X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=44d83147bbeceb4acae52d573749545dec3c4b99;hb=db1f7761ebf564221669137ae07476ea45d82a2c;hp=3cb7467119f2f7aabda97a90d243e448164b3529;hpb=bd94f055a5fdd9385c4014b74e115a7f5f0223fd;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 3cb7467..5bf5975 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -1,89 +1,581 @@ """ -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:: + + sage: from mjo.eja.eja_algebra import random_eja + +EXAMPLES:: + + sage: random_eja() + Euclidean Jordan algebra of dimension... """ -from itertools import izip, repeat +from itertools import repeat from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra from sage.categories.magmatic_algebras import MagmaticAlgebras +from sage.categories.sets_cat import cartesian_product from sage.combinat.free_module import CombinatorialFreeModule from sage.matrix.constructor import matrix from sage.matrix.matrix_space import MatrixSpace from sage.misc.cachefunc import cached_method -from sage.misc.prandom import choice from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace -from sage.rings.integer_ring import ZZ -from sage.rings.number_field.number_field import NumberField, QuadraticField -from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing -from sage.rings.rational_field import QQ -from sage.rings.real_lazy import CLF, RLF - -from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement -from mjo.eja.eja_utils import _mat2vec - -class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): - # This is an ugly hack needed to prevent the category framework - # from implementing a coercion from our base ring (e.g. the - # rationals) into the algebra. First of all -- such a coercion is - # nonsense to begin with. But more importantly, it tries to do so - # in the category of rings, and since our algebras aren't - # associative they generally won't be rings. - _no_generic_basering_coercion = True +from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, + PolynomialRing, + QuadraticField) +from mjo.eja.eja_element import FiniteDimensionalEJAElement +from mjo.eja.eja_operator import FiniteDimensionalEJAOperator +from mjo.eja.eja_utils import _all2list, _mat2vec + +class FiniteDimensionalEJA(CombinatorialFreeModule): + r""" + A finite-dimensional Euclidean Jordan algebra. + + INPUT: + + - ``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 def __init__(self, - field, - mult_table, - rank, - prefix='e', - category=None, - natural_basis=None): + basis, + jordan_product, + inner_product, + field=AA, + orthonormalize=True, + associative=None, + cartesian_product=False, + check_field=True, + check_axioms=True, + prefix="b"): + + n = len(basis) + + if check_field: + if not field.is_subring(RR): + # Note: this does return true for the real algebraic + # field, the rationals, and any quadratic field where + # we've specified a real embedding. + raise ValueError("scalar field is not real") + + if check_axioms: + # Check commutativity of the Jordan and inner-products. + # This has to be done before we build the multiplication + # and inner-product tables/matrices, because we take + # advantage of symmetry in the process. + if not all( jordan_product(bi,bj) == jordan_product(bj,bi) + for bi in basis + for bj in basis ): + raise ValueError("Jordan product is not commutative") + + if not all( inner_product(bi,bj) == inner_product(bj,bi) + for bi in basis + for bj in basis ): + raise ValueError("inner-product is not commutative") + + + 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. + 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, + # as well as a subspace W of V spanned by those (vectorized) + # basis elements. The W-coordinates are the coefficients that + # we see in things like x = 1*b1 + 2*b2. + vector_basis = basis + + degree = 0 + if n > 0: + degree = len(_all2list(basis[0])) + + # Build an ambient space that fits our matrix basis when + # written out as "long vectors." + V = VectorSpace(field, degree) + + # The matrix that will hole the orthonormal -> unorthonormal + # coordinate transformation. + self._deortho_matrix = None + + if orthonormalize: + # 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(_all2list(b)) for b in basis ) + + from mjo.eja.eja_utils import gram_schmidt + basis = tuple(gram_schmidt(basis, inner_product)) + + # Save the (possibly orthonormalized) matrix basis for + # later... + self._matrix_basis = basis + + # 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(_all2list(b)) for b in basis ) + W = V.span_of_basis( vector_basis, check=check_axioms) + + if orthonormalize: + # Now "W" is the vector space of our algebra coordinates. The + # variables "X1", "X2",... refer to the entries of vectors in + # W. Thus to convert back and forth between the orthonormal + # coordinates and the given ones, we need to stick the original + # basis in W. + U = V.span_of_basis( deortho_vector_basis, check=check_axioms) + self._deortho_matrix = matrix( U.coordinate_vector(q) + for q in vector_basis ) + + + # Now we actually compute the multiplication and inner-product + # tables/matrices using the possibly-orthonormalized basis. + self._inner_product_matrix = matrix.identity(field, n) + self._multiplication_table = [ [0 for j in range(i+1)] + for i in range(n) ] + + # Note: the Jordan and inner-products are defined in terms + # of the ambient basis. It's important that their arguments + # are in ambient coordinates as well. + for i in range(n): + for j in range(i+1): + # ortho basis w.r.t. ambient coords + q_i = basis[i] + q_j = basis[j] + + # 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(_all2list(elt))) + self._multiplication_table[i][j] = self.from_vector(elt) + + if not orthonormalize: + # If we're orthonormalizing the basis with respect + # to an inner-product, then the inner-product + # matrix with respect to the resulting basis is + # just going to be the identity. + ip = inner_product(q_i, q_j) + self._inner_product_matrix[i,j] = ip + self._inner_product_matrix[j,i] = ip + + self._inner_product_matrix._cache = {'hermitian': True} + self._inner_product_matrix.set_immutable() + + if check_axioms: + if not self._is_jordanian(): + raise ValueError("Jordan identity does not hold") + if not self._inner_product_is_associative(): + raise ValueError("inner product is not associative") + + + def _coerce_map_from_base_ring(self): + """ + Disable the map from the base ring into the algebra. + + Performing a nonsense conversion like this automatically + is counterpedagogical. The fallback is to try the usual + element constructor, which should also fail. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J(1) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + """ + return None + + + 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: bi = J.zero() + sage: bj = J.zero() + sage: bi_bj = J.zero()*J.zero() + sage: if n > 0: + ....: i = ZZ.random_element(n) + ....: j = ZZ.random_element(n) + ....: bi = J.monomial(i) + ....: bj = J.monomial(j) + ....: bi_bj = J.product_on_basis(i,j) + sage: bi*bj == bi_bj + True + + """ + # We only stored the lower-triangular portion of the + # multiplication table. + if j <= i: + return self._multiplication_table[i][j] + else: + return self._multiplication_table[j][i] + + def inner_product(self, x, y): + """ + The inner product associated with this Euclidean Jordan algebra. + + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: BilinearFormEJA) + EXAMPLES: - By definition, Jordan multiplication commutes:: + Our inner product is "associative," which means the following for + a symmetric bilinear form:: sage: set_random_seed() sage: J = random_eja() + sage: x,y,z = J.random_elements(3) + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() sage: x,y = J.random_elements(2) - sage: x*y == y*x + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + + Ensure that this is one-half of the trace inner-product in a + BilinearFormEJA that isn't just the reals (when ``n`` isn't + one). This is in Faraut and Koranyi, and also my "On the + symmetry..." paper:: + + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() + sage: x = J.random_element() + 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 """ - self._rank = rank - self._natural_basis = natural_basis + 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( x*y == y*x for x in self.gens() for y in self.gens() ) + + def _is_jordanian(self): + r""" + Whether or not this algebra's multiplication table respects the + Jordan identity `(x^{2})(xy) = x(x^{2}y)`. + + 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 + return ``True``, unless this algebra was constructed with + ``check_axioms=False`` and passed an invalid multiplication table. + """ + return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j)) + == + (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j)) + 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) - if category is None: - category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() + 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: - fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) - fda.__init__(field, - range(len(mult_table)), - prefix=prefix, - category=category) - self.print_options(bracket='') + The values we've presupplied to the constructors agree with + the computation:: - # The multiplication table we're given is necessarily in terms - # of vectors, because we don't have an algebra yet for - # anything to be an element of. However, it's faster in the - # long run to have the multiplication table be in terms of - # algebra elements. We do this after calling the superclass - # constructor so that from_vector() knows what to do. - self._multiplication_table = [ map(lambda x: self.from_vector(x), ls) - for ls in mult_table ] + 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 + associative; that is, whether or not `B(xy,z) = B(x,yz)`. + + This method should of course always return ``True``, unless + 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. + 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()): + for k in range(self.dimension()): + x = self.monomial(i) + y = self.monomial(j) + z = self.monomial(k) + diff = (x*y).inner_product(z) - x.inner_product(y*z) + + if diff.abs() > epsilon: + return False + + return True 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 @@ -91,8 +583,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: HadamardEJA, ....: RealSymmetricEJA) EXAMPLES: @@ -111,46 +604,86 @@ 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 + + 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]) ) + b1 + b5 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:: + Ensure that we can convert any element back and forth + faithfully between its matrix and algebra representations:: sage: set_random_seed() - sage: J = RealCartesianProductEJA.