,9 +153,10 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # we've specified a real embedding. raise ValueError("scalar field is not real") + from mjo.eja.eja_utils import _change_ring # If the basis given to us wasn't over the field that it's # supposed to be over, fix that. Or, you know, crash. - basis = tuple( b.change_ring(field) for b in basis ) + basis = tuple( _change_ring(b, field) for b in basis ) if check_axioms: # Check commutativity of the Jordan and inner-products. @@ -94,21 +174,38 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): raise ValueError("inner-product is not commutative") - if category is None: - category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() - if associative: - # Element subalgebras can take advantage of this. - category = category.Associative() + category = MagmaticAlgebras(field).FiniteDimensional() + 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() + if cartesian_product: + # Use join() here because otherwise we only get the + # "Cartesian product of..." and not the things themselves. + category = category.join([category, + category.CartesianProducts()]) # Call the superclass constructor so that we can use its from_vector() # method to build our multiplication table. - n = len(basis) - super().__init__(field, - range(n), - prefix=prefix, - category=category, - bracket=False) + CombinatorialFreeModule.__init__(self, + field, + range(n), + prefix=prefix, + category=category, + bracket=False) # Now comes all of the hard work. We'll be constructing an # ambient vector space V that our (vectorized) basis lives in, @@ -119,8 +216,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): degree = 0 if n > 0: - # Works on both column and square matrices... - degree = len(basis[0].list()) + degree = len(_all2list(basis[0])) # Build an ambient space that fits our matrix basis when # written out as "long vectors." @@ -134,7 +230,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # Save a copy of the un-orthonormalized basis for later. # Convert it to ambient V (vector) coordinates while we're # at it, because we'd have to do it later anyway. - deortho_vector_basis = tuple( V(b.list()) for b in basis ) + deortho_vector_basis = tuple( V(_all2list(b)) for b in basis ) from mjo.eja.eja_utils import gram_schmidt basis = tuple(gram_schmidt(basis, inner_product)) @@ -146,7 +242,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # Now create the vector space for the algebra, which will have # its own set of non-ambient coordinates (in terms of the # supplied basis). - vector_basis = tuple( V(b.list()) for b in basis ) + vector_basis = tuple( V(_all2list(b)) for b in basis ) W = V.span_of_basis( vector_basis, check=check_axioms) if orthonormalize: @@ -178,7 +274,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # The jordan product returns a matrixy answer, so we # have to convert it to the algebra coordinates. elt = jordan_product(q_i, q_j) - elt = W.coordinate_vector(V(elt.list())) + elt = W.coordinate_vector(V(_all2list(elt))) self._multiplication_table[i][j] = self.from_vector(elt) if not orthonormalize: @@ -226,6 +322,35 @@ 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.monomial(i) + ....: ej = J.monomial(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: @@ -283,11 +408,33 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): sage: y = J.random_element() sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) True + """ B = self._inner_product_matrix return (B*x.to_vector()).inner_product(y.to_vector()) + def is_associative(self): + r""" + Return whether or not this algebra's Jordan product is associative. + + SETUP:: + + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + + EXAMPLES:: + + sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False) + sage: J.is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A.is_associative() + True + + """ + return "Associative" in self.category().axioms() + def _is_commutative(self): r""" Whether or not this algebra's multiplication table is commutative. @@ -296,9 +443,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): 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()) ) + return all( x*y == y*x for x in self.gens() for y in self.gens() ) def _is_jordanian(self): r""" @@ -317,6 +462,92 @@ 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.monomial(i) + y = self.monomial(j) + z = self.monomial(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 @@ -326,11 +557,14 @@ 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()): @@ -340,12 +574,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): z = self.monomial(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 @@ -359,7 +589,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, ....: HadamardEJA, ....: RealSymmetricEJA) @@ -381,29 +612,42 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): ... ValueError: not an element of this algebra + Tuples work as well, provided that the matrix basis for the + algebra consists of them:: + + sage: J1 = HadamardEJA(3) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) ) + e1 + e5 + 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 @@ -411,9 +655,11 @@ 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(): @@ -426,14 +672,20 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # closure whereas the base ring of the 3-by-3 identity matrix # could be QQ instead of QQbar. # + # And, we also have to handle Cartesian product bases (when + # 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. + # # We pass check=False because the matrix basis is "guaranteed" # to be linearly independent... right? Ha ha. - V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols()) - W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()), + elt = _all2list(elt) + V = VectorSpace(self.base_ring(), len(elt)) + W = V.span_of_basis( (V(_all2list(s)) for s in self.matrix_basis()), check=False) try: - coords = W.coordinate_vector(_mat2vec(elt)) + coords = W.coordinate_vector(V(elt)) except ArithmeticError: # vector is not in free module raise ValueError(msg) @@ -660,8 +912,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # And to each subsequent row, prepend an entry that belongs to # the left-side "header column." - M += [ [self.monomial(i)] + [ self.product_on_basis(i,j) - for j in range(n) ] + M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j) + for j in range(n) ] for i in range(n) ] return table(M, header_row=True, header_column=True, frame=True) @@ -731,12 +983,49 @@ 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) @@ -997,14 +1286,12 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): if not c.is_idempotent(): raise ValueError("element is not idempotent: %s" % c) - from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra - # Default these to what they should be if they turn out to be # trivial, because eigenspaces_left() won't return eigenvalues # corresponding to trivial spaces (e.g. it returns only the # eigenspace corresponding to lambda=1 if you take the # decomposition relative to the identity element). - trivial = FiniteDimensionalEJASubalgebra(self, ()) + trivial = self.subalgebra(()) J0 = trivial # eigenvalue zero J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half J1 = trivial # eigenvalue one @@ -1014,9 +1301,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): J5 = eigspace else: gens = tuple( self.from_vector(b) for b in eigspace.basis() ) - subalg = FiniteDimensionalEJASubalgebra(self, - gens, - check_axioms=False) + subalg = self.subalgebra(gens, check_axioms=False) if eigval == 0: J0 = subalg elif eigval == 1: @@ -1235,6 +1520,14 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): return len(self._charpoly_coefficients()) + def subalgebra(self, basis, **kwargs): + r""" + Create a subalgebra of this algebra from the given basis. + """ + from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + return FiniteDimensionalEJASubalgebra(self, basis, **kwargs) + + def vector_space(self): """ Return the vector space that underlies this algebra. @@ -1253,7 +1546,6 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): return self.zero().to_vector().parent().ambient_vector_space() - Element = FiniteDimensionalEJAElement class RationalBasisEJA(FiniteDimensionalEJA): r""" @@ -1291,6 +1583,13 @@ 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 @@ -1304,17 +1603,11 @@ 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""" @@ -1587,9 +1880,9 @@ 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 @@ -1678,10 +1971,15 @@ 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 @@ -1771,7 +2069,7 @@ class ComplexMatrixEJA(MatrixEJA): True """ - super(ComplexMatrixEJA,cls).real_embed(M) + super().real_embed(M) n = M.nrows() # We don't need any adjoined elements... @@ -1818,7 +2116,7 @@ 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()) @@ -1855,9 +2153,9 @@ 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 @@ -1966,10 +2264,15 @@ 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(). @@ -2052,7 +2355,7 @@ class QuaternionMatrixEJA(MatrixEJA): True """ - super(QuaternionMatrixEJA,cls).real_embed(M) + super().real_embed(M) quaternions = M.base_ring() n = M.nrows() @@ -2107,7 +2410,7 @@ class QuaternionMatrixEJA(MatrixEJA): True """ - super(QuaternionMatrixEJA,cls).real_unembed(M) + super().real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() @@ -2152,9 +2455,9 @@ 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 @@ -2272,10 +2575,16 @@ 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(). @@ -2362,7 +2671,11 @@ class HadamardEJA(ConcreteEJA): if "check_axioms" not in kwargs: kwargs["check_axioms"] = False column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() ) - super().__init__(column_basis, jordan_product, inner_product, **kwargs) + super().__init__(column_basis, + jordan_product, + inner_product, + associative=True, + **kwargs) self.rank.set_cache(n) if n == 0: @@ -2497,10 +2810,17 @@ 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 @@ -2605,7 +2925,7 @@ 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(): @@ -2663,10 +2983,12 @@ 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) @@ -2679,8 +3001,7 @@ class TrivialEJA(ConcreteEJA): return cls(**kwargs) -class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, - FiniteDimensionalEJA): +class CartesianProductEJA(FiniteDimensionalEJA): r""" The external (orthogonal) direct sum of two or more Euclidean Jordan algebras. Every Euclidean Jordan algebra decomposes into an @@ -2690,7 +3011,8 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, SETUP:: - sage: from mjo.eja.eja_algebra import (CartesianProductEJA, + sage: from mjo.eja.eja_algebra import (random_eja, + ....: CartesianProductEJA, ....: HadamardEJA, ....: JordanSpinEJA, ....: RealSymmetricEJA) @@ -2729,6 +3051,79 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, Real Field (+) Euclidean Jordan algebra of dimension 6 over Algebraic Real Field + Rank is additive on a Cartesian product:: + + sage: J1 = HadamardEJA(1) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J1.rank.clear_cache() + sage: J2.rank.clear_cache() + sage: J.rank.clear_cache() + sage: J.rank() + 3 + sage: J.rank() == J1.rank() + J2.rank() + True + + The same rank computation works over the rationals, with whatever + basis you like:: + + sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False) + sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False) + sage: J = cartesian_product([J1,J2]) + sage: J1.rank.clear_cache() + sage: J2.rank.clear_cache() + sage: J.rank.clear_cache() + sage: J.rank() + 3 + sage: J.rank() == J1.rank() + J2.rank() + True + + The product algebra will be associative if and only if all of its + components are associative:: + + sage: J1 = HadamardEJA(2) + sage: J1.is_associative() + True + sage: J2 = HadamardEJA(3) + sage: J2.is_associative() + True + sage: J3 = RealSymmetricEJA(3) + sage: J3.is_associative() + False + sage: CP1 = cartesian_product([J1,J2]) + sage: CP1.is_associative() + True + sage: CP2 = cartesian_product([J1,J3]) + sage: CP2.is_associative() + False + + Cartesian products of Cartesian products work:: + + sage: J1 = JordanSpinEJA(1) + sage: J2 = JordanSpinEJA(1) + sage: J3 = JordanSpinEJA(1) + sage: J = cartesian_product([J1,cartesian_product([J2,J3])]) + sage: J.multiplication_table() + +----++----+----+----+ + | * || e0 | e1 | e2 | + +====++====+====+====+ + | e0 || e0 | 0 | 0 | + +----++----+----+----+ + | e1 || 0 | e1 | 0 | + +----++----+----+----+ + | e2 || 0 | 0 | e2 | + +----++----+----+----+ + sage: HadamardEJA(3).multiplication_table() + +----++----+----+----+ + | * || e0 | e1 | e2 | + +====++====+====+====+ + | e0 || e0 | 0 | 0 | + +----++----+----+----+ + | e1 || 0 | e1 | 0 | + +----++----+----+----+ + | e2 || 0 | 0 | e2 | + +----++----+----+----+ + TESTS: All factors must share the same base field:: @@ -2740,42 +3135,120 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, ... ValueError: all factors must share the same base field + The cached unit element is the same one that would be computed:: + + sage: set_random_seed() # long time + sage: J1 = random_eja() # long time + sage: J2 = random_eja() # long time + sage: J = cartesian_product([J1,J2]) # long time + sage: actual = J.one() # long time + sage: J.one.clear_cache() # long time + sage: expected = J.one() # long time + sage: actual == expected # long time + True + """ - def __init__(self, modules, **kwargs): - CombinatorialFreeModule_CartesianProduct.