X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=204a537df12d2ab4d3d27764c20c830ec324a68a;hb=ad25c5b8995a1cacefbf4d677316b9e7069521ff;hp=ad2afbda465ca37bb1f0e940b8e5fb72b89cdba0;hpb=207dd5b181cee27054080be2c0e6c300850358bf;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index ad2afbd..1fc3618 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -5,76 +5,67 @@ are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. """ +from itertools import repeat - -from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra -from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra -from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis -from sage.functions.other import sqrt +from sage.categories.magmatic_algebras import MagmaticAlgebras +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.lazy_import import lazy_import from sage.misc.prandom import choice -from sage.modules.free_module import VectorSpace -from sage.modules.free_module_element import vector -from sage.rings.integer_ring import ZZ -from sage.rings.number_field.number_field import QuadraticField -from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing -from sage.rings.rational_field import QQ -from sage.structure.element import is_Matrix -from sage.structure.category_object import normalize_names +from sage.misc.table import table +from sage.modules.free_module import FreeModule, VectorSpace +from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, + PolynomialRing, + QuadraticField) +from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement +lazy_import('mjo.eja.eja_subalgebra', + 'FiniteDimensionalEuclideanJordanSubalgebra') +from mjo.eja.eja_utils import _mat2vec + +class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): + + def _coerce_map_from_base_ring(self): + """ + Disable the map from the base ring into the algebra. -from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + Performing a nonsense conversion like this automatically + is counterpedagogical. The fallback is to try the usual + element constructor, which should also fail. + SETUP:: -class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): - @staticmethod - def __classcall_private__(cls, - field, - mult_table, - rank, - names='e', - assume_associative=False, - category=None, - natural_basis=None): - n = len(mult_table) - mult_table = [b.base_extend(field) for b in mult_table] - for b in mult_table: - b.set_immutable() - if not (is_Matrix(b) and b.dimensions() == (n, n)): - raise ValueError("input is not a multiplication table") - mult_table = tuple(mult_table) - - cat = FiniteDimensionalAlgebrasWithBasis(field) - cat.or_subcategory(category) - if assume_associative: - cat = cat.Associative() - - names = normalize_names(n, names) - - fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls) - return fda.__classcall__(cls, - field, - mult_table, - rank=rank, - assume_associative=assume_associative, - names=names, - category=cat, - natural_basis=natural_basis) + 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 a naturally-represented algebra element + """ + return None def __init__(self, field, mult_table, - rank, - names='e', - assume_associative=False, + prefix='e', category=None, - natural_basis=None): + natural_basis=None, + check_field=True, + check_axioms=True): """ SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import ( + ....: FiniteDimensionalEuclideanJordanAlgebra, + ....: JordanSpinEJA, + ....: random_eja) EXAMPLES: @@ -82,173 +73,260 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() + sage: x,y = J.random_elements(2) sage: x*y == y*x True + TESTS: + + The ``field`` we're given must be real with ``check_field=True``:: + + sage: JordanSpinEJA(2,QQbar) + Traceback (most recent call last): + ... + ValueError: scalar field is not real + + The multiplication table must be square with ``check_axioms=True``:: + + sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),())) + Traceback (most recent call last): + ... + ValueError: multiplication table is not square + """ - self._rank = rank + 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") + + # The multiplication table had better be square + n = len(mult_table) + if check_axioms: + if not all( len(l) == n for l in mult_table ): + raise ValueError("multiplication table is not square") + self._natural_basis = natural_basis - self._multiplication_table = mult_table + + if category is None: + category = MagmaticAlgebras(field).FiniteDimensional() + category = category.WithBasis().Unital() + fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, - mult_table, - names=names, + range(n), + prefix=prefix, category=category) - - - def _repr_(self): + self.print_options(bracket='') + + # 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 = [ + list(map(lambda x: self.from_vector(x), ls)) + for ls in mult_table + ] + + if check_axioms: + if not self._is_commutative(): + raise ValueError("algebra is not commutative") + 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 _element_constructor_(self, elt): """ - Return a string representation of ``self``. + Construct an element of this algebra from its natural + representation. + + This gets called only after the parent element _call_ method + fails to find a coercion for the argument. SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + The identity in `S^n` is converted to the identity in the EJA:: + + sage: J = RealSymmetricEJA(3) + sage: I = matrix.identity(QQ,3) + sage: J(I) == J.one() + True + + This skew-symmetric matrix can't be represented in the EJA:: + + sage: J = RealSymmetricEJA(3) + sage: A = matrix(QQ,3, lambda i,j: i-j) + sage: J(A) + Traceback (most recent call last): + ... + ArithmeticError: vector is not in free module TESTS: - Ensure that it says what we think it says:: + 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:: - sage: JordanSpinEJA(2, field=QQ) - Euclidean Jordan algebra of degree 2 over Rational Field - sage: JordanSpinEJA(3, field=RDF) - Euclidean Jordan algebra of degree 3 over Real Double Field + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x = J.random_element() + sage: J(x.to_vector().column()) == x + True + sage: J = JordanSpinEJA.random_instance() + sage: x = J.random_element() + sage: J(x.to_vector().column()) == x + True """ - fmt = "Euclidean Jordan algebra of degree {} over {}" - return fmt.format(self.degree(), self.base_ring()) + msg = "not a naturally-represented algebra element" + if elt == 0: + # The superclass implementation of random_element() + # needs to be able to coerce "0" into the algebra. + return self.zero() + elif elt in self.base_ring(): + # 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(msg) + + # Thanks for nothing! Matrix spaces aren't vector spaces in + # Sage, so we have to figure out its natural-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) + @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 - def _a_regular_element(self): + def _repr_(self): """ - Guess a regular element. Needed to compute the basis for our - characteristic polynomial coefficients. + Return a string representation of ``self``. SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import JordanSpinEJA 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 + Ensure that it says what we think it says:: + 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 - @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() - V = self.vector_space() - V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) ) - b = (V1.basis() + V1.complement().basis()) - return V.span_of_basis(b) + 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] - @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) - + def _is_commutative(self): + r""" + Whether or not this algebra's multiplication table is commutative. - @cached_method - def _charpoly_matrix_system(self): + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. """ - 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. + return all( self.product_on_basis(i,j) == self.product_on_basis(i,j) + for i in range(self.dimension()) + for j in range(self.dimension()) ) + + def _is_jordanian(self): + r""" + Whether or not this algebra's multiplication table respects the + 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 _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 multiplication table. """ - r = self.rank() - n = self.dimension() - - # Construct a new algebra over a multivariate polynomial ring... - names = ['X' + str(i) for i in range(1,n+1)] - R = PolynomialRing(self.base_ring(), names) - J = FiniteDimensionalEuclideanJordanAlgebra(R, - self._multiplication_table, - rank=r) - idmat = matrix.identity(J.base_ring(), n) + # Used to check whether or not something is zero in an inexact + # ring. This number is sufficient to allow the construction of + # QuaternionHermitianEJA(2, RDF) with check_axioms=True. + epsilon = 1e-16 - W = self._charpoly_basis_space() - W = W.change_ring(R.fraction_field()) + 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) - # 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 = J(W(R.gens())) - l1 = [matrix.column(W.coordinates((x**k).vector())) for k in range(r)] - l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)] - A_of_x = matrix.block(R, 1, n, (l1 + l2)) - xr = W.coordinates((x**r).vector()) - return (A_of_x, x, xr, A_of_x.det()) + if self.base_ring().is_exact(): + if diff != 0: + return False + else: + if diff.abs() > epsilon: + return False + return True @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 @@ -260,7 +338,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA EXAMPLES: @@ -268,41 +346,41 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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().vector() + 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 sum( a[k]*(t**k) for k in range(len(a)) ) + return (t**r + sum( a[k]*(t**k) for k in range(r) )) def inner_product(self, x, y): @@ -319,21 +397,80 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES: - The inner product must satisfy its axiom for this algebra to truly - be a Euclidean Jordan Algebra:: + Our inner product is "associative," which means the following for + a symmetric bilinear form:: sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() + sage: x,y,z = J.random_elements(3) sage: (x*y).inner_product(z) == y.inner_product(x*z) True """ - if (not x in self) or (not y in self): - raise TypeError("arguments must live in this algebra") - return x.trace_inner_product(y) + X = x.natural_representation() + Y = y.natural_representation() + return self.natural_inner_product(X,Y) + + + def is_trivial(self): + """ + Return whether or not this algebra is trivial. + + A trivial algebra contains only the zero element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: TrivialEJA) + + EXAMPLES:: + + sage: J = ComplexHermitianEJA(3) + sage: J.is_trivial() + False + + :: + + sage: J = TrivialEJA() + sage: J.is_trivial() + True + + """ + return