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 """ - if elt == 0: - # The superclass implementation of random_element() - # needs to be able to coerce "0" into the algebra. - return self.zero() + msg = "not an element of this algebra" + 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 + # 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: - raise ValueError("not a naturally-represented algebra element") + try: + # Try to convert a vector into a column-matrix... + elt = elt.column() + except (AttributeError, TypeError): + # and ignore failure, because we weren't really expecting + # a vector as an argument anyway. + pass + + 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)) - return self.from_vector(coords) + # + # 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. + 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(V(elt)) + except ArithmeticError: # vector is not in free module + raise ValueError(msg) + return self.from_vector(coords) def _repr_(self): """ @@ -164,8 +697,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Ensure that it says what we think it says:: - sage: JordanSpinEJA(2, field=QQ) - Euclidean Jordan algebra of dimension 2 over Rational Field + sage: JordanSpinEJA(2, field=AA) + Euclidean Jordan algebra of dimension 2 over Algebraic Real Field sage: JordanSpinEJA(3, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field @@ -173,168 +706,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): fmt = "Euclidean Jordan algebra of dimension {} over {}" return fmt.format(self.dimension(), self.base_ring()) - def product_on_basis(self, i, j): - return self._multiplication_table[i][j] - - def _a_regular_element(self): - """ - Guess a regular element. Needed to compute the basis for our - characteristic polynomial coefficients. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS: - - Ensure that this hacky method succeeds for every algebra that we - know how to construct:: - - sage: set_random_seed() - sage: J = random_eja() - sage: J._a_regular_element().is_regular() - True - - """ - gs = self.gens() - z = self.sum( (i+1)*gs[i] for i in range(len(gs)) ) - if not z.is_regular(): - raise ValueError("don't know a regular element") - return z - - - @cached_method - def _charpoly_basis_space(self): - """ - Return the vector space spanned by the basis used in our - characteristic polynomial coefficients. This is used not only to - compute those coefficients, but also any time we need to - evaluate the coefficients (like when we compute the trace or - determinant). - """ - z = self._a_regular_element() - # Don't use the parent vector space directly here in case this - # happens to be a subalgebra. In that case, we would be e.g. - # two-dimensional but span_of_basis() would expect three - # coordinates. - V = VectorSpace(self.base_ring(), self.vector_space().dimension()) - basis = [ (z**k).to_vector() for k in range(self.rank()) ] - V1 = V.span_of_basis( basis ) - b = (V1.basis() + V1.complement().basis()) - return V.span_of_basis(b) - - - - @cached_method - def _charpoly_coeff(self, i): - """ - Return the coefficient polynomial "a_{i}" of this algebra's - general characteristic polynomial. - - Having this be a separate cached method lets us compute and - store the trace/determinant (a_{r-1} and a_{0} respectively) - separate from the entire characteristic polynomial. - """ - (A_of_x, x, xr, detA) = self._charpoly_matrix_system() - R = A_of_x.base_ring() - if i >= self.rank(): - # Guaranteed by theory - return R.zero() - - # Danger: the in-place modification is done for performance - # reasons (reconstructing a matrix with huge polynomial - # entries is slow), but I don't know how cached_method works, - # so it's highly possible that we're modifying some global - # list variable by reference, here. In other words, you - # probably shouldn't call this method twice on the same - # algebra, at the same time, in two threads - Ai_orig = A_of_x.column(i) - A_of_x.set_column(i,xr) - numerator = A_of_x.det() - A_of_x.set_column(i,Ai_orig) - - # We're relying on the theory here to ensure that each a_i is - # indeed back in R, and the added negative signs are to make - # the whole charpoly expression sum to zero. - return R(-numerator/detA) - - - @cached_method - def _charpoly_matrix_system(self): - """ - Compute the matrix whose entries A_ij are polynomials in - X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector - corresponding to `x^r` and the determinent of the matrix A = - [A_ij]. In other words, all of the fixed (cachable) data needed - to compute the coefficients of the characteristic polynomial. - """ - r = self.rank() - n = self.dimension() - - # Turn my vector space into a module so that "vectors" can - # have multivatiate polynomial entries. - names = tuple('X' + str(i) for i in range(1,n+1)) - R = PolynomialRing(self.base_ring(), names) - - # Using change_ring() on the parent's vector space doesn't work - # here because, in a subalgebra, that vector space has a basis - # and change_ring() tries to bring the basis along with it. And - # that doesn't work unless the new ring is a PID, which it usually - # won't be. - V = FreeModule(R,n) - - # Now let x = (X1,X2,...,Xn) be the vector whose entries are - # indeterminates... - x = V(names) - - # And figure out the "left multiplication by x" matrix in - # that setting. - lmbx_cols = [] - monomial_matrices = [ self.monomial(i).operator().matrix() - for i in range(n) ] # don't recompute these! - for k in range(n): - ek = self.monomial(k).to_vector() - lmbx_cols.append( - sum( x[i]*(monomial_matrices[i]*ek) - for i in range(n) ) ) - Lx = matrix.column(R, lmbx_cols) - - # Now we can compute powers of x "symbolically" - x_powers = [self.one().to_vector(), x] - for d in range(2, r+1): - x_powers.append( Lx*(x_powers[-1]) ) - - idmat = matrix.identity(R, n) - - W = self._charpoly_basis_space() - W = W.change_ring(R.fraction_field()) - - # Starting with the standard coordinates x = (X1,X2,...,Xn) - # and then converting the entries to W-coordinates allows us - # to pass in the standard coordinates to the charpoly and get - # back the right answer. Specifically, with x = (X1,X2,...,Xn), - # we have - # - # W.coordinates(x^2) eval'd at (standard z-coords) - # = - # W-coords of (z^2) - # = - # W-coords of (standard coords of x^2 eval'd at std-coords of z) - # - # We want the middle equivalent thing in our matrix, but use - # the first equivalent thing instead so that we can pass in - # standard coordinates. - x_powers = [ W.coordinate_vector(xp) for xp in x_powers ] - l2 = [idmat.column(k-1) for k in range(r+1, n+1)] - A_of_x = matrix.column(R, n, (x_powers[:r] + l2)) - return (A_of_x, x, x_powers[r], A_of_x.det()) - @cached_method - def characteristic_polynomial(self): + def characteristic_polynomial_of(self): """ - Return a characteristic polynomial that works for all elements - of this algebra. + Return the algebra's "characteristic polynomial of" function, + which is itself a multivariate polynomial that, when evaluated + at the coordinates of some algebra element, returns that + element's characteristic polynomial. The resulting polynomial has `n+1` variables, where `n` is the dimension of this algebra. The first `n` variables correspond to @@ -346,7 +725,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA EXAMPLES: @@ -354,42 +733,64 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Alizadeh, Example 11.11:: sage: J = JordanSpinEJA(3) - sage: p = J.characteristic_polynomial(); p + sage: p = J.characteristic_polynomial_of(); p X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 sage: xvec = J.one().to_vector() sage: p(*xvec) t^2 - 2*t + 1 + By definition, the characteristic polynomial is a monic + degree-zero polynomial in a rank-zero algebra. Note that + Cayley-Hamilton is indeed satisfied since the polynomial + ``1`` evaluates to the identity element of the algebra on + any argument:: + + sage: J = TrivialEJA() + sage: J.characteristic_polynomial_of() + 1 + """ r = self.rank() n = self.dimension() - # The list of coefficient polynomials a_1, a_2, ..., a_n. - a = [ self._charpoly_coeff(i) for i in range(n) ] + # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1). + a = self._charpoly_coefficients() # We go to a bit of trouble here to reorder the # indeterminates, so that it's easier to evaluate the # characteristic polynomial at x's coordinates and get back # something in terms of t, which is what we want. - R = a[0].parent() S = PolynomialRing(self.base_ring(),'t') t = S.gen(0) - S = PolynomialRing(S, R.variable_names()) - t = S(t) - - # Note: all entries past the rth should be zero. The - # coefficient of the highest power (x^r) is 1, but it doesn't - # appear in the solution vector which contains coefficients - # for the other powers (to make them sum to x^r). - if (r < n): - a[r] = 1 # corresponds to x^r - else: - # When the rank is equal to the dimension, trying to - # assign a[r] goes out-of-bounds. - a.append(1) # corresponds to x^r + if r > 0: + R = a[0].parent() + S = PolynomialRing(S, R.variable_names()) + t = S(t) + + return (t**r + sum( a[k]*(t**k) for k in range(r) )) - return sum( a[k]*(t**k) for k in xrange(len(a)) ) + def coordinate_polynomial_ring(self): + r""" + The multivariate polynomial ring in which this algebra's + :meth:`characteristic_polynomial_of` lives. + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J = HadamardEJA(2) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2... + sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... + + """ + var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) ) + return PolynomialRing(self.base_ring(), var_names) def inner_product(self, x, y): """ @@ -401,7 +802,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: BilinearFormEJA) EXAMPLES: @@ -414,10 +817,34 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: (x*y).inner_product(z) == y.inner_product(x*z) True + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + + Ensure that this is one-half of the trace inner-product in a + BilinearFormEJA that isn't just the reals (when ``n`` isn't + one). This is in Faraut and Koranyi, and also my "On the + symmetry..." paper:: + + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) + True """ - X = x.natural_representation() - Y = y.natural_representation() - return self.natural_inner_product(X,Y) + B = self._inner_product_matrix + return (B*x.to_vector()).inner_product(y.to_vector()) def is_trivial(self): @@ -428,15 +855,19 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: TrivialEJA) EXAMPLES:: sage: J = ComplexHermitianEJA(3) sage: J.is_trivial() False - sage: A = J.zero().subalgebra_generated_by() - sage: A.is_trivial() + + :: + + sage: J = TrivialEJA() + sage: J.is_trivial() True """ @@ -457,38 +888,58 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = JordanSpinEJA(4) sage: J.multiplication_table() +----++----+----+----+----+ - | * || e0 | e1 | e2 | e3 | + | * || b0 | b1 | b2 | b3 | +====++====+====+====+====+ - | e0 || e0 | e1 | e2 | e3 | + | b0 || b0 | b1 | b2 | b3 | +----++----+----+----+----+ - | e1 || e1 | e0 | 0 | 0 | + | b1 || b1 | b0 | 0 | 0 | +----++----+----+----+----+ - | e2 || e2 | 0 | e0 | 0 | + | b2 || b2 | 0 | b0 | 0 | +----++----+----+----+----+ - | e3 || e3 | 0 | 0 | e0 | + | b3 || b3 | 0 | 0 | b0 | +----++----+----+----+----+ """ - M = list(self._multiplication_table) # copy - for i in xrange(len(M)): - # M had better be "square" - M[i] = [self.monomial(i)] + M[i] - M = [["*"] + list(self.gens())] + M + n = self.dimension() + # Prepend the header row. + M = [["*"] + list(self.gens())] + + # And to each subsequent row, prepend an entry that belongs to + # the left-side "header column." + 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) - 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:`CartesianProductEJA` can be displayed + nicely, without having to have special classes for direct sums + one of whose components was a matrix algebra. SETUP:: @@ -499,55 +950,81 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = RealSymmetricEJA(2) sage: J.basis() - Finite family {0: e0, 1: e1, 2: e2} - sage: J.natural_basis() + Finite family {0: b0, 1: b1, 2: b2} + sage: J.matrix_basis() ( - [1 0] [ 0 1/2*sqrt2] [0 0] - [0 0], [1/2*sqrt2 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ) :: sage: J = JordanSpinEJA(2) sage: J.basis() - Finite family {0: e0, 1: e1} - sage: J.natural_basis() + Finite family {0: b0, 1: b1} + sage: J.matrix_basis() ( [1] [0] [0], [1] ) - """ - if self._natural_basis is None: - M = self.natural_basis_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. - """ - if self._natural_basis is None or len(self._natural_basis) == 0: - return MatrixSpace(self.base_ring(), self.dimension(), 1) - else: - return self._natural_basis[0].matrix_space() + 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). + "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. For algebras of + matrices, this returns the space in which their + real embeddings live. - @staticmethod - def natural_inner_product(X,Y): - """ - Compute the inner product of two naturally-represented elements. + 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 - For example in the real symmetric matrix EJA, this will compute - the trace inner-product of two n-by-n symmetric matrices. The - default should work for the real cartesian product EJA, the - Jordan spin EJA, and the real symmetric matrices. The others - will have to be overridden. """ - return (X.conjugate_transpose()*Y).trace() + if self.is_trivial(): + return MatrixSpace(self.base_ring(), 0) + else: + return self.matrix_basis()[0].parent() @cached_method @@ -557,26 +1034,60 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + 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() + b0 + b1 + b2 + b3 + b4 + + The unit element in the Hadamard EJA is inherited in the + subalgebras generated by its elements:: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one() - e0 + e1 + e2 + e3 + e4 + b0 + b1 + b2 + b3 + b4 + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A.one() + c0 + sage: A.one().superalgebra_element() + b0 + b1 + b2 + b3 + b4 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() @@ -584,45 +1095,272 @@ class FiniteDimensionalEuclideanJordanAlgebra(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 - """ - # We can brute-force compute the matrices of the operators - # that correspond to the basis elements of this algebra. - # If some linear combination of those basis elements is the - # algebra identity, then the same linear combination of - # their matrices has to be the identity matrix. - # - # Of course, matrices aren't vectors in sage, so we have to - # appeal to the "long vectors" isometry. - oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ] + :: + + 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:: + + sage: set_random_seed() + sage: J = random_eja() + sage: cached = J.one() + sage: J.one.clear_cache() + 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. + # If some linear combination of those basis elements is the + # algebra identity, then the same linear combination of + # their matrices has to be the identity matrix. + # + # Of course, matrices aren't vectors in sage, so we have to + # appeal to the "long vectors" isometry. + oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ] - # Now we use basis linear algebra to find the coefficients, + # Now we use basic linear algebra to find the coefficients, # of the matrices-as-vectors-linear-combination, which should # work for the original algebra basis too. - A = matrix.column(self.base_ring(), oper_vecs) + A = matrix(self.base_ring(), oper_vecs) # We used the isometry on the left-hand side already, but we # still need to do it for the right-hand side. Recall that we # wanted something that summed to the identity matrix. b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) ) - # Now if there's an identity element in the algebra, this should work. - coeffs = A.solve_right(b) - return self.linear_combination(zip(self.gens(), coeffs)) + # Now if there's an identity element in the algebra, this + # should work. We solve on the left to avoid having to + # transpose the matrix "A". + return self.from_vector(A.solve_left(b)) - def random_element(self): - # Temporary workaround for https://trac.sagemath.org/ticket/28327 - if self.is_trivial(): - return self.zero() - else: - s = super(FiniteDimensionalEuclideanJordanAlgebra, self) - return s.random_element() + def peirce_decomposition(self, c): + """ + The Peirce decomposition of this algebra relative to the + idempotent ``c``. + + In the future, this can be extended to a complete system of + orthogonal idempotents. + + INPUT: + + - ``c`` -- an idempotent of this algebra. + + OUTPUT: + + A triple (J0, J5, J1) containing two subalgebras and one subspace + of this algebra, + + - ``J0`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue zero. + + - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()`` + corresponding to the eigenvalue one-half. + + - ``J1`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue one. + + These are the only possible eigenspaces for that operator, and this + algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are + orthogonal, and are subalgebras of this algebra with the appropriate + restrictions. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA + + EXAMPLES: + + The canonical example comes from the symmetric matrices, which + decompose into diagonal and off-diagonal parts:: + + sage: J = RealSymmetricEJA(3) + sage: C = matrix(QQ, [ [1,0,0], + ....: [0,1,0], + ....: [0,0,0] ]) + sage: c = J(C) + sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: J0 + Euclidean Jordan algebra of dimension 1... + sage: J5 + Vector space of degree 6 and dimension 2... + sage: J1 + Euclidean Jordan algebra of dimension 3... + 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]).to_matrix() + [ 0 0 1/2*sqrt(2)] + [ 0 0 0] + [1/2*sqrt(2) 0 0] + 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().to_matrix() + [1 0 0] + [0 1 0] + [0 0 0] + + TESTS: + + Every algebra decomposes trivially with respect to its identity + element:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J0,J5,J1 = J.peirce_decomposition(J.one()) + sage: J0.dimension() == 0 and J5.dimension() == 0 + True + sage: J1.superalgebra() == J and J1.dimension() == J.dimension() + True + + The decomposition is into eigenspaces, and its components are + therefore necessarily orthogonal. Moreover, the identity + elements in the two subalgebras are the projections onto their + respective subspaces of the superalgebra's identity element:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: if not J.is_trivial(): + ....: while x.is_nilpotent(): + ....: x = J.random_element() + sage: c = x.subalgebra_idempotent() + sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: ipsum = 0 + sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()): + ....: w = w.superalgebra_element() + ....: y = J.from_vector(y) + ....: z = z.superalgebra_element() + ....: ipsum += w.inner_product(y).abs() + ....: ipsum += w.inner_product(z).abs() + ....: ipsum += y.inner_product(z).abs() + sage: ipsum + 0 + sage: J1(c) == J1.one() + True + sage: J0(J.one() - c) == J0.one() + True + + """ + if not c.is_idempotent(): + raise ValueError("element is not idempotent: %s" % c) + + # 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 = self.subalgebra(()) + J0 = trivial # eigenvalue zero + J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half + J1 = trivial # eigenvalue one + + for (eigval, eigspace) in c.operator().matrix().right_eigenspaces(): + if eigval == ~(self.base_ring()(2)): + J5 = eigspace + else: + gens = tuple( self.from_vector(b) for b in eigspace.basis() ) + subalg = self.subalgebra(gens, check_axioms=False) + if eigval == 0: + J0 = subalg + elif eigval == 1: + J1 = subalg + else: + raise ValueError("unexpected eigenvalue: %s" % eigval) + + return (J0, J5, J1) + + + def random_element(self, thorough=False): + r""" + Return a random element of this algebra. + + Our algebra superclass method only returns a linear + combination of at most two basis elements. We instead + want the vector space "random element" method that + returns a more diverse selection. + + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + element when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + + """ + # For a general base ring... maybe we can trust this to do the + # right thing? Unlikely, but. + V = self.vector_space() + v = V.random_element() + + if self.base_ring() is AA: + # The "random element" method of the algebraic reals is + # stupid at the moment, and only returns integers between + # -2 and 2, inclusive: + # + # https://trac.sagemath.org/ticket/30875 + # + # Instead, we implement our own "random vector" method, + # and then coerce that into the algebra. We use the vector + # space degree here instead of the dimension because a + # subalgebra could (for example) be spanned by only two + # vectors, each with five coordinates. We need to + # generate all five coordinates. + if thorough: + v *= QQbar.random_element().real() + else: + v *= QQ.random_element() + + return self.from_vector(V.coordinate_vector(v)) - def random_elements(self, count): + def random_elements(self, count, thorough=False): """ Return ``count`` random elements as a tuple. + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + elements when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA @@ -637,23 +1375,91 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in xrange(count) ) + return tuple( self.random_element(thorough) + for idx in range(count) ) - def rank(self): + @cached_method + def _charpoly_coefficients(self): + r""" + The `r` polynomial coefficients of the "characteristic polynomial + of" function. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS: + + The theory shows that these are all homogeneous polynomials of + a known degree:: + + sage: set_random_seed() + sage: J = random_eja() + sage: all(p.is_homogeneous() for p in J._charpoly_coefficients()) + True + """ - Return the rank of this EJA. + n = self.dimension() + R = self.coordinate_polynomial_ring() + vars = R.gens() + F = R.fraction_field() + + def L_x_i_j(i,j): + # From a result in my book, these are the entries of the + # basis representation of L_x. + return sum( vars[k]*self.monomial(k).operator().matrix()[i,j] + for k in range(n) ) + + L_x = matrix(F, n, n, L_x_i_j) + + r = None + if self.rank.is_in_cache(): + r = self.rank() + # There's no need to pad the system with redundant + # columns if we *know* they'll be redundant. + n = r + + # Compute an extra power in case the rank is equal to + # the dimension (otherwise, we would stop at x^(r-1)). + x_powers = [ (L_x**k)*self.one().to_vector() + for k in range(n+1) ] + A = matrix.column(F, x_powers[:n]) + AE = A.extended_echelon_form() + E = AE[:,n:] + A_rref = AE[:,:n] + if r is None: + r = A_rref.rank() + b = x_powers[r] + + # The theory says that only the first "r" coefficients are + # nonzero, and they actually live in the original polynomial + # ring and not the fraction field. We negate them because in + # the actual characteristic polynomial, they get moved to the + # other side where x^r lives. We don't bother to trim A_rref + # down to a square matrix and solve the resulting system, + # because the upper-left r-by-r portion of A_rref is + # guaranteed to be the identity matrix, so e.g. + # + # A_rref.solve_right(Y) + # + # would just be returning Y. + return (-E*b)[:r].change_ring(R) - ALGORITHM: + @cached_method + def rank(self): + r""" + Return the rank of this EJA. - The author knows of no algorithm to compute the rank of an EJA - where only the multiplication table is known. In lieu of one, we - require the rank to be specified when the algebra is created, - and simply pass along that number here. + This is a cached method because we know the rank a priori for + all of the algebras we can construct. Thus we can avoid the + expensive ``_charpoly_coefficients()`` call unless we truly + need to compute the whole characteristic polynomial. SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, ....: RealSymmetricEJA, ....: ComplexHermitianEJA, ....: QuaternionHermitianEJA, @@ -683,15 +1489,37 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): TESTS: Ensure that every EJA that we know how to construct has a - positive integer rank:: + positive integer rank, unless the algebra is trivial in + which case its rank will be zero:: sage: set_random_seed() - sage: r = random_eja().rank() - sage: r in ZZ and r > 0 + sage: J = random_eja() + sage: r = J.rank() + sage: r in ZZ + True + sage: r > 0 or (r == 0 and J.is_trivial()) True + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: + + sage: set_random_seed() # long time + sage: J = random_eja() # long time + sage: cached = J.rank() # long time + sage: J.rank.clear_cache() # long time + sage: J.rank() == cached # long time + True + + """ + return len(self._charpoly_coefficients()) + + + def subalgebra(self, basis, **kwargs): + r""" + Create a subalgebra of this algebra from the given basis. """ - return self._rank + from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + return FiniteDimensionalEJASubalgebra(self, basis, **kwargs) def vector_space(self): @@ -712,286 +1540,223 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.zero().to_vector().parent().ambient_vector_space() - Element = FiniteDimensionalEuclideanJordanAlgebraElement - - -class KnownRankEJA(object): - """ - A class for algebras that we actually know we can construct. The - main issue is that, for most of our methods to make sense, we need - to know the rank of our algebra. Thus we can't simply generate a - "random" algebra, or even check that a given basis and product - satisfy the axioms; because even if everything looks OK, we wouldn't - know the rank we need to actuallty build the thing. - - Not really a subclass of FDEJA because doing that causes method - resolution errors, e.g. - - TypeError: Error when calling the metaclass bases - Cannot create a consistent method resolution - order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA - - """ - @staticmethod - def _max_test_case_size(): - """ - Return an integer "size" that is an upper bound on the size of - this algebra when it is used in a random test - case. Unfortunately, the term "size" is quite vague -- when - dealing with `R^n` under either the Hadamard or Jordan spin - product, the "size" refers to the dimension `n`. When dealing - with a matrix algebra (real symmetric or complex/quaternion - Hermitian), it refers to the size of the matrix, which is - far less than the dimension of the underlying vector space. - - We default to five in this class, which is safe in `R^n`. The - matrix algebra subclasses (or any class where the "size" is - interpreted to be far less than the dimension) should override - with a smaller number. - """ - return 5 - - @classmethod - def random_instance(cls, field=QQ, **kwargs): - """ - Return a random instance of this type of algebra. - - Beware, this will crash for "most instances" because the - constructor below looks wrong. - """ - n = ZZ.random_element(cls._max_test_case_size()) + 1 - return cls(n, field, **kwargs) - - -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA): - """ - Return the Euclidean Jordan Algebra corresponding to the set - `R^n` under the Hadamard product. - Note: this is nothing more than the Cartesian product of ``n`` - copies of the spin algebra. Once Cartesian product algebras - are implemented, this can go. +class RationalBasisEJA(FiniteDimensionalEJA): + r""" + New class for algebras whose supplied basis elements have all rational entries. SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import BilinearFormEJA EXAMPLES: - This multiplication table can be verified by hand:: - - sage: J = RealCartesianProductEJA(3) - sage: e0,e1,e2 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - 0 - sage: e0*e2 - 0 - sage: e1*e1 - e1 - sage: e1*e2 - 0 - sage: e2*e2 - e2 - - TESTS: - - We can change the generator prefix:: + The supplied basis is orthonormalized by default:: - sage: RealCartesianProductEJA(3, prefix='r').gens() - (r0, r1, r2) + sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]]) + sage: J = BilinearFormEJA(B) + sage: J.matrix_basis() + ( + [1] [ 0] [ 0] + [0] [1/5] [32/5] + [0], [ 0], [ 5] + ) """ - def __init__(self, n, field=QQ, **kwargs): - V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in xrange(n) ] - for i in xrange(n) ] - - fdeja = super(RealCartesianProductEJA, self) - return fdeja.__init__(field, mult_table, rank=n, **kwargs) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. + def __init__(self, + basis, + jordan_product, + inner_product, + field=AA, + check_field=True, + **kwargs): + + if check_field: + # Abuse the check_field parameter to check that the entries of + # out basis (in ambient coordinates) are in the field QQ. + 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 + # one is already rational. + # + # Note: the same Jordan and inner-products work here, + # because they are necessarily defined with respect to + # ambient coordinates and not any particular basis. + self._rational_algebra = FiniteDimensionalEJA( + basis, + jordan_product, + inner_product, + field=QQ, + associative=self.is_associative(), + orthonormalize=False, + check_field=False, + check_axioms=False) + @cached_method + def _charpoly_coefficients(self): + r""" SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA - - TESTS: - - Ensure that this is the usual inner product for the algebras - over `R^n`:: - - sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.natural_inner_product(X,Y) - True - - """ - return x.to_vector().inner_product(y.to_vector()) - - -def random_eja(): - """ - Return a "random" finite-dimensional Euclidean Jordan Algebra. - - ALGORITHM: + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) - For now, we choose a random natural number ``n`` (greater than zero) - and then give you back one of the following: - - * The cartesian product of the rational numbers ``n`` times; this is - ``QQ^n`` with the Hadamard product. - - * The Jordan spin algebra on ``QQ^n``. - - * The ``n``-by-``n`` rational symmetric matrices with the symmetric - product. - - * The ``n``-by-``n`` complex-rational Hermitian matrices embedded - in the space of ``2n``-by-``2n`` real symmetric matrices. + EXAMPLES: - * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded - in the space of ``4n``-by-``4n`` real symmetric matrices. + The base ring of the resulting polynomial coefficients is what + it should be, and not the rationals (unless the algebra was + already over the rationals):: - Later this might be extended to return Cartesian products of the - EJAs above. + sage: J = JordanSpinEJA(3) + sage: J._charpoly_coefficients() + (X1^2 - X2^2 - X3^2, -2*X1) + sage: a0 = J._charpoly_coefficients()[0] + sage: J.base_ring() + Algebraic Real Field + sage: a0.base_ring() + Algebraic Real Field + + """ + if self._rational_algebra is None: + # There's no need to construct *another* algebra over the + # rationals if this one is already over the + # rationals. Likewise, if we never orthonormalized our + # basis, we might as well just use the given one. + return super()._charpoly_coefficients() + + # Do the computation over the rationals. The answer will be + # the same, because all we've done is a change of basis. + # Then, change back from QQ to our real base ring + a = ( a_i.change_ring(self.base_ring()) + for a_i in self._rational_algebra._charpoly_coefficients() ) + + if self._deortho_matrix is None: + # This can happen if our base ring was, say, AA and we + # chose not to (or didn't need to) orthonormalize. It's + # still faster to do the computations over QQ even if + # the numbers in the boxes stay the same. + return tuple(a) + + # Otherwise, convert the coordinate variables back to the + # deorthonormalized ones. + R = self.coordinate_polynomial_ring() + from sage.modules.free_module_element import vector + X = vector(R, R.gens()) + BX = self._deortho_matrix*X + + subs_dict = { X[i]: BX[i] for i in range(len(X)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) + +class ConcreteEJA(RationalBasisEJA): + r""" + A class for the Euclidean Jordan algebras that we know by name. + + These are the Jordan algebras whose basis, multiplication table, + rank, and so on are known a priori. More to the point, they are + the Euclidean Jordan algebras for which we are able to conjure up + a "random instance." SETUP:: - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: random_eja() - Euclidean Jordan algebra of dimension... - - """ - classname = choice(KnownRankEJA.