__init__(self, modules) - field = modules[0].base_ring() - if not all( J.base_ring() == field for J in modules ): + Element = FiniteDimensionalEJAElement + + + def __init__(self, factors, **kwargs): + m = len(factors) + if m == 0: + return TrivialEJA() + + self._sets = factors + + field = factors[0].base_ring() + if not all( J.base_ring() == field for J in factors ): raise ValueError("all factors must share the same base field") - basis = tuple( b.to_vector().column() for b in self.basis() ) + associative = all( f.is_associative() for f in factors ) - # Define jordan/inner products that operate on the basis. - def jordan_product(x_mat,y_mat): - x = self.from_vector(_mat2vec(x_mat)) - y = self.from_vector(_mat2vec(y_mat)) - return self.cartesian_jordan_product(x,y).to_vector().column() + MS = self.matrix_space() + basis = [] + zero = MS.zero() + for i in range(m): + for b in factors[i].matrix_basis(): + z = list(zero) + z[i] = b + basis.append(z) - def inner_product(x_mat, y_mat): - x = self.from_vector(_mat2vec(x_mat)) - y = self.from_vector(_mat2vec(y_mat)) - return self.cartesian_inner_product(x,y) + basis = tuple( MS(b) for b in basis ) - # Use whatever category the superclass came up with. Usually - # some join of the EJA and Cartesian product - # categories. There's no need to check the field since it - # already came from an EJA. Likewise the axioms are guaranteed - # to be satisfied. + # Define jordan/inner products that operate on that matrix_basis. + def jordan_product(x,y): + return MS(tuple( + (factors[i](x[i])*factors[i](y[i])).to_matrix() + for i in range(m) + )) + + def inner_product(x, y): + return sum( + factors[i](x[i]).inner_product(factors[i](y[i])) + for i in range(m) + ) + + # There's no need to check the field since it already came + # from an EJA. Likewise the axioms are guaranteed to be + # satisfied, unless the guy writing this class sucks. + # + # If you want the basis to be orthonormalized, orthonormalize + # the factors. FiniteDimensionalEJA.__init__(self, basis, jordan_product, inner_product, field=field, + orthonormalize=False, + associative=associative, + cartesian_product=True, check_field=False, - check_axioms=False, - category=self.category(), - **kwargs) + check_axioms=False) + + ones = tuple(J.one().to_matrix() for J in factors) + self.one.set_cache(self(ones)) + self.rank.set_cache(sum(J.rank() for J in factors)) + + def cartesian_factors(self): + # Copy/pasted from CombinatorialFreeModule_CartesianProduct. + return self._sets + + def cartesian_factor(self, i): + r""" + Return the ``i``th factor of this algebra. + """ + return self._sets[i] + + def _repr_(self): + # Copy/pasted from CombinatorialFreeModule_CartesianProduct. + from sage.categories.cartesian_product import cartesian_product + return cartesian_product.symbol.join("%s" % factor + for factor in self._sets) + + def matrix_space(self): + r""" + Return the space that our matrix basis lives in as a Cartesian + product. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(1) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.matrix_space() + The Cartesian product of (Full MatrixSpace of 1 by 1 dense + matrices over Algebraic Real Field, Full MatrixSpace of 2 + by 2 dense matrices over Algebraic Real Field) - self.rank.set_cache(sum(J.rank() for J in modules)) + """ + from sage.categories.cartesian_product import cartesian_product + return cartesian_product( [J.matrix_space() + for J in self.cartesian_factors()] ) @cached_method def cartesian_projection(self, i): @@ -2847,9 +3320,12 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, True """ - Ji = self.cartesian_factors()[i] - # Requires the fix on Trac 31421/31422 to work! - Pi = super().cartesian_projection(i) + offset = sum( self.cartesian_factor(k).dimension() + for k in range(i) ) + Ji = self.cartesian_factor(i) + Pi = self._module_morphism(lambda j: Ji.monomial(j - offset), + codomain=Ji) + return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix()) @cached_method @@ -2955,108 +3431,64 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, True """ - Ji = self.cartesian_factors()[i] - # Requires the fix on Trac 31421/31422 to work! - Ei = super().cartesian_embedding(i) + offset = sum( self.cartesian_factor(k).dimension() + for k in range(i) ) + Ji = self.cartesian_factor(i) + Ei = Ji._module_morphism(lambda j: self.monomial(j + offset), + codomain=self) return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix()) - def cartesian_jordan_product(self, x, y): - r""" - The componentwise Jordan product. - - We project ``x`` and ``y`` onto our factors, and add up the - Jordan products from the subalgebras. This may still be useful - after (if) the default Jordan product in the Cartesian product - algebra is overridden. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (HadamardEJA, - ....: JordanSpinEJA) - EXAMPLE:: - - sage: J1 = HadamardEJA(3) - sage: J2 = JordanSpinEJA(3) - sage: J = cartesian_product([J1,J2]) - sage: x1 = J1.from_vector(vector(QQ,(1,2,1))) - sage: y1 = J1.from_vector(vector(QQ,(1,0,2))) - sage: x2 = J2.from_vector(vector(QQ,(1,2,3))) - sage: y2 = J2.from_vector(vector(QQ,(1,1,1))) - sage: z1 = J.from_vector(vector(QQ,(1,2,1,1,2,3))) - sage: z2 = J.from_vector(vector(QQ,(1,0,2,1,1,1))) - sage: (x1*y1).to_vector() - (1, 0, 2) - sage: (x2*y2).to_vector() - (6, 3, 4) - sage: J.cartesian_jordan_product(z1,z2).to_vector() - (1, 0, 2, 6, 3, 4) - - """ - m = len(self.cartesian_factors()) - projections = ( self.cartesian_projection(i) for i in range(m) ) - products = ( P(x)*P(y) for P in projections ) - return self._cartesian_product_of_elements(tuple(products)) - - def cartesian_inner_product(self, x, y): - r""" - The standard componentwise Cartesian inner-product. - - We project ``x`` and ``y`` onto our factors, and add up the - inner-products from the subalgebras. This may still be useful - after (if) the default inner product in the Cartesian product - algebra is overridden. - - SETUP:: +FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA - sage: from mjo.eja.eja_algebra import (HadamardEJA, - ....: QuaternionHermitianEJA) +class RationalBasisCartesianProductEJA(CartesianProductEJA, + RationalBasisEJA): + r""" + A separate class for products of algebras for which we know a + rational basis. - EXAMPLE:: + SETUP:: - sage: J1 = HadamardEJA(3,field=QQ) - sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False) - sage: J = cartesian_product([J1,J2]) - sage: x1 = J1.one() - sage: x2 = x1 - sage: y1 = J2.one() - sage: y2 = y1 - sage: x1.inner_product(x2) - 3 - sage: y1.inner_product(y2) - 2 - sage: z1 = J._cartesian_product_of_elements((x1,y1)) - sage: z2 = J._cartesian_product_of_elements((x2,y2)) - sage: J.cartesian_inner_product(z1,z2) - 5 + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA) - """ - m = len(self.cartesian_factors()) - projections = ( self.cartesian_projection(i) for i in range(m) ) - return sum( P(x).inner_product(P(y)) for P in projections ) + EXAMPLES: + This gives us fast characteristic polynomial computations in + product algebras, too:: - Element = FiniteDimensionalEJAElement + 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 -class FiniteDimensionalEJA_CartesianProduct(CartesianProductEJA): - r""" - A wrapper around the :class:`CartesianProductEJA` class that gets - used by the ``cartesian_product`` functor. Its one job is to set - ``orthonormalize=False``, since ``cartesian_product()`` can't be - made to pass that option through. And if we try to orthonormalize - over the rationals, we get conversion errors. If you want a non- - standard Jordan product or inner product, or if you want to - orthonormalize the basis, use :class:`CartesianProductEJA` - directly. """ - def __init__(self, modules, **options): - CombinatorialFreeModule_CartesianProduct.__init__(self, - modules, - **options) - CartesianProductEJA.__init__(self, modules, orthonormalize=False) + def __init__(self, algebras, **kwargs): + CartesianProductEJA.__init__(self, algebras, **kwargs) - -FiniteDimensionalEJA.CartesianProduct = FiniteDimensionalEJA_CartesianProduct -random_eja = ConcreteEJA.random_instance + 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 + +def random_eja(*args, **kwargs): + J1 = ConcreteEJA.random_instance(*args, **kwargs) + + # This might make Cartesian products appear roughly as often as + # any other ConcreteEJA. + if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0: + # Use random_eja() again so we can get more than two factors. + J2 = random_eja(*args, **kwargs) + J = cartesian_product([J1,J2]) + return J + else: + return J1