self.dimension() == 0 + + + def multiplication_table(self): + """ + Return a visual representation of this algebra's multiplication + table (on basis elements). + + SETUP:: + + sage: from mjo.eja.eja_algebra import JordanSpinEJA + + EXAMPLES:: + + sage: J = JordanSpinEJA(4) + sage: J.multiplication_table() + +----++----+----+----+----+ + | * || e0 | e1 | e2 | e3 | + +====++====+====+====+====+ + | e0 || e0 | e1 | e2 | e3 | + +----++----+----+----+----+ + | e1 || e1 | e0 | 0 | 0 | + +----++----+----+----+----+ + | e2 || e2 | 0 | e0 | 0 | + +----++----+----+----+----+ + | e3 || e3 | 0 | 0 | e0 | + +----++----+----+----+----+ + + """ + M = list(self._multiplication_table) # copy + for i in range(len(M)): + # M had better be "square" + M[i] = [self.monomial(i)] + M[i] + M = [["*"] + list(self.gens())] + M + return table(M, header_row=True, header_column=True, frame=True) def natural_basis(self): @@ -358,18 +495,18 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J = RealSymmetricEJA(2) sage: J.basis() - Family (e0, e1, e2) + Finite family {0: e0, 1: e1, 2: e2} sage: J.natural_basis() ( - [1 0] [0 1] [0 0] - [0 0], [1 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ) :: sage: J = JordanSpinEJA(2) sage: J.basis() - Family (e0, e1) + Finite family {0: e0, 1: e1} sage: J.natural_basis() ( [1] [0] @@ -378,2013 +515,1677 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ if self._natural_basis is None: - return tuple( b.vector().column() for b in self.basis() ) + M = self.natural_basis_space() + return tuple( M(b.to_vector()) for b in self.basis() ) else: return self._natural_basis - def rank(self): + def natural_basis_space(self): """ - Return the rank of this EJA. + Return the matrix space in which this algebra's natural basis + elements live. - ALGORITHM: + Generally this will be an `n`-by-`1` column-vector space, + except when the algebra is trivial. There it's `n`-by-`n` + (where `n` is zero), to ensure that two elements of the + natural basis space (empty matrices) can be multiplied. + """ + if self.is_trivial(): + return MatrixSpace(self.base_ring(), 0) + elif 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() - 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. - SETUP:: + @staticmethod + def natural_inner_product(X,Y): + """ + Compute the inner product of two naturally-represented elements. - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealSymmetricEJA, - ....: ComplexHermitianEJA, - ....: QuaternionHermitianEJA, - ....: random_eja) + 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() - EXAMPLES: - The rank of the Jordan spin algebra is always two:: + @cached_method + def one(self): + """ + Return the unit element of this algebra. - sage: JordanSpinEJA(2).rank() - 2 - sage: JordanSpinEJA(3).rank() - 2 - sage: JordanSpinEJA(4).rank() - 2 + SETUP:: - The rank of the `n`-by-`n` Hermitian real, complex, or - quaternion matrices is `n`:: + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: random_eja) - sage: RealSymmetricEJA(2).rank() - 2 - sage: ComplexHermitianEJA(2).rank() - 2 - sage: QuaternionHermitianEJA(2).rank() - 2 - sage: RealSymmetricEJA(5).rank() - 5 - sage: ComplexHermitianEJA(5).rank() - 5 - sage: QuaternionHermitianEJA(5).rank() - 5 + EXAMPLES:: + + sage: J = HadamardEJA(5) + sage: J.one() + e0 + e1 + e2 + e3 + e4 TESTS: - Ensure that every EJA that we know how to construct has a - positive integer rank:: + The identity element acts like the identity:: sage: set_random_seed() - sage: r = random_eja().rank() - sage: r in ZZ and r > 0 + sage: J = random_eja() + sage: x = J.random_element() + sage: J.one()*x == x and x*J.one() == x True - """ - return self._rank + The matrix of the unit element's operator is the identity:: + sage: set_random_seed() + sage: J = random_eja() + sage: actual = J.one().operator().matrix() + sage: expected = matrix.identity(J.base_ring(), J.dimension()) + sage: actual == expected + True - def vector_space(self): """ - Return the vector space that underlies this algebra. + # 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() ] - SETUP:: + # Now we use basis 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) - sage: from mjo.eja.eja_algebra import RealSymmetricEJA + # 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()) ) - EXAMPLES:: + # 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)) - sage: J = RealSymmetricEJA(2) - sage: J.vector_space() - Vector space of dimension 3 over Rational Field + def peirce_decomposition(self, c): """ - return self.zero().vector().parent().ambient_vector_space() + 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. - class Element(FiniteDimensionalAlgebraElement): - """ - An element of a Euclidean Jordan algebra. - """ + INPUT: - def __dir__(self): - """ - Oh man, I should not be doing this. This hides the "disabled" - methods ``left_matrix`` and ``matrix`` from introspection; - in particular it removes them from tab-completion. - """ - return filter(lambda s: s not in ['left_matrix', 'matrix'], - dir(self.__class__) ) + - ``c`` -- an idempotent of this algebra. + OUTPUT: - def __init__(self, A, elt=None): - """ + A triple (J0, J5, J1) containing two subalgebras and one subspace + of this algebra, - SETUP:: + - ``J0`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue zero. - sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, - ....: random_eja) + - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()`` + corresponding to the eigenvalue one-half. - EXAMPLES: + - ``J1`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue one. - The identity in `S^n` is converted to the identity in the EJA:: + 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. - sage: J = RealSymmetricEJA(3) - sage: I = matrix.identity(QQ,3) - sage: J(I) == J.one() - True + SETUP:: - This skew-symmetric matrix can't be represented in the EJA:: + sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA - sage: J = RealSymmetricEJA(3) - sage: A = matrix(QQ,3, lambda i,j: i-j) - sage: J(A) - Traceback (most recent call last): - ... - ArithmeticError: vector is not in free module + EXAMPLES: - TESTS: + 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().natural_representation() + [0 0 0] + [0 0 0] + [0 0 1] + sage: orig_df = AA.options.display_format + sage: AA.options.display_format = 'radical' + sage: J.from_vector(J5.basis()[0]).natural_representation() + [ 0 0 1/2*sqrt(2)] + [ 0 0 0] + [1/2*sqrt(2) 0 0] + sage: J.from_vector(J5.basis()[1]).natural_representation() + [ 0 0 0] + [ 0 0 1/2*sqrt(2)] + [ 0 1/2*sqrt(2) 0] + sage: AA.options.display_format = orig_df + sage: J1.one().natural_representation() + [1 0 0] + [0 1 0] + [0 0 0] - Ensure that we can convert any element of the parent's - underlying vector space back into an algebra element whose - vector representation is what we started with:: + TESTS: - sage: set_random_seed() - sage: J = random_eja() - sage: v = J.vector_space().random_element() - sage: J(v).vector() == v - True + Every algebra decomposes trivially with respect to its identity + element:: - """ - # Goal: if we're given a matrix, and if it lives in our - # parent algebra's "natural ambient space," convert it - # into an algebra element. - # - # The catch is, we make a recursive call after converting - # the given matrix into a vector that lives in the algebra. - # This we need to try the parent class initializer first, - # to avoid recursing forever if we're given something that - # already fits into the algebra, but also happens to live - # in the parent's "natural ambient space" (this happens with - # vectors in R^n). - try: - FiniteDimensionalAlgebraElement.__init__(self, A, elt) - except ValueError: - natural_basis = A.natural_basis() - if elt in natural_basis[0].matrix_space(): - # Thanks for nothing! Matrix spaces aren't vector - # spaces in Sage, so we have to figure out its - # natural-basis coordinates ourselves. - V = VectorSpace(elt.base_ring(), elt.nrows()**2) - W = V.span( _mat2vec(s) for s in natural_basis ) - coords = W.coordinates(_mat2vec(elt)) - FiniteDimensionalAlgebraElement.__init__(self, A, coords) - - def __pow__(self, n): - """ - Return ``self`` raised to the power ``n``. - - Jordan algebras are always power-associative; see for - example Faraut and Koranyi, Proposition II.1.2 (ii). - - We have to override this because our superclass uses row - vectors instead of column vectors! We, on the other hand, - assume column vectors everywhere. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS: - - The definition of `x^2` is the unambiguous `x*x`:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x*x == (x^2) - True - - A few examples of power-associativity:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x*(x*x)*(x*x) == x^5 - True - sage: (x*x)*(x*x*x) == x^5 - True - - We also know that powers operator-commute (Koecher, Chapter - III, Corollary 1):: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: m = ZZ.random_element(0,10) - sage: n = ZZ.random_element(0,10) - sage: Lxm = (x^m).operator() - sage: Lxn = (x^n).operator() - sage: Lxm*Lxn == Lxn*Lxm - True - - """ - if n == 0: - return self.parent().one() - elif n == 1: - return self - else: - return (self.operator()**(n-1))(self) + 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:: - def apply_univariate_polynomial(self, p): - """ - Apply the univariate polynomial ``p`` to this 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 - A priori, SageMath won't allow us to apply a univariate - polynomial to an element of an EJA, because we don't know - that EJAs are rings (they are usually not associative). Of - course, we know that EJAs are power-associative, so the - operation is ultimately kosher. This function sidesteps - the CAS to get the answer we want and expect. + """ + 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 = FiniteDimensionalEuclideanJordanSubalgebra(self, ()) + 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 = FiniteDimensionalEuclideanJordanSubalgebra(self, + gens, + check_axioms=False) + if eigval == 0: + J0 = subalg + elif eigval == 1: + J1 = subalg + else: + raise ValueError("unexpected eigenvalue: %s" % eigval) - SETUP:: + return (J0, J5, J1) - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, - ....: random_eja) - EXAMPLES:: + def random_element(self, thorough=False): + r""" + Return a random element of this algebra. - sage: R = PolynomialRing(QQ, 't') - sage: t = R.gen(0) - sage: p = t^4 - t^3 + 5*t - 2 - sage: J = RealCartesianProductEJA(5) - sage: J.one().apply_univariate_polynomial(p) == 3*J.one() - True + 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. - TESTS: + INPUT: - We should always get back an element of the algebra:: + - ``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 - sage: set_random_seed() - sage: p = PolynomialRing(QQ, 't').random_element() - sage: J = random_eja() - sage: x = J.random_element() - sage: x.apply_univariate_polynomial(p) in J - True + """ + # 