__subclasses__()) - return classname.random_instance() + sage: from mjo.eja.eja_algebra import ConcreteEJA + TESTS: + Our basis is normalized with respect to the algebra's inner + product, unless we specify otherwise:: + sage: set_random_seed() + sage: J = ConcreteEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True + 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:: + sage: set_random_seed() + sage: J = ConcreteEJA.random_instance() + sage: x = J.random_element() + sage: x.operator().is_self_adjoint() + True + """ -class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): @staticmethod - def _max_test_case_size(): - # Play it safe, since this will be squared and the underlying - # field can have dimension 4 (quaternions) too. - return 2 - - def __init__(self, field, basis, rank, normalize_basis=True, **kwargs): + def _max_random_instance_size(): """ - Compared to the superclass constructor, we take a basis instead of - a multiplication table because the latter can be computed in terms - of the former when the product is known (like it is here). + Return an integer "size" that is an upper bound on the size of + this algebra when it is used in a random test + case. Unfortunately, the term "size" is ambiguous -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is far + less than the dimension of the underlying vector space. + + This method must be implemented in each subclass. """ - # Used in this class's fast _charpoly_coeff() override. - self._basis_normalizers = None + raise NotImplementedError - # We're going to loop through this a few times, so now's a good - # time to ensure that it isn't a generator expression. - basis = tuple(basis) + @classmethod + def random_instance(cls, *args, **kwargs): + """ + Return a random instance of this type of algebra. - if rank > 1 and normalize_basis: - # We'll need sqrt(2) to normalize the basis, and this - # winds up in the multiplication table, so the whole - # algebra needs to be over the field extension. - R = PolynomialRing(field, 'z') - z = R.gen() - p = z**2 - 2 - if p.is_irreducible(): - field = NumberField(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 ) - basis = tuple(s*c for (s,c) in izip(basis,self._basis_normalizers)) + This method should be implemented in each subclass. + """ + from sage.misc.prandom import choice + eja_class = choice(cls.__subclasses__()) - Qs = self.multiplication_table_from_matrix_basis(basis) + # These all bubble up to the RationalBasisEJA superclass + # constructor, so any (kw)args valid there are also valid + # here. + return eja_class.random_instance(*args, **kwargs) - fdeja = super(MatrixEuclideanJordanAlgebra, self) - return fdeja.__init__(field, - Qs, - rank=rank, - natural_basis=basis, - **kwargs) +class MatrixEJA: + @staticmethod + def jordan_product(X,Y): + return (X*Y + Y*X)/2 - @cached_method - def _charpoly_coeff(self, i): - """ - Override the parent method with something that tries to compute - over a faster (non-extension) field. + @staticmethod + def trace_inner_product(X,Y): + r""" + A trace inner-product for matrices that aren't embedded in the + reals. """ - if self._basis_normalizers is None: - # We didn't normalize, so assume that the basis we started - # with had entries in a nice field. - return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i) - else: - basis = ( (b/n) for (b,n) in izip(self.natural_basis(), - self._basis_normalizers) ) - - # Do this over the rationals and convert back at the end. - J = MatrixEuclideanJordanAlgebra(QQ, - basis, - self.rank(), - normalize_basis=False) - (_,x,_,_) = J._charpoly_matrix_system() - p = J._charpoly_coeff(i) - # p might be missing some vars, have to substitute "optionally" - pairs = izip(x.base_ring().gens(), self._basis_normalizers) - substitutions = { v: v*c for (v,c) in pairs } - result = p.subs(substitutions) - - # The result of "subs" can be either a coefficient-ring - # element or a polynomial. Gotta handle both cases. - if result in QQ: - return self.base_ring()(result) - else: - return result.change_ring(self.base_ring()) - + # We take the norm (absolute value) because Octonions() isn't + # smart enough yet to coerce its one() into the base field. + return (X*Y).trace().abs() +class RealEmbeddedMatrixEJA(MatrixEJA): @staticmethod - def multiplication_table_from_matrix_basis(basis): - """ - At least three of the five simple Euclidean Jordan algebras have the - symmetric multiplication (A,B) |-> (AB + BA)/2, where the - multiplication on the right is matrix multiplication. Given a basis - for the underlying matrix space, this function returns a - multiplication table (obtained by looping through the basis - elements) for an algebra of those matrices. - """ - # In S^2, for example, we nominally have four coordinates even - # though the space is of dimension three only. The vector space V - # is supposed to hold the entire long vector, and the subspace W - # of V will be spanned by the vectors that arise from symmetric - # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. - field = basis[0].base_ring() - dimension = basis[0].nrows() - - V = VectorSpace(field, dimension**2) - W = V.span_of_basis( _mat2vec(s) for s in basis ) - n = len(basis) - mult_table = [[W.zero() for j in xrange(n)] for i in xrange(n)] - for i in xrange(n): - for j in xrange(n): - mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2 - mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry)) + def dimension_over_reals(): + r""" + The dimension of this matrix's base ring over the reals. - return mult_table + The reals are dimension one over themselves, obviously; that's + just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi` + have dimension two. Finally, the quaternions have dimension + four over the reals. + This is used to determine the size of the matrix returned from + :meth:`real_embed`, among other things. + """ + raise NotImplementedError - @staticmethod - def real_embed(M): + @classmethod + def real_embed(cls,M): """ Embed the matrix ``M`` into a space of real matrices. @@ -1004,56 +1769,71 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): real_embed(M*N) = real_embed(M)*real_embed(N) """ - raise NotImplementedError + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + return M - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of :meth:`real_embed`. """ - raise NotImplementedError + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero(): + raise ValueError("the matrix 'M' must be a real embedding") + return M @classmethod - def natural_inner_product(cls,X,Y): - Xu = cls.real_unembed(X) - Yu = cls.real_unembed(Y) - tr = (Xu*Yu).trace() - if tr in RLF: - # It's real already. - return tr - - # Otherwise, try the thing that works for complex numbers; and - # if that doesn't work, the thing that works for quaternions. - try: - return tr.vector()[0] # real part, imag part is index 1 - except AttributeError: - # A quaternions doesn't have a vector() method, but does - # have coefficient_tuple() method that returns the - # coefficients of 1, i, j, and k -- in that order. - return tr.coefficient_tuple()[0] + def trace_inner_product(cls,X,Y): + r""" + Compute the trace inner-product of two real-embeddings. + SETUP:: -class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): - @staticmethod - def real_embed(M): - """ - The identity function, for embedding real matrices into real - matrices. - """ - return M + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: QuaternionHermitianEJA) - @staticmethod - def real_unembed(M): - """ - The identity function, for unembedding real matrices from real - matrices. - """ - return M + EXAMPLES:: + + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace().real() + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True + :: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace().coefficient_tuple()[0] + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True -class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): + """ + # This does in fact compute the real part of the trace. + # If we compute the trace of e.g. a complex matrix M, + # then we do so by adding up its diagonal entries -- + # call them z_1 through z_n. The real embedding of z_1 + # will be a 2-by-2 REAL matrix [a, b; -b, a] whose trace + # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth. + return (X*Y).trace()/cls.dimension_over_reals() + +class RealSymmetricEJA(ConcreteEJA, MatrixEJA): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1066,20 +1846,28 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): EXAMPLES:: sage: J = RealSymmetricEJA(2) - sage: e0, e1, e2 = J.gens() - sage: e0*e0 - e0 - sage: e1*e1 - 1/2*e0 + 1/2*e2 - sage: e2*e2 - e2 + sage: b0, b1, b2 = J.gens() + sage: b0*b0 + b0 + sage: b1*b1 + 1/2*b0 + 1/2*b2 + sage: b2*b2 + b2 + + In theory, our "field" can be any subfield of the reals:: + + sage: RealSymmetricEJA(2, field=RDF, check_axioms=True) + Euclidean Jordan algebra of dimension 3 over Real Double Field + sage: RealSymmetricEJA(2, field=RR, check_axioms=True) + Euclidean Jordan algebra of dimension 3 over Real Field with + 53 bits of precision TESTS: The dimension of this algebra is `(n^2 + n) / 2`:: sage: set_random_seed() - sage: n_max = RealSymmetricEJA._max_test_case_size() + sage: n_max = RealSymmetricEJA._max_random_instance_size() sage: n = ZZ.random_element(1, n_max) sage: J = RealSymmetricEJA(n) sage: J.dimension() == (n^2 + n)/2 @@ -1090,9 +1878,9 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): 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 @@ -1104,24 +1892,10 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: RealSymmetricEJA(3, prefix='q').gens() (q0, q1, q2, q3, q4, q5) - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: - - sage: set_random_seed() - sage: J = RealSymmetricEJA.random_instance() - sage: all( b.norm() == 1 for b in J.gens() ) - True + We can construct the (trivial) algebra of rank zero:: - Since our natural basis is normalized with respect to the natural - 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:: - - sage: set_random_seed() - sage: x = RealSymmetricEJA.random_instance().random_element() - sage: x.operator().matrix().is_symmetric() - True + sage: RealSymmetricEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @classmethod @@ -1137,7 +1911,7 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = RealSymmetricEJA._denormalized_basis(n,QQ) + sage: B = RealSymmetricEJA._denormalized_basis(n,ZZ) sage: all( M.is_symmetric() for M in B) True @@ -1145,30 +1919,93 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): # The basis of symmetric matrices, as matrices, in their R^(n-by-n) # coordinates. S = [] - for i in xrange(n): - for j in xrange(i+1): + for i in range(n): + for j in range(i+1): Eij = matrix(field, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij else: Sij = Eij + Eij.transpose() S.append(Sij) - return S + return tuple(S) @staticmethod - def _max_test_case_size(): + def _max_random_instance_size(): return 4 # Dimension 10 + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + def __init__(self, n, field=AA, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n,field), + self.jordan_product, + self.trace_inner_product, + field=field, + 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(). + self.rank.set_cache(n) + idV = self.matrix_space().one() + self.one.set_cache(self(idV)) - def __init__(self, n, field=QQ, **kwargs): - basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs) -class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): +class ComplexMatrixEJA(RealEmbeddedMatrixEJA): + # A manual dictionary-cache for the complex_extension() method, + # since apparently @classmethods