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 len(p.variables()) > 1: - raise ValueError("not a univariate polynomial") - P = self.parent() - R = P.base_ring() - # Convert the coeficcients to the parent's base ring, - # because a priori they might live in an (unnecessarily) - # larger ring for which P.sum() would fail below. - cs = [ R(c) for c in p.coefficients(sparse=False) ] - return P.sum( cs[k]*(self**k) for k in range(len(cs)) ) + 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 characteristic_polynomial(self): - """ - Return the characteristic polynomial of this element. + def random_elements(self, count, thorough=False): + """ + Return ``count`` random elements as a tuple. - SETUP:: + INPUT: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + - ``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 - EXAMPLES: + SETUP:: - The rank of `R^3` is three, and the minimal polynomial of - the identity element is `(t-1)` from which it follows that - the characteristic polynomial should be `(t-1)^3`:: + sage: from mjo.eja.eja_algebra import JordanSpinEJA - sage: J = RealCartesianProductEJA(3) - sage: J.one().characteristic_polynomial() - t^3 - 3*t^2 + 3*t - 1 + EXAMPLES:: - Likewise, the characteristic of the zero element in the - rank-three algebra `R^{n}` should be `t^{3}`:: + sage: J = JordanSpinEJA(3) + sage: x,y,z = J.random_elements(3) + sage: all( [ x in J, y in J, z in J ]) + True + sage: len( J.random_elements(10) ) == 10 + True - sage: J = RealCartesianProductEJA(3) - sage: J.zero().characteristic_polynomial() - t^3 + """ + return tuple( self.random_element(thorough) + for idx in range(count) ) - TESTS: + @classmethod + def random_instance(cls, field=AA, **kwargs): + """ + Return a random instance of this type of algebra. - The characteristic polynomial of an element should evaluate - to zero on that element:: + Beware, this will crash for "most instances" because the + constructor below looks wrong. + """ + if cls is TrivialEJA: + # The TrivialEJA class doesn't take an "n" argument because + # there's only one. + return cls(field) - sage: set_random_seed() - sage: x = RealCartesianProductEJA(3).random_element() - sage: p = x.characteristic_polynomial() - sage: x.apply_univariate_polynomial(p) - 0 + n = ZZ.random_element(cls._max_test_case_size() + 1) + return cls(n, field, **kwargs) - """ - p = self.parent().characteristic_polynomial() - return p(*self.vector()) + @cached_method + def _charpoly_coefficients(self): + r""" + The `r` polynomial coefficients of the "characteristic polynomial + of" function. + """ + n = self.dimension() + var_names = [ "X" + str(z) for z in range(1,n+1) ] + R = PolynomialRing(self.base_ring(), var_names) + 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. + return -A_rref.solve_right(E*b).change_ring(R)[:r] + @cached_method + def rank(self): + r""" + Return the rank of this EJA. - def inner_product(self, other): - """ - Return the parent algebra's inner product of myself and ``other``. + 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:: + SETUP:: - sage: from mjo.eja.eja_algebra import ( - ....: ComplexHermitianEJA, - ....: JordanSpinEJA, - ....: QuaternionHermitianEJA, - ....: RealSymmetricEJA, - ....: random_eja) + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA, + ....: random_eja) - EXAMPLES: + EXAMPLES: - The inner product in the Jordan spin algebra is the usual - inner product on `R^n` (this example only works because the - basis for the Jordan algebra is the standard basis in `R^n`):: + The rank of the Jordan spin algebra is always two:: - sage: J = JordanSpinEJA(3) - sage: x = vector(QQ,[1,2,3]) - sage: y = vector(QQ,[4,5,6]) - sage: x.inner_product(y) - 32 - sage: J(x).inner_product(J(y)) - 32 + sage: JordanSpinEJA(2).rank() + 2 + sage: JordanSpinEJA(3).rank() + 2 + sage: JordanSpinEJA(4).rank() + 2 - The inner product on `S^n` is ` = trace(X*Y)`, where - multiplication is the usual matrix multiplication in `S^n`, - so the inner product of the identity matrix with itself - should be the `n`:: - - sage: J = RealSymmetricEJA(3) - sage: J.one().inner_product(J.one()) - 3 + The rank of the `n`-by-`n` Hermitian real, complex, or + quaternion matrices is `n`:: - Likewise, the inner product on `C^n` is ` = - Re(trace(X*Y))`, where we must necessarily take the real - part because the product of Hermitian matrices may not be - Hermitian:: - - sage: J = ComplexHermitianEJA(3) - sage: J.one().inner_product(J.one()) - 3 - - Ditto for the quaternions:: - - sage: J = QuaternionHermitianEJA(3) - sage: J.one().inner_product(J.one()) - 3 - - TESTS: - - Ensure that we can always compute an inner product, and that - it gives us back a real number:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: x.inner_product(y) in RR - True - - """ - P = self.parent() - if not other in P: - raise TypeError("'other' must live in the same algebra") - - return P.inner_product(self, other) - - - def operator_commutes_with(self, other): - """ - Return whether or not this element operator-commutes - with ``other``. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - EXAMPLES: - - The definition of a Jordan algebra says that any element - operator-commutes with its square:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.operator_commutes_with(x^2) - True - - TESTS: - - Test Lemma 1 from Chapter III of Koecher:: - - sage: set_random_seed() - sage: J = random_eja() - sage: u = J.random_element() - sage: v = J.random_element() - sage: lhs = u.operator_commutes_with(u*v) - sage: rhs = v.operator_commutes_with(u^2) - sage: lhs == rhs - True - - Test the first polarization identity from my notes, Koecher - Chapter III, or from Baes (2.3):: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: Lx = x.operator() - sage: Ly = y.operator() - sage: Lxx = (x*x).operator() - sage: Lxy = (x*y).operator() - sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) - True - - Test the second polarization identity from my notes or from - Baes (2.4):: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: Lx = x.operator() - sage: Ly = y.operator() - sage: Lz = z.operator() - sage: Lzy = (z*y).operator() - sage: Lxy = (x*y).operator() - sage: Lxz = (x*z).operator() - sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) - True - - Test the third polarization identity from my notes or from - Baes (2.5):: - - sage: set_random_seed() - sage: J = random_eja() - sage: u = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: Lu = u.operator() - sage: Ly = y.operator() - sage: Lz = z.operator() - sage: Lzy = (z*y).operator() - sage: Luy = (u*y).operator() - sage: Luz = (u*z).operator() - sage: Luyz = (u*(y*z)).operator() - sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz - sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly - sage: bool(lhs == rhs) - True - - """ - if not other in self.parent(): - raise TypeError("'other' must live in the same algebra") - - A = self.operator() - B = other.operator() - return (A*B == B*A) - - - def det(self): - """ - Return my determinant, the product of my eigenvalues. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - EXAMPLES:: - - sage: J = JordanSpinEJA(2) - sage: e0,e1 = J.gens() - sage: x = sum( J.gens() ) - sage: x.det() - 0 - - :: - - sage: J = JordanSpinEJA(3) - sage: e0,e1,e2 = J.gens() - sage: x = sum( J.gens() ) - sage: x.det() - -1 - - TESTS: - - An element is invertible if and only if its determinant is - non-zero:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.is_invertible() == (x.det() != 0) - True - - """ - P = self.parent() - r = P.rank() - p = P._charpoly_coeff(0) - # The _charpoly_coeff function already adds the factor of - # -1 to ensure that _charpoly_coeff(0) is really what - # appears in front of t^{0} in the charpoly. However, - # we want (-1)^r times THAT for the determinant. - return ((-1)**r)*p(*self.vector()) - - - def inverse(self): - """ - Return the Jordan-multiplicative inverse of this element. - - ALGORITHM: + sage: RealSymmetricEJA(4).rank() + 4 + sage: ComplexHermitianEJA(3).rank() + 3 + sage: QuaternionHermitianEJA(2).rank() + 2 - We appeal to the quadratic representation as in Koecher's - Theorem 12 in Chapter III, Section 5. - - SETUP:: + TESTS: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - EXAMPLES: + Ensure that every EJA that we know how to construct has a + positive integer rank, unless the algebra is trivial in + which case its rank will be zero:: - The inverse in the spin factor algebra is given in Alizadeh's - Example 11.11:: + sage: set_random_seed() + sage: J = random_eja() + sage: r = J.rank() + sage: r in ZZ + True + sage: r > 0 or (r == 0 and J.is_trivial()) + True - sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: while not x.is_invertible(): - ....: x = J.random_element() - sage: x_vec = x.vector() - sage: x0 = x_vec[0] - sage: x_bar = x_vec[1:] - sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar)) - sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) - sage: x_inverse = coeff*inv_vec - sage: x.inverse() == J(x_inverse) - True + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: - TESTS: + sage: J = HadamardEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 4 - The identity element is its own inverse:: + :: - sage: set_random_seed() - sage: J = random_eja() - sage: J.one().inverse() == J.one() - True + sage: J = JordanSpinEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 2 - If an element has an inverse, it acts like one:: + :: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: (not x.is_invertible()) or (x.inverse()*x == J.one()) - True + sage: J = RealSymmetricEJA(3) + sage: J.rank.clear_cache() + sage: J.rank() + 3 - The inverse of the inverse is what we started with:: + :: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: (not x.is_invertible()) or (x.inverse().inverse() == x) - True + sage: J = ComplexHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 - The zero element is never invertible:: + :: - sage: set_random_seed() - sage: J = random_eja().zero().inverse() - Traceback (most recent call last): - ... - ValueError: element is not invertible + sage: J = QuaternionHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + """ + return len(self._charpoly_coefficients()) - """ - if not self.is_invertible(): - raise ValueError("element is not invertible") - return (~self.quadratic_representation())(self) + def vector_space(self): + """ + Return the vector space that underlies this algebra. + SETUP:: - def is_invertible(self): - """ - Return whether or not this element is invertible. + sage: from mjo.eja.eja_algebra import RealSymmetricEJA - ALGORITHM: + EXAMPLES:: - The usual way to do this is to check if the determinant is - zero, but we need the characteristic polynomial for the - determinant. The minimal polynomial is a lot easier to get, - so we use Corollary 2 in Chapter V of Koecher to check - whether or not the paren't algebra's zero element is a root - of this element's minimal polynomial. + sage: J = RealSymmetricEJA(2) + sage: J.vector_space() + Vector space of dimension 3 over... - Beware that we can't use the superclass method, because it - relies on the algebra being associative. + """ + return self.zero().to_vector().parent().ambient_vector_space() - SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + Element = FiniteDimensionalEuclideanJordanAlgebraElement - TESTS: - The identity element is always invertible:: - sage: set_random_seed() - sage: J = random_eja() - sage: J.one().is_invertible() - True +def random_eja(field=AA): + """ + Return a "random" finite-dimensional Euclidean Jordan Algebra. - The zero element is never invertible:: + SETUP:: - sage: set_random_seed() - sage: J = random_eja() - sage: J.zero().is_invertible() - False + sage: from mjo.eja.eja_algebra import random_eja - """ - zero = self.parent().zero() - p = self.minimal_polynomial() - return not (p(zero) == zero) + TESTS:: + sage: random_eja() + Euclidean Jordan algebra of dimension... - def is_nilpotent(self): - """ - Return whether or not some power of this element is zero. + """ + classname = choice([TrivialEJA, + HadamardEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) + return classname.random_instance(field=field) - ALGORITHM: - We use Theorem 5 in Chapter III of Koecher, which says that - an element ``x`` is nilpotent if and only if ``x.operator()`` - is nilpotent. And it is a basic fact of linear algebra that - an operator on an `n`-dimensional space is nilpotent if and - only if, when raised to the `n`th power, it equals the zero - operator (for example, see Axler Corollary 8.8). - SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) +class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): + r""" + Algebras whose basis consists of vectors with rational + entries. Equivalently, algebras whose multiplication tables + contain only rational coefficients. - EXAMPLES:: + When an EJA has a basis that can be made rational, we can speed up + the computation of its characteristic polynomial by doing it over + ``QQ``. All of the named EJA constructors that we provide fall + into this category. + """ + @cached_method + def _charpoly_coefficients(self): + r""" + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + if self.base_ring() is QQ: + # There's no need to construct *another* algebra over the + # rationals if this one is already over the rationals. + superclass = super(RationalBasisEuclideanJordanAlgebra, self) + return superclass._charpoly_coefficients() + + mult_table = tuple( + map(lambda x: x.to_vector(), ls) + for ls in self._multiplication_table + ) + + # Do the computation over the rationals. The answer will be + # the same, because our basis coordinates are (essentially) + # rational. + J = FiniteDimensionalEuclideanJordanAlgebra(QQ, + mult_table, + check_field=False, + check_axioms=False) + return J._charpoly_coefficients() + + +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 - sage: J = JordanSpinEJA(3) - sage: x = sum(J.gens()) - sage: x.is_nilpotent() - False + def __init__(self, field, basis, normalize_basis=True, **kwargs): + """ + 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). + """ + # Used in this class's fast _charpoly_coefficients() override. + self._basis_normalizers = None + + # 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) + + if len(basis) > 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 = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt()) + basis = tuple( s.change_ring(field) for s in basis ) + self._basis_normalizers = tuple( + ~(self.natural_inner_product(s,s).sqrt()) for s in basis ) + basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers)) + + Qs = self.multiplication_table_from_matrix_basis(basis) + + super(MatrixEuclideanJordanAlgebra, self).__init__(field, + Qs, + natural_basis=basis, + **kwargs) - TESTS: - The identity element is never nilpotent:: + @cached_method + def _charpoly_coefficients(self): + r""" + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + 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_coefficients() + else: + basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + self._basis_normalizers) ) + + # Do this over the rationals and convert back at the end. + # Only works because we know the entries of the basis are + # integers. The argument ``check_axioms=False`` is required + # because the trace inner-product method for this + # class is a stub and can't actually be checked. + J = MatrixEuclideanJordanAlgebra(QQ, + basis, + normalize_basis=False, + check_field=False, + check_axioms=False) + a = J._charpoly_coefficients() + + # Unfortunately, changing the basis does change the + # coefficients of the characteristic polynomial, but since + # these are really the coefficients of the "characteristic + # polynomial of" function, everything is still nice and + # unevaluated. It's therefore "obvious" how scaling the + # basis affects the coordinate variables X1, X2, et + # cetera. Scaling the first basis vector up by "n" adds a + # factor of 1/n into every "X1" term, for example. So here + # we simply undo the basis_normalizer scaling that we + # performed earlier. + # + # The a[0] access here is safe because trivial algebras + # won't have any basis normalizers and therefore won't + # make it to this "else" branch. + XS = a[0].parent().gens() + subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i] + for i in range(len(XS)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) - sage: set_random_seed() - sage: random_eja().one().is_nilpotent() - False - The additive identity is always nilpotent:: + @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. + if len(basis) == 0: + return [] - sage: set_random_seed() - sage: random_eja().zero().is_nilpotent() - True + field = basis[0].base_ring() + dimension = basis[0].nrows() - """ - P = self.parent() - zero_operator = P.zero().operator() - return self.operator()**P.dimension() == zero_operator + 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 range(n)] for i in range(n)] + for i in range(n): + for j in range(n): + mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2 + mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry)) + return mult_table - def is_regular(self): - """ - Return whether or not this is a regular element. - SETUP:: + @staticmethod + def real_embed(M): + """ + Embed the matrix ``M`` into a space of real matrices. - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) + The matrix ``M`` can have entries in any field at the moment: + the real numbers, complex numbers, or quaternions. And although + they are not a field, we can probably support octonions at some + point, too. This function returns a real matrix that "acts like" + the original with respect to matrix multiplication; i.e. - EXAMPLES: + real_embed(M*N) = real_embed(M)*real_embed(N) - The identity element always has degree one, but any element - linearly-independent from it is regular:: + """ + raise NotImplementedError - sage: J = JordanSpinEJA(5) - sage: J.one().is_regular() - False - sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity - sage: for x in J.gens(): - ....: (J.one() + x).is_regular() - False - True - True - True - True - TESTS: + @staticmethod + def real_unembed(M): + """ + The inverse of :meth:`real_embed`. + """ + raise NotImplementedError - The zero element should never be regular:: - sage: set_random_seed() - sage: J = random_eja() - sage: J.zero().is_regular() - False + @classmethod + def natural_inner_product(cls,X,Y): + Xu = cls.real_unembed(X) + Yu = cls.real_unembed(Y) + tr = (Xu*Yu).trace() - The unit element isn't regular unless the algebra happens to - consist of only its scalar multiples:: + try: + # Works in QQ, AA, RDF, et cetera. + return tr.real() + except AttributeError: + # A quaternion doesn't have a real() 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] - sage: set_random_seed() - sage: J = random_eja() - sage: J.dimension() == 1 or not J.one().is_regular() - True - """ - return self.degree() == self.parent().rank() +class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): + @staticmethod + def real_embed(M): + """ + The identity function, for embedding real matrices into real + matrices. + """ + return M + @staticmethod + def real_unembed(M): + """ + The identity function, for unembedding real matrices from real + matrices. + """ + return M - def degree(self): - """ - Return the degree of this element, which is defined to be - the degree of its minimal polynomial. - ALGORITHM: +class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): + """ + The rank-n simple EJA consisting of real symmetric n-by-n + matrices, the usual symmetric Jordan product, and the trace inner + product. It has dimension `(n^2 + n)/2` over the reals. - For now, we skip the messy minimal polynomial computation - and instead return the dimension of the vector space spanned - by the powers of this element. The latter is a bit more - straightforward to compute. + SETUP:: - SETUP:: + sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) + EXAMPLES:: - 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: J = JordanSpinEJA(4) - sage: J.one().degree() - 1 - sage: e0,e1,e2,e3 = J.gens() - sage: (e0 - e1).degree() - 2 + In theory, our "field" can be any subfield of the reals:: - In the spin factor algebra (of rank two), all elements that - aren't multiples of the identity are regular:: + sage: RealSymmetricEJA(2, RDF) + Euclidean Jordan algebra of dimension 3 over Real Double Field + sage: RealSymmetricEJA(2, RR) + Euclidean Jordan algebra of dimension 3 over Real Field with + 53 bits of precision - sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: x == x.coefficient(0)*J.one() or x.degree() == 2 - True + TESTS: - TESTS: + The dimension of this algebra is `(n^2 + n) / 2`:: - The zero and unit elements are both of degree one:: + sage: set_random_seed() + sage: n_max = RealSymmetricEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = RealSymmetricEJA(n) + sage: J.dimension() == (n^2 + n)/2 + True - sage: set_random_seed() - sage: J = random_eja() - sage: J.zero().degree() - 1 - sage: J.one().degree() - 1 + The Jordan multiplication is what we think it is:: - Our implementation agrees with the definition:: + 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: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True + We can change the generator prefix:: - """ - return self.span_of_powers().dimension() + 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:: - def left_matrix(self): - """ - Our parent class defines ``left_matrix`` and ``matrix`` - methods whose names are misleading. We don't want them. - """ - raise NotImplementedError("use operator().matrix() instead") + sage: set_random_seed() + sage: J = RealSymmetricEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True - matrix = left_matrix + 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 - def