can't also be @cached_methods. + _complex_extension = {} + + @classmethod + def complex_extension(cls,field): + r""" + The complex field that we embed/unembed, as an extension + of the given ``field``. + """ + if field in cls._complex_extension: + return cls._complex_extension[field] + + # Sage doesn't know how to adjoin the complex "i" (the root of + # x^2 + 1) to a field in a general way. Here, we just enumerate + # all of the cases that I have cared to support so far. + if field is AA: + # Sage doesn't know how to embed AA into QQbar, i.e. how + # to adjoin sqrt(-1) to AA. + F = QQbar + elif not field.is_exact(): + # RDF or RR + F = field.complex_field() + else: + # Works for QQ and... maybe some other fields. + R = PolynomialRing(field, 'z') + z = R.gen() + F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) + + cls._complex_extension[field] = F + return F + @staticmethod - def real_embed(M): + def dimension_over_reals(): + return 2 + + @classmethod + def real_embed(cls,M): """ Embed the n-by-n complex matrix ``M`` into the space of real matrices of size 2n-by-2n via the map the sends each entry `z = a + @@ -1176,18 +2013,17 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) sage: x4 = F(6) sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) - sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M) + sage: ComplexMatrixEJA.real_embed(M) [ 4 -2| 1 2] [ 2 4|-2 1] [-----+-----] @@ -1199,41 +2035,41 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() - sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size() - sage: n = ZZ.random_element(n_max) - sage: F = QuadraticField(-1, 'i') + sage: n = ZZ.random_element(3) + sage: F = QuadraticField(-1, 'I') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) - sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X) - sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y) - sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y) + sage: Xe = ComplexMatrixEJA.real_embed(X) + sage: Ye = ComplexMatrixEJA.real_embed(Y) + sage: XYe = ComplexMatrixEJA.real_embed(X*Y) sage: Xe*Ye == XYe True """ + super().real_embed(M) n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - field = M.base_ring() + + # We don't need any adjoined elements... + field = M.base_ring().base_ring() + blocks = [] for z in M.list(): - a = z.vector()[0] # real part, I guess - b = z.vector()[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] ])) - # We can drop the imaginaries here. - return matrix.block(field.base_ring(), n, blocks) + return matrix.block(field, n, blocks) - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of _embed_complex_matrix(). SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: @@ -1241,42 +2077,34 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [-2, 1, -4, 3], ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) - sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + sage: ComplexMatrixEJA.real_unembed(A) + [ 2*I + 1 4*I + 3] + [ 10*I + 9 12*I + 11] TESTS: Unembedding is the inverse of embedding:: sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: M = random_matrix(F, 3) - sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) - sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + sage: Me = ComplexMatrixEJA.real_embed(M) + sage: ComplexMatrixEJA.real_unembed(Me) == M True """ + super().real_unembed(M) n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(2).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") - - # 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() - F = NumberField(z**2 + 1,'i', embedding=CLF(-1).sqrt()) + d = cls.dimension_over_reals() + F = cls.complex_extension(M.base_ring()) i = F.gen() # Go top-left to bottom-right (reading order), converting every # 2-by-2 block we see to a single complex element. elements = [] - for k in xrange(n/2): - for j in xrange(n/2): - submat = M[2*k:2*k+2,2*j:2*j+2] + for k in range(n/d): + for j in range(n/d): + submat = M[d*k:d*k+d,d*j:d*j+d] if submat[0,0] != submat[1,1]: raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: @@ -1284,41 +2112,10 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): z = submat[0,0] + submat[0,1]*i elements.append(z) - return matrix(F, n/2, elements) - - - @classmethod - def natural_inner_product(cls,X,Y): - """ - Compute a natural inner product in this algebra directly from - its real embedding. - - SETUP:: - - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA - - TESTS: - - This gives the same answer as the slow, default method implemented - in :class:`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: X = ComplexHermitianEJA.real_unembed(Xe) - sage: Y = ComplexHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().vector()[0] - sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) - sage: actual == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 + return matrix(F, n/d, elements) -class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): +class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1329,12 +2126,22 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + EXAMPLES: + + In theory, our "field" can be any subfield of the reals:: + + sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True) + Euclidean Jordan algebra of dimension 4 over Real Double Field + sage: ComplexHermitianEJA(2, field=RR, check_axioms=True) + Euclidean Jordan algebra of dimension 4 over Real Field with + 53 bits of precision + TESTS: The dimension of this algebra is `n^2`:: sage: set_random_seed() - sage: n_max = ComplexHermitianEJA._max_test_case_size() + sage: n_max = ComplexHermitianEJA._max_random_instance_size() sage: n = ZZ.random_element(1, n_max) sage: J = ComplexHermitianEJA(n) sage: J.dimension() == n^2 @@ -1345,9 +2152,9 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): 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 @@ -1359,24 +2166,10 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: ComplexHermitianEJA(2, prefix='z').gens() (z0, z1, z2, z3) - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: - - sage: set_random_seed() - sage: J = ComplexHermitianEJA.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 - 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:: + We can construct the (trivial) algebra of rank zero:: - sage: set_random_seed() - sage: x = ComplexHermitianEJA.random_instance().random_element() - sage: x.operator().matrix().is_symmetric() - True + sage: ComplexHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @@ -1399,16 +2192,15 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: field = QuadraticField(2, 'sqrt2') - sage: B = ComplexHermitianEJA._denormalized_basis(n, field) + sage: B = ComplexHermitianEJA._denormalized_basis(n,ZZ) sage: all( M.is_symmetric() for M in B) True """ - R = PolynomialRing(field, 'z') + R = PolynomialRing(ZZ, 'z') z = R.gen() - F = NumberField(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) - I = F.gen() + F = ZZ.extension(z**2 + 1, 'I') + I = F.gen(1) # This is like the symmetric case, but we need to be careful: # @@ -1416,32 +2208,93 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): # * The diagonal will (as a result) be real. # S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(F, n, lambda k,l: k==i and l==j) + Eij = matrix.zero(F,n) + for i in range(n): + for j in range(i+1): + # "build" E_ij + Eij[i,j] = 1 if i == j: Sij = cls.real_embed(Eij) S.append(Sij) 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 the entries, we can drop back to the + # desired real "field" instead of the extension "F". + return tuple( s.change_ring(field) for s in S ) + + + def __init__(self, n, field=AA, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n,field), + self.jordan_product, + self.trace_inner_product, + field=field, + 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(). + self.rank.set_cache(n) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) - # Since we embedded these, we can drop back to the "field" that we - # started with instead of the complex extension "F". - return ( s.change_ring(field) for s in S ) + @staticmethod + def _max_random_instance_size(): + return 3 # Dimension 9 + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + +class QuaternionMatrixEJA(RealEmbeddedMatrixEJA): + + # A manual dictionary-cache for the quaternion_extension() method, + # since apparently @classmethods can't also be @cached_methods. + _quaternion_extension = {} - def __init__(self, n, field=QQ, **kwargs): - basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs) + @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 -class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @staticmethod - def real_embed(M): + def dimension_over_reals(): + return 4 + + @classmethod + def real_embed(cls,M): """ Embed the n-by-n quaternion matrix ``M`` into the space of real matrices of size 4n-by-4n by first sending each quaternion entry `z @@ -1451,8 +2304,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -1460,7 +2312,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: i,j,k = Q.gens() sage: x = 1 + 2*i + 3*j + 4*k sage: M = matrix(Q, 1, [[x]]) - sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M) + sage: QuaternionMatrixEJA.real_embed(M) [ 1 2 3 4] [-2 1 -4 3] [-3 4 1 -2] @@ -1469,24 +2321,22 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() - sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size() - sage: n = ZZ.random_element(n_max) + sage: n = ZZ.random_element(2) sage: Q = QuaternionAlgebra(QQ,-1,-1) sage: X = random_matrix(Q, n) sage: Y = random_matrix(Q, n) - sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X) - sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y) - sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y) + sage: Xe = QuaternionMatrixEJA.real_embed(X) + sage: Ye = QuaternionMatrixEJA.real_embed(Y) + sage: XYe = QuaternionMatrixEJA.real_embed(X*Y) sage: Xe*Ye == XYe True """ + super().real_embed(M) quaternions = M.base_ring() n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - F = QuadraticField(-1, 'i') + F = QuadraticField(-1, 'I') i = F.gen() blocks = [] @@ -1498,7 +2348,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): d = t[3] cplxM = matrix(F, 2, [[ a + b*i, c + d*i], [-c + d*i, a - b*i]]) - realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM) + realM = ComplexMatrixEJA.real_embed(cplxM) blocks.append(realM) # We should have real entries by now, so use the realest field @@ -1507,15 +2357,14 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of _embed_quaternion_matrix(). SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -1523,7 +2372,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [-2, 1, -4, 3], ....: [-3, 4, 1, -2], ....: [-4, -3, 2, 1]]) - sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M) + sage: QuaternionMatrixEJA.real_unembed(M) [1 + 2*i + 3*j + 4*k] TESTS: @@ -1533,78 +2382,43 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: Q = QuaternionAlgebra(QQ, -1, -1) sage: M = random_matrix(Q, 3) - sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M) - sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + sage: Me = QuaternionMatrixEJA.real_embed(M) + sage: QuaternionMatrixEJA.real_unembed(Me) == M True """ + super().real_unembed(M) n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(4).is_zero(): - raise ValueError("the matrix 'M' must be a quaternion embedding") + d = cls.dimension_over_reals() # Use the base ring of the matrix to ensure that its entries can be # multiplied by elements of the quaternion algebra. - 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 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 # quaternion block. elements = [] - for l in xrange(n/4): - for m in xrange(n/4): - submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed( - M[4*l:4*l+4,4*m:4*m+4] ) + for l in range(n/d): + for m in range(n/d): + submat = ComplexMatrixEJA.real_unembed( + M[d*l:d*l+d,d*m:d*m+d] ) if submat[0,0] != submat[1,1].conjugate(): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): raise ValueError('bad off-diagonal submatrix') - z = submat[0,0].vector()[0] # real part - z += submat[0,0].vector()[1]*i # imag part - z += submat[0,1].vector()[0]*j # real part - z += submat[0,1].vector()[1]*k # imag part + z = submat[0,0].real() + z += submat[0,0].imag()*i + z += submat[0,1].real()*j + z += submat[0,1].imag()*k elements.append(z) - return matrix(Q, n/4, elements) - - - @classmethod - def natural_inner_product(cls,X,Y): - """ - Compute a natural inner product in this algebra directly from - its real embedding. + return matrix(Q, n/d, elements) - SETUP:: - - sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - TESTS: - - This gives the same answer as the slow, default method implemented - in :class:`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: 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 == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 - - -class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, - KnownRankEJA): - """ +class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): + r""" The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the real-part-of-trace inner product. It has dimension `2n^2 - n` over @@ -1614,12 +2428,22 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA + EXAMPLES: + + In theory, our "field" can be any subfield of the reals:: + + sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True) + Euclidean Jordan algebra of dimension 6 over Real Double Field + sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True) + Euclidean Jordan algebra of dimension 6 over Real Field with + 53 bits of precision + TESTS: The dimension of this algebra is `2*n^2 - n`:: sage: set_random_seed() - sage: n_max = QuaternionHermitianEJA._max_test_case_size() + sage: n_max = QuaternionHermitianEJA._max_random_instance_size() sage: n = ZZ.random_element(1, n_max) sage: J = QuaternionHermitianEJA(n) sage: J.dimension() == 2*(n^2) - n @@ -1630,9 +2454,9 @@ 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 @@ -1644,24 +2468,10 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: QuaternionHermitianEJA(2, prefix='a').gens() (a0, a1, a2, a3, a4, a5) - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: + We can construct the (trivial) algebra of rank zero:: - sage: set_random_seed() - sage: J = QuaternionHermitianEJA.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 - 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:: - - sage: set_random_seed() - sage: x = QuaternionHermitianEJA.random_instance().random_element() - sage: x.operator().matrix().is_symmetric() - True + sage: QuaternionHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @classmethod @@ -1683,7 +2493,7 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ) + sage: B = QuaternionHermitianEJA._denormalized_basis(n,ZZ) sage: all( M.is_symmetric() for M in B ) True @@ -1697,39 +2507,349 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, # * The diagonal will (as a result) be real. # S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(Q, n, lambda k,l: k==i and l==j) + Eij = matrix.zero(Q,n) + for i in range(n): + for j in range(i+1): + # "build" E_ij + Eij[i,j] = 1 if i == j: Sij = cls.real_embed(Eij) S.append(Sij) 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 the entries, we can drop back to the + # desired real "field" instead of the quaternion algebra "Q". + return tuple( s.change_ring(field) for s in S ) + + + def __init__(self, n, field=AA, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n,field), + self.jordan_product, + self.trace_inner_product, + field=field, + 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(). + self.rank.set_cache(n) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) + + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum rank of a random QuaternionHermitianEJA. + """ + return 2 # Dimension 6 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class HadamardEJA(ConcreteEJA): + """ + Return the Euclidean Jordan Algebra corresponding to the set + `R^n` under the Hadamard product. + + Note: this is nothing more than the Cartesian product of ``n`` + copies of the spin algebra. Once Cartesian product algebras + are implemented, this can go. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + EXAMPLES: + + This multiplication table can be verified by hand:: + + sage: J = HadamardEJA(3) + sage: b0,b1,b2 = J.gens() + sage: b0*b0 + b0 + sage: b0*b1 + 0 + sage: b0*b2 + 0 + sage: b1*b1 + b1 + sage: b1*b2 + 0 + sage: b2*b2 + b2 + + TESTS: + + We can change the generator prefix:: + + sage: HadamardEJA(3, prefix='r').gens() + (r0, r1, r2) + + """ + def __init__(self, n, field=AA, **kwargs): + if n == 0: + jordan_product = lambda x,y: x + inner_product = lambda x,y: x + else: + def jordan_product(x,y): + P = x.parent() + return P( xi*yi for (xi,yi) in zip(x,y) ) + + def inner_product(x,y): + return (x.T*y)[0,0] + + # New defaults for keyword arguments. Don't orthonormalize + # because our basis is already orthonormal with respect to our + # inner-product. Don't check the axioms, because we know this + # is a valid EJA... but do double-check if the user passes + # check_axioms=True. Note: we DON'T override the "check_field" + # default here, because the user can pass in a field! + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + column_basis = tuple( b.column() + for b in FreeModule(field, n).basis() ) + super().__init__(column_basis, + jordan_product, + inner_product, + field=field, + associative=True, + **kwargs) + self.rank.set_cache(n) + + if n == 0: + self.one.set_cache( self.zero() ) + else: + self.one.set_cache( sum(self.gens()) ) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random HadamardEJA. + """ + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class BilinearFormEJA(ConcreteEJA): + r""" + The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` + with the half-trace inner product and jordan product ``x*y = + (,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is + a symmetric positive-definite "bilinear form" matrix. Its + dimension is the size of `B`, and it has rank two in dimensions + larger than two. It reduces to the ``JordanSpinEJA`` when `B` is + the identity matrix of order ``n``. + + We insist that the one-by-one upper-left identity block of `B` be + passed in as well so that we can be passed a matrix of size zero + to construct a trivial algebra. - # Since we embedded these, we can drop back to the "field" that we - # started with instead of the quaternion algebra "Q". - return ( s.change_ring(field) for s in S ) + SETUP:: + + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) + + EXAMPLES: + + When no bilinear form is specified, the identity matrix is used, + and the resulting algebra is the Jordan spin algebra:: + + sage: B = matrix.identity(AA,3) + sage: J0 = BilinearFormEJA(B) + sage: J1 = JordanSpinEJA(3) + sage: J0.multiplication_table() == J0.multiplication_table() + True + + An error is raised if the matrix `B` does not correspond to a + positive-definite bilinear form:: + + sage: B = matrix.random(QQ,2,3) + sage: J = BilinearFormEJA(B) + Traceback (most recent call last): + ... + ValueError: bilinear form is not positive-definite + sage: B = matrix.zero(QQ,3) + sage: J = BilinearFormEJA(B) + Traceback (most recent call last): + ... + ValueError: bilinear form is not positive-definite + + TESTS: + We can create a zero-dimensional algebra:: - def __init__(self, n, field=QQ, **kwargs): - basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) + sage: B = matrix.identity(AA,0) + sage: J = BilinearFormEJA(B) + sage: J.basis() + Finite family {} + We can check the multiplication condition given in the Jordan, von + Neumann, and Wigner paper (and also discussed on my "On the + symmetry..." paper). Note that this relies heavily on the standard + choice of basis, as does anything utilizing the bilinear form + matrix. We opt not to orthonormalize the basis, because if we + did, we would have to normalize the `s_{i}` in a similar manner:: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B11 = matrix.identity(QQ,1) + sage: B22 = M.transpose()*M + sage: B = block_matrix(2,2,[ [B11,0 ], + ....: [0, B22 ] ]) + sage: J = BilinearFormEJA(B, orthonormalize=False) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() ) + ....: for ei in eis ] + sage: actual = [ sis[i]*sis[j] + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: expected = [ J.one() if i == j else J.zero() + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: actual == expected + True -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): + """ + def __init__(self, B, field=AA, **kwargs): + # The matrix "B" is supplied by the user in most cases, + # so it makes sense to check whether or not its positive- + # definite unless we are specifically asked not to... + if ("check_axioms" not in kwargs) or kwargs["check_axioms"]: + if not B.is_positive_definite(): + raise ValueError("bilinear form is not positive-definite") + + # However, all of the other data for this EJA is computed + # by us in manner that guarantees the axioms are + # satisfied. So, again, unless we are specifically asked to + # verify things, we'll skip the rest of the checks. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + def inner_product(x,y): + return (y.T*B*x)[0,0] + + def jordan_product(x,y): + P = x.parent() + x0 = x[0,0] + xbar = x[1:,0] + y0 = y[0,0] + ybar = y[1:,0] + z0 = inner_product(y,x) + zbar = y0*xbar + x0*ybar + return P([z0] + zbar.list()) + + n = B.nrows() + column_basis = tuple( b.column() + for b in FreeModule(field, n).basis() ) + + # 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, + field=field, + associative=associative, + **kwargs) + + # The rank of this algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded + # by the ambient dimension). + self.rank.set_cache(min(n,2)) + + if n == 0: + self.one.set_cache( self.zero() ) + else: + self.one.set_cache( self.monomial(0) ) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random BilinearFormEJA. + """ + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + if n.is_zero(): + B = matrix.identity(ZZ, n) + return cls(B, **kwargs) + + B11 = matrix.identity(ZZ, 1) + M = matrix.random(ZZ, n-1) + I = matrix.identity(ZZ, n-1) + alpha = ZZ.zero() + while alpha.is_zero(): + alpha = ZZ.random_element().abs() + B22 = M.transpose()*M + alpha*I + + from sage.matrix.special import block_matrix + B = block_matrix(2,2, [ [B11, ZZ(0) ], + [ZZ(0), B22 ] ]) + + return cls(B, **kwargs) + + +class JordanSpinEJA(BilinearFormEJA): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = - (, x0*y_bar + y0*x_bar)``. It has dimension `n` over + (, x0*y_bar + y0*x_bar)``. It has dimension `n` over the reals. SETUP:: @@ -1741,20 +2861,20 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): This multiplication table can be verified by hand:: sage: J = JordanSpinEJA(4) - sage: e0,e1,e2,e3 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - e1 - sage: e0*e2 - e2 - sage: e0*e3 - e3 - sage: e1*e2 + sage: b0,b1,b2,b3 = J.gens() + sage: b0*b0 + b0 + sage: b0*b1 + b1 + sage: b0*b2 + b2 + sage: b0*b3 + b3 + sage: b1*b2 0 - sage: e1*e3 + sage: b1*b3 0 - sage: e2*e3 + sage: b2*b3 0 We can change the generator prefix:: @@ -1762,50 +2882,594 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: JordanSpinEJA(2, prefix='B').gens() (B0, B1) + TESTS: + + Ensure that we have the usual inner product on `R^n`:: + + sage: set_random_seed() + sage: J = JordanSpinEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + """ - def __init__(self, n, field=QQ, **kwargs): - V = VectorSpace(field, n) - mult_table = [[V.zero() for j in xrange(n)] for i in xrange(n)] - for i in xrange(n): - for j in xrange(n): - x = V.gen(i) - y = V.gen(j) - x0 = x[0] - xbar = x[1:] - y0 = y[0] - ybar = y[1:] - # z = x*y - z0 = x.inner_product(y) - zbar = y0*xbar + x0*ybar - z = V([z0] + zbar.list()) - mult_table[i][j] = z - - # The rank of the spin algebra is two, unless we're in a - # one-dimensional ambient space (because the rank is bounded by - # the ambient dimension). - fdeja = super(JordanSpinEJA, self) - return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) + def __init__(self, n, *args, **kwargs): + # This is a special case of the BilinearFormEJA with the + # identity matrix as its bilinear form. + B = matrix.identity(ZZ, n) - def inner_product(self, x, y): + # Don't orthonormalize because our basis is already + # orthonormal with respect to our inner-product. + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + + # But also don't pass check_field=False here, because the user + # can pass in a field! + super().__init__(B, *args, **kwargs) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random JordanSpinEJA. """ - Faster to reimplement than to use natural representations. + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + + Needed here to override the implementation for ``BilinearFormEJA``. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class TrivialEJA(ConcreteEJA): + """ + The trivial Euclidean Jordan algebra consisting of only a zero element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import TrivialEJA + + EXAMPLES:: + + sage: J = TrivialEJA() + sage: J.dimension() + 0 + sage: J.zero() + 0 + sage: J.one() + 0 + sage: 7*J.one()*12*J.one() + 0 + sage: J.one().inner_product(J.one()) + 0 + sage: J.one().norm() + 0 + sage: J.one().subalgebra_generated_by() + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + sage: J.rank() + 0 + + """ + def __init__(self, **kwargs): + jordan_product = lambda x,y: x + inner_product = lambda x,y: 0 + basis = () + + # New defaults for keyword arguments + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + 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) + self.one.set_cache( self.zero() ) + + @classmethod + def random_instance(cls, **kwargs): + # We don't take a "size" argument so the superclass method is + # inappropriate for us. + return cls(**kwargs) + + +class CartesianProductEJA(FiniteDimensionalEJA): + r""" + The external (orthogonal) direct sum of two or more Euclidean + Jordan algebras. Every Euclidean Jordan algebra decomposes into an + orthogonal direct sum of simple Euclidean Jordan algebras which is + then isometric to a Cartesian product, so no generality is lost by + providing only this construction. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: CartesianProductEJA, + ....: HadamardEJA, + ....: JordanSpinEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + The Jordan product is inherited from our factors and implemented by + our CombinatorialFreeModule Cartesian product superclass:: + + sage: set_random_seed() + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: x,y = J.random_elements(2) + sage: x*y in J + True + + The ability to retrieve the original factors is implemented by our + CombinatorialFreeModule Cartesian product superclass:: + + sage: J1 = HadamardEJA(2, field=QQ) + sage: J2 = JordanSpinEJA(3, field=QQ) + sage: J = cartesian_product([J1,J2]) + sage: J.cartesian_factors() + (Euclidean Jordan algebra of dimension 2 over Rational Field, + Euclidean Jordan algebra of dimension 3 over Rational Field) + + You can provide more than two factors:: + + sage: J1 = HadamardEJA(2) + sage: J2 = JordanSpinEJA(3) + sage: J3 = RealSymmetricEJA(3) + sage: cartesian_product([J1,J2,J3]) + Euclidean Jordan algebra of dimension 2 over Algebraic Real + Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic + 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() + +----++----+----+----+ + | * || b0 | b1 | b2 | + +====++====+====+====+ + | b0 || b0 | 0 | 0 | + +----++----+----+----+ + | b1 || 0 | b1 | 0 | + +----++----+----+----+ + | b2 || 0 | 0 | b2 | + +----++----+----+----+ + sage: HadamardEJA(3).multiplication_table() + +----++----+----+----+ + | * || b0 | b1 | b2 | + +====++====+====+====+ + | b0 || b0 | 0 | 0 | + +----++----+----+----+ + | b1 || 0 | b1 | 0 | + +----++----+----+----+ + | b2 || 0 | 0 | b2 | + +----++----+----+----+ + + TESTS: + + All factors must share the same base field:: + + sage: J1 = HadamardEJA(2, field=QQ) + sage: J2 = RealSymmetricEJA(2) + sage: CartesianProductEJA((J1,J2)) + Traceback (most recent call last): + ... + 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 + + """ + 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") + + associative = all( f.is_associative() for f in factors ) + + 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) + + basis = tuple( MS(b) for b in basis ) + + # 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) + + 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 JordanSpinEJA + 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) + + """ + 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): + r""" + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: HadamardEJA, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA) + + EXAMPLES: + + The projection morphisms are Euclidean Jordan algebra + operators:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.cartesian_projection(0) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [1 0 0 0 0] + [0 1 0 0 0] + Domain: Euclidean Jordan algebra of dimension 2 over Algebraic + Real Field (+) Euclidean Jordan algebra of dimension 3 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 2 over Algebraic + Real Field + sage: J.cartesian_projection(1) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [0 0 1 0 0] + [0 0 0 1 0] + [0 0 0 0 1] + Domain: Euclidean Jordan algebra of dimension 2 over Algebraic + Real Field (+) Euclidean Jordan algebra of dimension 3 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic + Real Field + + The projections work the way you'd expect on the vector + representation of an element:: + + sage: J1 = JordanSpinEJA(2) + sage: J2 = ComplexHermitianEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: pi_left = J.cartesian_projection(0) + sage: pi_right = J.cartesian_projection(1) + sage: pi_left(J.one()).to_vector() + (1, 0) + sage: pi_right(J.one()).to_vector() + (1, 0, 0, 1) + sage: J.one().to_vector() + (1, 0, 1, 0, 0, 1) TESTS: - Ensure that this is the usual inner product for the algebras - over `R^n`:: + The answer never changes:: sage: set_random_seed() - sage: J = JordanSpinEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.natural_inner_product(X,Y) + sage: J1 = random_eja() + sage: J2 = random_eja() + sage: J = cartesian_product([J1,J2]) + sage: P0 = J.cartesian_projection(0) + sage: P1 = J.cartesian_projection(0) + sage: P0 == P1 + True + + """ + 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 + def cartesian_embedding(self, i): + r""" + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + The embedding morphisms are Euclidean Jordan algebra + operators:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.cartesian_embedding(0) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [1 0] + [0 1] + [0 0] + [0 0] + [0 0] + Domain: Euclidean Jordan algebra of dimension 2 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 2 over + Algebraic Real Field (+) Euclidean Jordan algebra of + dimension 3 over Algebraic Real Field + sage: J.cartesian_embedding(1) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [0 0 0] + [0 0 0] + [1 0 0] + [0 1 0] + [0 0 1] + Domain: Euclidean Jordan algebra of dimension 3 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 2 over + Algebraic Real Field (+) Euclidean Jordan algebra of + dimension 3 over Algebraic Real Field + + The embeddings work the way you'd expect on the vector + representation of an element:: + + sage: J1 = JordanSpinEJA(3) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: iota_left = J.cartesian_embedding(0) + sage: iota_right = J.cartesian_embedding(1) + sage: iota_left(J1.zero()) == J.zero() + True + sage: iota_right(J2.zero()) == J.zero() + True + sage: J1.one().to_vector() + (1, 0, 0) + sage: iota_left(J1.one()).to_vector() + (1, 0, 0, 0, 0, 0) + sage: J2.one().to_vector() + (1, 0, 1) + sage: iota_right(J2.one()).to_vector() + (0, 0, 0, 1, 0, 1) + sage: J.one().to_vector() + (1, 0, 0, 1, 0, 1) + + TESTS: + + The answer never changes:: + + sage: set_random_seed() + sage: J1 = random_eja() + sage: J2 = random_eja() + sage: J = cartesian_product([J1,J2]) + sage: E0 = J.cartesian_embedding(0) + sage: E1 = J.cartesian_embedding(0) + sage: E0 == E1 + True + + Composing a projection with the corresponding inclusion should + produce the identity map, and mismatching them should produce + the zero map:: + + sage: set_random_seed() + sage: J1 = random_eja() + sage: J2 = random_eja() + sage: J = cartesian_product([J1,J2]) + sage: iota_left = J.cartesian_embedding(0) + sage: iota_right = J.cartesian_embedding(1) + sage: pi_left = J.cartesian_projection(0) + sage: pi_right = J.cartesian_projection(1) + sage: pi_left*iota_left == J1.one().operator() + True + sage: pi_right*iota_right == J2.one().operator() + True + sage: (pi_left*iota_right).is_zero() + True + sage: (pi_right*iota_left).is_zero() True """ - return x.to_vector().inner_product(y.to_vector()) + 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()) + + + +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 + +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