minimal_polynomial(self): - """ - Return the minimal polynomial of this element, - as a function of the variable `t`. - - ALGORITHM: + We can construct the (trivial) algebra of rank zero:: - We restrict ourselves to the associative subalgebra - generated by this element, and then return the minimal - polynomial of this element's operator matrix (in that - subalgebra). This works by Baes Proposition 2.3.16. + sage: RealSymmetricEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field - SETUP:: - - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - TESTS: - - The minimal polynomial of the identity and zero elements are - always the same:: - - sage: set_random_seed() - sage: J = random_eja() - sage: J.one().minimal_polynomial() - t - 1 - sage: J.zero().minimal_polynomial() - t - - The degree of an element is (by one definition) the degree - of its minimal polynomial:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True - - The minimal polynomial and the characteristic polynomial coincide - and are known (see Alizadeh, Example 11.11) for all elements of - the spin factor algebra that aren't scalar multiples of the - identity:: - - sage: set_random_seed() - sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinEJA(n) - sage: y = J.random_element() - sage: while y == y.coefficient(0)*J.one(): - ....: y = J.random_element() - sage: y0 = y.vector()[0] - sage: y_bar = y.vector()[1:] - sage: actual = y.minimal_polynomial() - sage: t = PolynomialRing(J.base_ring(),'t').gen(0) - sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2) - sage: bool(actual == expected) - True - - The minimal polynomial should always kill its element:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: p = x.minimal_polynomial() - sage: x.apply_univariate_polynomial(p) - 0 - - """ - V = self.span_of_powers() - assoc_subalg = self.subalgebra_generated_by() - # Mis-design warning: the basis used for span_of_powers() - # and subalgebra_generated_by() must be the same, and in - # the same order! - elt = assoc_subalg(V.coordinates(self.vector())) - return elt.operator().minimal_polynomial() - - - - def natural_representation(self): - """ - Return a more-natural representation of this element. - - 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" representation of this element as a Hermitian - matrix, if it has one. If not, you get the usual representation. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, - ....: QuaternionHermitianEJA) - - EXAMPLES:: - - sage: J = ComplexHermitianEJA(3) - sage: J.one() - e0 + e5 + e8 - sage: J.one().natural_representation() - [1 0 0 0 0 0] - [0 1 0 0 0 0] - [0 0 1 0 0 0] - [0 0 0 1 0 0] - [0 0 0 0 1 0] - [0 0 0 0 0 1] - - :: - - sage: J = QuaternionHermitianEJA(3) - sage: J.one() - e0 + e9 + e14 - sage: J.one().natural_representation() - [1 0 0 0 0 0 0 0 0 0 0 0] - [0 1 0 0 0 0 0 0 0 0 0 0] - [0 0 1 0 0 0 0 0 0 0 0 0] - [0 0 0 1 0 0 0 0 0 0 0 0] - [0 0 0 0 1 0 0 0 0 0 0 0] - [0 0 0 0 0 1 0 0 0 0 0 0] - [0 0 0 0 0 0 1 0 0 0 0 0] - [0 0 0 0 0 0 0 1 0 0 0 0] - [0 0 0 0 0 0 0 0 1 0 0 0] - [0 0 0 0 0 0 0 0 0 1 0 0] - [0 0 0 0 0 0 0 0 0 0 1 0] - [0 0 0 0 0 0 0 0 0 0 0 1] - - """ - B = self.parent().natural_basis() - W = B[0].matrix_space() - return W.linear_combination(zip(self.vector(), B)) - - - def operator(self): - """ - Return the left-multiplication-by-this-element - operator on the ambient algebra. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: x.operator()(y) == x*y - True - sage: y.operator()(x) == x*y - True - - """ - P = self.parent() - fda_elt = FiniteDimensionalAlgebraElement(P, self) - return FiniteDimensionalEuclideanJordanAlgebraOperator( - P, - P, - fda_elt.matrix().transpose() ) - - - def quadratic_representation(self, other=None): - """ - Return the quadratic representation of this element. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - EXAMPLES: - - The explicit form in the spin factor algebra is given by - Alizadeh's Example 11.12:: - - sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: x_vec = x.vector() - sage: x0 = x_vec[0] - sage: x_bar = x_vec[1:] - sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) - sage: B = 2*x0*x_bar.row() - sage: C = 2*x0*x_bar.column() - sage: D = matrix.identity(QQ, n-1) - sage: D = (x0^2 - x_bar.inner_product(x_bar))*D - sage: D = D + 2*x_bar.tensor_product(x_bar) - sage: Q = matrix.block(2,2,[A,B,C,D]) - sage: Q == x.quadratic_representation().matrix() - True - - Test all of the properties from Theorem 11.2 in Alizadeh:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: Lx = x.operator() - sage: Lxx = (x*x).operator() - sage: Qx = x.quadratic_representation() - sage: Qy = y.quadratic_representation() - sage: Qxy = x.quadratic_representation(y) - sage: Qex = J.one().quadratic_representation(x) - sage: n = ZZ.random_element(10) - sage: Qxn = (x^n).quadratic_representation() - - Property 1: - - sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy - True - - Property 2 (multiply on the right for :trac:`28272`): - - sage: alpha = QQ.random_element() - sage: (alpha*x).quadratic_representation() == Qx*(alpha^2) - True - - Property 3: - - sage: not x.is_invertible() or ( Qx(x.inverse()) == x ) - True + """ + @classmethod + def _denormalized_basis(cls, n, field): + """ + Return a basis for the space of real symmetric n-by-n matrices. - sage: not x.is_invertible() or ( - ....: ~Qx - ....: == - ....: x.inverse().quadratic_representation() ) - True + SETUP:: - sage: Qxy(J.one()) == x*y - True + sage: from mjo.eja.eja_algebra import RealSymmetricEJA - Property 4: - - sage: not x.is_invertible() or ( - ....: x.quadratic_representation(x.inverse())*Qx - ....: == Qx*x.quadratic_representation(x.inverse()) ) - True + TESTS:: - sage: not x.is_invertible() or ( - ....: x.quadratic_representation(x.inverse())*Qx - ....: == - ....: 2*x.operator()*Qex - Qx ) - True - - sage: 2*x.operator()*Qex - Qx == Lxx - True - - Property 5: - - sage: Qy(x).quadratic_representation() == Qy*Qx*Qy - True - - Property 6: + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: B = RealSymmetricEJA._denormalized_basis(n,QQ) + sage: all( M.is_symmetric() for M in B) + True - sage: Qxn == (Qx)^n - True + """ + # The basis of symmetric matrices, as matrices, in their R^(n-by-n) + # coordinates. + S = [] + 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 - Property 7: - sage: not x.is_invertible() or ( - ....: Qx*x.inverse().operator() == Lx ) - True + @staticmethod + def _max_test_case_size(): + return 4 # Dimension 10 - Property 8: - - sage: not x.operator_commutes_with(y) or ( - ....: Qx(y)^n == Qxn(y^n) ) - True - - """ - if other is None: - other=self - elif not other in self.parent(): - raise TypeError("'other' must live in the same algebra") - - L = self.operator() - M = other.operator() - return ( L*M + M*L - (self*other).operator() ) - - - def span_of_powers(self): - """ - Return the vector space spanned by successive powers of - this element. - """ - # The dimension of the subalgebra can't be greater than - # the big algebra, so just put everything into a list - # and let span() get rid of the excess. - # - # We do the extra ambient_vector_space() in case we're messing - # with polynomials and the direct parent is a module. - V = self.parent().vector_space() - return V.span( (self**d).vector() for d in xrange(V.dimension()) ) - - - def subalgebra_generated_by(self): - """ - Return the associative subalgebra of the parent EJA generated - by this element. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.subalgebra_generated_by().is_associative() - True - - Squaring in the subalgebra should work the same as in - the superalgebra:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: u = x.subalgebra_generated_by().random_element() - sage: u.operator()(u) == u^2 - True - - """ - # First get the subspace spanned by the powers of myself... - V = self.span_of_powers() - F = self.base_ring() - - # Now figure out the entries of the right-multiplication - # matrix for the successive basis elements b0, b1,... of - # that subspace. - mats = [] - for b_right in V.basis(): - eja_b_right = self.parent()(b_right) - b_right_rows = [] - # The first row of the right-multiplication matrix by - # b1 is what we get if we apply that matrix to b1. The - # second row of the right multiplication matrix by b1 - # is what we get when we apply that matrix to b2... - # - # IMPORTANT: this assumes that all vectors are COLUMN - # vectors, unlike our superclass (which uses row vectors). - for b_left in V.basis(): - eja_b_left = self.parent()(b_left) - # Multiply in the original EJA, but then get the - # coordinates from the subalgebra in terms of its - # basis. - this_row = V.coordinates((eja_b_left*eja_b_right).vector()) - b_right_rows.append(this_row) - b_right_matrix = matrix(F, b_right_rows) - mats.append(b_right_matrix) - - # It's an algebra of polynomials in one element, and EJAs - # are power-associative. - # - # TODO: choose generator names intelligently. - # - # The rank is the highest possible degree of a minimal polynomial, - # and is bounded above by the dimension. We know in this case that - # there's an element whose minimal polynomial has the same degree - # as the space's dimension, so that must be its rank too. - return FiniteDimensionalEuclideanJordanAlgebra( - F, - mats, - V.dimension(), - assume_associative=True, - names='f') - - - def subalgebra_idempotent(self): - """ - Find an idempotent in the associative subalgebra I generate - using Proposition 2.3.5 in Baes. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: while x.is_nilpotent(): - ....: x = J.random_element() - sage: c = x.subalgebra_idempotent() - sage: c^2 == c - True - - """ - if self.is_nilpotent(): - raise ValueError("this only works with non-nilpotent elements!") - - V = self.span_of_powers() - J = self.subalgebra_generated_by() - # Mis-design warning: the basis used for span_of_powers() - # and subalgebra_generated_by() must be the same, and in - # the same order! - u = J(V.coordinates(self.vector())) - - # The image of the matrix of left-u^m-multiplication - # will be minimal for some natural number s... - s = 0 - minimal_dim = V.dimension() - for i in xrange(1, V.dimension()): - this_dim = (u**i).operator().matrix().image().dimension() - if this_dim < minimal_dim: - minimal_dim = this_dim - s = i - - # Now minimal_matrix should correspond to the smallest - # non-zero subspace in Baes's (or really, Koecher's) - # proposition. - # - # However, we need to restrict the matrix to work on the - # subspace... or do we? Can't we just solve, knowing that - # A(c) = u^(s+1) should have a solution in the big space, - # too? - # - # Beware, solve_right() means that we're using COLUMN vectors. - # Our FiniteDimensionalAlgebraElement superclass uses rows. - u_next = u**(s+1) - A = u_next.operator().matrix() - c_coordinates = A.solve_right(u_next.vector()) - - # Now c_coordinates is the idempotent we want, but it's in - # the coordinate system of the subalgebra. - # - # We need the basis for J, but as elements of the parent algebra. - # - basis = [self.parent(v) for v in V.basis()] - return self.parent().linear_combination(zip(c_coordinates, basis)) + def __init__(self, n, field=AA, **kwargs): + basis = self._denormalized_basis(n, field) + super(RealSymmetricEJA, self).__init__(field, + basis, + check_axioms=False, + **kwargs) + self.rank.set_cache(n) - def trace(self): - """ - Return my trace, the sum of my eigenvalues. - SETUP:: +class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): + @staticmethod + def real_embed(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 + + bi` to the block matrix ``[[a,b],[-b,a]]``. - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, - ....: random_eja) + SETUP:: - EXAMPLES:: + sage: from mjo.eja.eja_algebra import \ + ....: ComplexMatrixEuclideanJordanAlgebra - sage: J = JordanSpinEJA(3) - sage: x = sum(J.gens()) - sage: x.trace() - 2 + EXAMPLES:: - :: + 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) + [ 4 -2| 1 2] + [ 2 4|-2 1] + [-----+-----] + [ 0 -1| 6 0] + [ 1 0| 0 6] - sage: J = RealCartesianProductEJA(5) - sage: J.one().trace() - 5 + TESTS: - TESTS: + Embedding is a homomorphism (isomorphism, in fact):: - The trace of an element is a real number:: + 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: 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*Ye == XYe + True + + """ + n = M.nrows() + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + + # We don't need any adjoined elements... + field = M.base_ring().base_ring() + + blocks = [] + for z in M.list(): + a = z.list()[0] # real part, I guess + b = z.list()[1] # imag part, I guess + blocks.append(matrix(field, 2, [[a,b],[-b,a]])) + + return matrix.block(field, n, blocks) - sage: set_random_seed() - sage: J = random_eja() - sage: J.random_element().trace() in J.base_ring() - True - """ - P = self.parent() - r = P.rank() - p = P._charpoly_coeff(r-1) - # The _charpoly_coeff function already adds the factor of - # -1 to ensure that _charpoly_coeff(r-1) is really what - # appears in front of t^{r-1} in the charpoly. However, - # we want the negative of THAT for the trace. - return -p(*self.vector()) + @staticmethod + def real_unembed(M): + """ + The inverse of _embed_complex_matrix(). + + SETUP:: + sage: from mjo.eja.eja_algebra import \ + ....: ComplexMatrixEuclideanJordanAlgebra - def trace_inner_product(self, other): - """ - Return the trace inner product of myself and ``other``. + EXAMPLES:: - SETUP:: + sage: A = matrix(QQ,[ [ 1, 2, 3, 4], + ....: [-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: from mjo.eja.eja_algebra import random_eja + TESTS: - TESTS: + Unembedding is the inverse of embedding:: - The trace inner product is commutative:: + sage: set_random_seed() + sage: F = QuadraticField(-1, 'I') + sage: M = random_matrix(F, 3) + sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) + sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + True - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element(); y = J.random_element() - sage: x.trace_inner_product(y) == y.trace_inner_product(x) - True + """ + 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() + 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 + else: + F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) + 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 range(n/2): + for j in range(n/2): + submat = M[2*k:2*k+2,2*j:2*j+2] + if submat[0,0] != submat[1,1]: + raise ValueError('bad on-diagonal submatrix') + if submat[0,1] != -submat[1,0]: + raise ValueError('bad off-diagonal submatrix') + 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. - The trace inner product is bilinear:: + SETUP:: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: a = QQ.random_element(); - sage: actual = (a*(x+z)).trace_inner_product(y) - sage: expected = ( a*x.trace_inner_product(y) + - ....: a*z.trace_inner_product(y) ) - sage: actual == expected - True - sage: actual = x.trace_inner_product(a*(y+z)) - sage: expected = ( a*x.trace_inner_product(y) + - ....: a*x.trace_inner_product(z) ) - sage: actual == expected - True + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA - The trace inner product satisfies the compatibility - condition in the definition of a Euclidean Jordan algebra:: + TESTS: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z) - True + This gives the same answer as the slow, default method implemented + in :class:`MatrixEuclideanJordanAlgebra`:: - """ - if not other in self.parent(): - raise TypeError("'other' must live in the same algebra") + 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().real() + sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual == expected + True - return (self*other).trace() + """ + return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): +class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): """ - 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. + The rank-n simple EJA consisting of complex Hermitian n-by-n + matrices over the real numbers, the usual symmetric Jordan product, + and the real-part-of-trace inner product. It has dimension `n^2` over + the reals. SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA EXAMPLES: - This multiplication table can be verified by hand:: + In theory, our "field" can be any subfield of the reals:: - 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 + sage: ComplexHermitianEJA(2, RDF) + Euclidean Jordan algebra of dimension 4 over Real Double Field + sage: ComplexHermitianEJA(2, RR) + Euclidean Jordan algebra of dimension 4 over Real Field with + 53 bits of precision - """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - # The FiniteDimensionalAlgebra constructor takes a list of - # matrices, the ith representing right multiplication by the ith - # basis element in the vector space. So if e_1 = (1,0,0), then - # right (Hadamard) multiplication of x by e_1 picks out the first - # component of x; and likewise for the ith basis element e_i. - Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i)) - for i in xrange(n) ] - - fdeja = super(RealCartesianProductEJA, cls) - return fdeja.__classcall_private__(cls, field, Qs, rank=n) + TESTS: - def inner_product(self, x, y): - return _usual_ip(x,y) + The dimension of this algebra is `n^2`:: + sage: set_random_seed() + sage: n_max = ComplexHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = ComplexHermitianEJA(n) + sage: J.dimension() == n^2 + True -def random_eja(): - """ - Return a "random" finite-dimensional Euclidean Jordan Algebra. + The Jordan multiplication is what we think it is:: - ALGORITHM: + 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: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True - For now, we choose a random natural number ``n`` (greater than zero) - and then give you back one of the following: + We can change the generator prefix:: - * The cartesian product of the rational numbers ``n`` times; this is - ``QQ^n`` with the Hadamard product. + sage: ComplexHermitianEJA(2, prefix='z').gens() + (z0, z1, z2, z3) - * The Jordan spin algebra on ``QQ^n``. + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: - * The ``n``-by-``n`` rational symmetric matrices with the symmetric - product. + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True - * The ``n``-by-``n`` complex-rational Hermitian matrices embedded - in the space of ``2n``-by-``2n`` real symmetric matrices. + 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:: - * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded - in the space of ``4n``-by-``4n`` real symmetric matrices. + sage: set_random_seed() + sage: x = ComplexHermitianEJA.random_instance().random_element() + sage: x.operator().matrix().is_symmetric() + True - Later this might be extended to return Cartesian products of the - EJAs above. + We can construct the (trivial) algebra of rank zero:: - SETUP:: + sage: ComplexHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field - sage: from mjo.eja.eja_algebra import random_eja + """ - TESTS:: + @classmethod + def _denormalized_basis(cls, n, field): + """ + Returns a basis for the space of complex Hermitian n-by-n matrices. - sage: random_eja() - Euclidean Jordan algebra of degree... + Why do we embed these? Basically, because all of numerical linear + algebra assumes that you're working with vectors consisting of `n` + entries from a field and scalars from the same field. There's no way + to tell SageMath that (for example) the vectors contain complex + numbers, while the scalar field is real. - """ + SETUP:: - # The max_n component lets us choose different upper bounds on the - # value "n" that gets passed to the constructor. This is needed - # because e.g. R^{10} is reasonable to test, while the Hermitian - # 10-by-10 quaternion matrices are not. - (constructor, max_n) = choice([(RealCartesianProductEJA, 6), - (JordanSpinEJA, 6), - (RealSymmetricEJA, 5), - (ComplexHermitianEJA, 4), - (QuaternionHermitianEJA, 3)]) - n = ZZ.random_element(1, max_n) - return constructor(n, field=QQ) + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + TESTS:: + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: field = QuadraticField(2, 'sqrt2') + sage: B = ComplexHermitianEJA._denormalized_basis(n, field) + sage: all( M.is_symmetric() for M in B) + True -def _real_symmetric_basis(n, field=QQ): - """ - Return a basis for the space of real symmetric n-by-n matrices. - """ - # 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): - Eij = matrix(field, n, lambda k,l: k==i and l==j) - if i == j: - Sij = Eij - else: - # Beware, orthogonal but not normalized! - Sij = Eij + Eij.transpose() - S.append(Sij) - return tuple(S) + """ + R = PolynomialRing(field, 'z') + z = R.gen() + F = field.extension(z**2 + 1, 'I') + I = F.gen() + # This is like the symmetric case, but we need to be careful: + # + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. + # + S = [] + for i in range(n): + for j in range(i+1): + Eij = matrix(F, n, lambda k,l: k==i and l==j) + 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()) + S.append(Sij_real) + Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose()) + S.append(Sij_imag) + + # 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 ) + + + def __init__(self, n, field=AA, **kwargs): + basis = self._denormalized_basis(n,field) + super(ComplexHermitianEJA,self).__init__(field, + basis, + check_axioms=False, + **kwargs) + self.rank.set_cache(n) + + +class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): + @staticmethod + def real_embed(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 + = a + bi + cj + dk` to the block-complex matrix ``[[a + bi, + c+di],[-c + di, a-bi]]`, and then embedding those into a real + matrix. -def _complex_hermitian_basis(n, field=QQ): - """ - Returns a basis for the space of complex Hermitian n-by-n matrices. + SETUP:: - SETUP:: + sage: from mjo.eja.eja_algebra import \ + ....: QuaternionMatrixEuclideanJordanAlgebra - sage: from mjo.eja.eja_algebra import _complex_hermitian_basis + EXAMPLES:: - TESTS:: + sage: Q = QuaternionAlgebra(QQ,-1,-1) + 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) + [ 1 2 3 4] + [-2 1 -4 3] + [-3 4 1 -2] + [-4 -3 2 1] - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) - True + Embedding is a homomorphism (isomorphism, in fact):: - """ - F = QuadraticField(-1, 'I') - I = F.gen() - - # This is like the symmetric case, but we need to be careful: - # - # * We want conjugate-symmetry, not just symmetry. - # * The diagonal will (as a result) be real. - # - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) - if i == j: - Sij = _embed_complex_matrix(Eij) - S.append(Sij) - else: - # Beware, orthogonal but not normalized! The second one - # has a minus because it's conjugated. - Sij_real = _embed_complex_matrix(Eij + Eij.transpose()) - S.append(Sij_real) - Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) - S.append(Sij_imag) - return tuple(S) + sage: set_random_seed() + sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size() + sage: n = ZZ.random_element(n_max) + 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*Ye == XYe + True + """ + quaternions = M.base_ring() + n = M.nrows() + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") -def _quaternion_hermitian_basis(n, field=QQ): - """ - Returns a basis for the space of quaternion Hermitian n-by-n matrices. + F = QuadraticField(-1, 'I') + i = F.gen() - SETUP:: + blocks = [] + for z in M.list(): + t = z.coefficient_tuple() + a = t[0] + b = t[1] + c = t[2] + d = t[3] + cplxM = matrix(F, 2, [[ a + b*i, c + d*i], + [-c + d*i, a - b*i]]) + realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM) + blocks.append(realM) - sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis + # We should have real entries by now, so use the realest field + # we've got for the return value. + return matrix.block(quaternions.base_ring(), n, blocks) - TESTS:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) - True - """ - Q = QuaternionAlgebra(QQ,-1,-1) - I,J,K = Q.gens() - - # This is like the symmetric case, but we need to be careful: - # - # * We want conjugate-symmetry, not just symmetry. - # * 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) - if i == j: - Sij = _embed_quaternion_matrix(Eij) - S.append(Sij) - else: - # Beware, orthogonal but not normalized! The second, - # third, and fourth ones have a minus because they're - # conjugated. - Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose()) - S.append(Sij_real) - Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose()) - S.append(Sij_I) - Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose()) - S.append(Sij_J) - Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose()) - S.append(Sij_K) - return tuple(S) - - -def _mat2vec(m): - return vector(m.base_ring(), m.list()) - -def _vec2mat(v): - return matrix(v.base_ring(), sqrt(v.degree()), v.list()) - -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. A reordered copy - of the basis is also returned to work around the fact that - the ``span()`` in this function will change the order of the basis - from what we think it is, to... something else. - """ - # 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( _mat2vec(s) for s in basis ) - - # Taking the span above reorders our basis (thanks, jerk!) so we - # need to put our "matrix basis" in the same order as the - # (reordered) vector basis. - S = tuple( _vec2mat(b) for b in W.basis() ) - - Qs = [] - for s in S: - # Brute force the multiplication-by-s matrix by looping - # through all elements of the basis and doing the computation - # to find out what the corresponding row should be. BEWARE: - # these multiplication tables won't be symmetric! It therefore - # becomes REALLY IMPORTANT that the underlying algebra - # constructor uses ROW vectors and not COLUMN vectors. That's - # why we're computing rows here and not columns. - Q_rows = [] - for t in S: - this_row = _mat2vec((s*t + t*s)/2) - Q_rows.append(W.coordinates(this_row)) - Q = matrix(field, W.dimension(), Q_rows) - Qs.append(Q) - - return (Qs, S) - - -def _embed_complex_matrix(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 + - bi` to the block matrix ``[[a,b],[-b,a]]``. + @staticmethod + def real_unembed(M): + """ + The inverse of _embed_quaternion_matrix(). - SETUP:: + SETUP:: - sage: from mjo.eja.eja_algebra import _embed_complex_matrix + sage: from mjo.eja.eja_algebra import \ + ....: QuaternionMatrixEuclideanJordanAlgebra - EXAMPLES:: + EXAMPLES:: - 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: _embed_complex_matrix(M) - [ 4 -2| 1 2] - [ 2 4|-2 1] - [-----+-----] - [ 0 -1| 6 0] - [ 1 0| 0 6] + sage: M = matrix(QQ, [[ 1, 2, 3, 4], + ....: [-2, 1, -4, 3], + ....: [-3, 4, 1, -2], + ....: [-4, -3, 2, 1]]) + sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M) + [1 + 2*i + 3*j + 4*k] - TESTS: + TESTS: - Embedding is a homomorphism (isomorphism, in fact):: + Unembedding is the inverse of embedding:: - sage: set_random_seed() - sage: n = ZZ.random_element(5) - sage: F = QuadraticField(-1, 'i') - sage: X = random_matrix(F, n) - sage: Y = random_matrix(F, n) - sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) - sage: expected = _embed_complex_matrix(X*Y) - sage: actual == expected - True + 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 + True - """ - n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - field = M.base_ring() - blocks = [] - for z in M.list(): - a = z.real() - b = z.imag() - blocks.append(matrix(field, 2, [[a,b],[-b,a]])) + """ + 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") + + # 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) + 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 range(n/4): + for m in range(n/4): + submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed( + M[4*l:4*l+4,4*m:4*m+4] ) + 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].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. + + SETUP:: + + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - # We can drop the imaginaries here. - return matrix.block(field.base_ring(), n, blocks) + 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 -def _unembed_complex_matrix(M): + +class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): """ - The inverse of _embed_complex_matrix(). + 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 + the reals. SETUP:: - sage: from mjo.eja.eja_algebra import (_embed_complex_matrix, - ....: _unembed_complex_matrix) + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - EXAMPLES:: + EXAMPLES: + + In theory, our "field" can be any subfield of the reals:: - sage: A = matrix(QQ,[ [ 1, 2, 3, 4], - ....: [-2, 1, -4, 3], - ....: [ 9, 10, 11, 12], - ....: [-10, 9, -12, 11] ]) - sage: _unembed_complex_matrix(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + sage: QuaternionHermitianEJA(2, RDF) + Euclidean Jordan algebra of dimension 6 over Real Double Field + sage: QuaternionHermitianEJA(2, RR) + Euclidean Jordan algebra of dimension 6 over Real Field with + 53 bits of precision TESTS: - Unembedding is the inverse of embedding:: + The dimension of this algebra is `2*n^2 - n`:: sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') - sage: M = random_matrix(F, 3) - sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M + sage: n_max = QuaternionHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = QuaternionHermitianEJA(n) + sage: J.dimension() == 2*(n^2) - n True - """ - 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") - - F = QuadraticField(-1, 'i') - 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] - if submat[0,0] != submat[1,1]: - raise ValueError('bad on-diagonal submatrix') - if submat[0,1] != -submat[1,0]: - raise ValueError('bad off-diagonal submatrix') - z = submat[0,0] + submat[0,1]*i - elements.append(z) - - return matrix(F, n/2, elements) - - -def _embed_quaternion_matrix(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 = a + bi + cj + dk` to the block-complex matrix - ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into - a real matrix. + The Jordan multiplication is what we think it is:: - SETUP:: + 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: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True - sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix + We can change the generator prefix:: - EXAMPLES:: + sage: QuaternionHermitianEJA(2, prefix='a').gens() + (a0, a1, a2, a3, a4, a5) - sage: Q = QuaternionAlgebra(QQ,-1,-1) - sage: i,j,k = Q.gens() - sage: x = 1 + 2*i + 3*j + 4*k - sage: M = matrix(Q, 1, [[x]]) - sage: _embed_quaternion_matrix(M) - [ 1 2 3 4] - [-2 1 -4 3] - [-3 4 1 -2] - [-4 -3 2 1] + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: - Embedding is a homomorphism (isomorphism, in fact):: + 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: n = ZZ.random_element(5) - sage: Q = QuaternionAlgebra(QQ,-1,-1) - sage: X = random_matrix(Q, n) - sage: Y = random_matrix(Q, n) - sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y) - sage: expected = _embed_quaternion_matrix(X*Y) - sage: actual == expected + sage: x = QuaternionHermitianEJA.random_instance().random_element() + sage: x.operator().matrix().is_symmetric() True - """ - quaternions = M.base_ring() - n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - - F = QuadraticField(-1, 'i') - i = F.gen() - - blocks = [] - for z in M.list(): - t = z.coefficient_tuple() - a = t[0] - b = t[1] - c = t[2] - d = t[3] - cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i], - [-c + d*i, a - b*i]]) - blocks.append(_embed_complex_matrix(cplx_matrix)) - - # We should have real entries by now, so use the realest field - # we've got for the return value. - return matrix.block(quaternions.base_ring(), n, blocks) - - -def _unembed_quaternion_matrix(M): - """ - The inverse of _embed_quaternion_matrix(). + We can construct the (trivial) algebra of rank zero:: - SETUP:: + sage: QuaternionHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field - sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix, - ....: _unembed_quaternion_matrix) + """ + @classmethod + def _denormalized_basis(cls, n, field): + """ + Returns a basis for the space of quaternion Hermitian n-by-n matrices. - EXAMPLES:: + Why do we embed these? Basically, because all of numerical + linear algebra assumes that you're working with vectors consisting + of `n` entries from a field and scalars from the same field. There's + no way to tell SageMath that (for example) the vectors contain + complex numbers, while the scalar field is real. - sage: M = matrix(QQ, [[ 1, 2, 3, 4], - ....: [-2, 1, -4, 3], - ....: [-3, 4, 1, -2], - ....: [-4, -3, 2, 1]]) - sage: _unembed_quaternion_matrix(M) - [1 + 2*i + 3*j + 4*k] + SETUP:: - TESTS: + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - Unembedding is the inverse of embedding:: + TESTS:: - sage: set_random_seed() - sage: Q = QuaternionAlgebra(QQ, -1, -1) - sage: M = random_matrix(Q, 3) - sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M - True + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ) + sage: all( M.is_symmetric() for M in B ) + True + """ + Q = QuaternionAlgebra(QQ,-1,-1) + I,J,K = Q.gens() + + # This is like the symmetric case, but we need to be careful: + # + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. + # + S = [] + for i in range(n): + for j in range(i+1): + Eij = matrix(Q, n, lambda k,l: k==i and l==j) + 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()) + S.append(Sij_real) + Sij_I = cls.real_embed(I*Eij - I*Eij.transpose()) + S.append(Sij_I) + Sij_J = cls.real_embed(J*Eij - J*Eij.transpose()) + S.append(Sij_J) + Sij_K = cls.real_embed(K*Eij - K*Eij.transpose()) + S.append(Sij_K) + + # 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 ) + + + def __init__(self, n, field=AA, **kwargs): + basis = self._denormalized_basis(n,field) + super(QuaternionHermitianEJA,self).__init__(field, + basis, + check_axioms=False, + **kwargs) + self.rank.set_cache(n) + + +class HadamardEJA(RationalBasisEuclideanJordanAlgebra): """ - 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 complex embedding") - - Q = QuaternionAlgebra(QQ,-1,-1) - 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 = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4]) - 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].real() + submat[0,0].imag()*i - z += submat[0,1].real()*j + submat[0,1].imag()*k - elements.append(z) - - return matrix(Q, n/4, elements) - - -# The usual inner product on R^n. -def _usual_ip(x,y): - return x.vector().inner_product(y.vector()) - -# The inner product used for the real symmetric simple EJA. -# We keep it as a separate function because e.g. the complex -# algebra uses the same inner product, except divided by 2. -def _matrix_ip(X,Y): - X_mat = X.natural_representation() - Y_mat = Y.natural_representation() - return (X_mat*Y_mat).trace() - - -class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - The rank-n simple EJA consisting of real symmetric n-by-n - matrices, the usual symmetric Jordan product, and the trace inner - product. It has dimension `(n^2 + n)/2` over the reals. + 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 RealSymmetricEJA + sage: from mjo.eja.eja_algebra import HadamardEJA - EXAMPLES:: + EXAMPLES: - sage: J = RealSymmetricEJA(2) - sage: e0, e1, e2 = J.gens() + This multiplication table can be verified by hand:: + + sage: J = HadamardEJA(3) + sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 + sage: e0*e1 + 0 + sage: e0*e2 + 0 sage: e1*e1 - e0 + e2 + e1 + sage: e1*e2 + 0 sage: e2*e2 e2 TESTS: - The degree of this algebra is `(n^2 + n) / 2`:: - - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) - sage: J.degree() == (n^2 + n)/2 - True - - The Jordan multiplication is what we think it is:: + We can change the generator prefix:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 - sage: actual == expected - True - sage: J(expected) == x*y - True + sage: HadamardEJA(3, prefix='r').gens() + (r0, r1, r2) """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - S = _real_symmetric_basis(n, field=field) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + def __init__(self, n, field=AA, **kwargs): + V = VectorSpace(field, n) + mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] + for i in range(n) ] - fdeja = super(RealSymmetricEJA, cls) - return fdeja.__classcall_private__(cls, - field, - Qs, - rank=n, - natural_basis=T) + super(HadamardEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + self.rank.set_cache(n) def inner_product(self, x, y): - return _matrix_ip(x,y) - - -class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - The rank-n simple EJA consisting of complex Hermitian n-by-n - matrices over the real numbers, the usual symmetric Jordan product, - and the real-part-of-trace inner product. It has dimension `n^2` over - the reals. + """ + Faster to reimplement than to use natural representations. - SETUP:: + SETUP:: - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + sage: from mjo.eja.eja_algebra import HadamardEJA - TESTS: + TESTS: - The degree of this algebra is `n^2`:: + Ensure that this is the usual inner product for the algebras + over `R^n`:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) - sage: J.degree() == n^2 - True - - The Jordan multiplication is what we think it is:: + sage: set_random_seed() + sage: J = HadamardEJA.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 - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 - sage: actual == expected - True - sage: J(expected) == x*y - True + """ + return x.to_vector().inner_product(y.to_vector()) - """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - S = _complex_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) - fdeja = super(ComplexHermitianEJA, cls) - return fdeja.__classcall_private__(cls, - field, - Qs, - rank=n, - natural_basis=T) +class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra): + 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 = + (x0*y0 + , x0*y_bar + y0*x_bar)`` where ``B`` is a + symmetric positive-definite "bilinear form" matrix. It has + dimension `n` over the reals, and reduces to the ``JordanSpinEJA`` + when ``B`` is the identity matrix of order ``n-1``. - def inner_product(self, x, y): - # Since a+bi on the diagonal is represented as - # - # a + bi = [ a b ] - # [ -b a ], - # - # we'll double-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/2 + SETUP:: + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) -class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - 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 - the reals. + EXAMPLES: - SETUP:: + When no bilinear form is specified, the identity matrix is used, + and the resulting algebra is the Jordan spin algebra:: - sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA + sage: J0 = BilinearFormEJA(3) + sage: J1 = JordanSpinEJA(3) + sage: J0.multiplication_table() == J0.multiplication_table() + True TESTS: - The degree of this algebra is `n^2`:: + We can create a zero-dimensional algebra:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) - sage: J.degree() == 2*(n^2) - n - True + sage: J = BilinearFormEJA(0) + sage: J.basis() + Finite family {} - The Jordan multiplication is what we think it is:: + 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:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B = M.transpose()*M + sage: J = BilinearFormEJA(n, B=B) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list())) + ....: 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 - sage: J(expected) == x*y - True - """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - S = _quaternion_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) - - fdeja = super(QuaternionHermitianEJA, cls) - return fdeja.__classcall_private__(cls, - field, - Qs, - rank=n, - natural_basis=T) + def __init__(self, n, field=AA, B=None, **kwargs): + if B is None: + self._B = matrix.identity(field, max(0,n-1)) + else: + self._B = B + + V = VectorSpace(field, n) + mult_table = [[V.zero() for j in range(n)] for i in range(n)] + for i in range(n): + for j in range(n): + x = V.gen(i) + y = V.gen(j) + x0 = x[0] + xbar = x[1:] + y0 = y[0] + ybar = y[1:] + z0 = x0*y0 + (self._B*xbar).inner_product(ybar) + zbar = y0*xbar + x0*ybar + z = V([z0] + zbar.list()) + mult_table[i][j] = z + + # 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). + super(BilinearFormEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + self.rank.set_cache(min(n,2)) def inner_product(self, x, y): - # Since a+bi+cj+dk on the diagonal is represented as - # - # a + bi +cj + dk = [ a b c d] - # [ -b a -d c] - # [ -c d a -b] - # [ -d -c b a], - # - # we'll quadruple-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/4 + r""" + Half of the trace inner product. + + This is defined so that the special case of the Jordan spin + algebra gets the usual inner product. + + SETUP:: + + sage: from mjo.eja.eja_algebra import BilinearFormEJA + + TESTS: + Ensure that this is one-half of the trace inner-product when + the algebra isn't just the reals (when ``n`` isn't one). This + is in Faraut and Koranyi, and also my "On the symmetry..." + paper:: -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): + sage: set_random_seed() + sage: n = ZZ.random_element(2,5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B = M.transpose()*M + sage: J = BilinearFormEJA(n, B=B) + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.inner_product(y) == (x*y).trace()/2 + True + + """ + xvec = x.to_vector() + xbar = xvec[1:] + yvec = y.to_vector() + ybar = yvec[1:] + return x[0]*y[0] + (self._B*xbar).inner_product(ybar) + + +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:: @@ -2412,27 +2213,114 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: e2*e3 0 + We can change the generator prefix:: + + 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: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: x.inner_product(y) == J.natural_inner_product(X,Y) + True + """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - Qs = [] - id_matrix = matrix.identity(field, n) - for i in xrange(n): - ei = id_matrix.column(i) - Qi = matrix.zero(field, n) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += matrix.diagonal(n, [ei[0]]*n) - # The addition of the diagonal matrix adds an extra ei[0] in the - # upper-left corner of the matrix. - Qi[0,0] = Qi[0,0] * ~field(2) - Qs.append(Qi) - - # 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, cls) - return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2)) + def __init__(self, n, field=AA, **kwargs): + # This is a special case of the BilinearFormEJA with the identity + # matrix as its bilinear form. + super(JordanSpinEJA, self).__init__(n, field, **kwargs) - def inner_product(self, x, y): - return _usual_ip(x,y) + +class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): + """ + 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, field=AA, **kwargs): + mult_table = [] + super(TrivialEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + # The rank is zero using my definition, namely the dimension of the + # largest subalgebra generated by any element. + self.rank.set_cache(0) + + +class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): + r""" + The external (orthogonal) direct sum of two other Euclidean Jordan + algebras. Essentially the Cartesian product of its two factors. + Every Euclidean Jordan algebra decomposes into an orthogonal + direct sum of simple Euclidean Jordan algebras, so no generality + is lost by providing only this construction. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA, + ....: DirectSumEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(3) + sage: J = DirectSumEJA(J1,J2) + sage: J.dimension() + 8 + sage: J.rank() + 5 + + """ + def __init__(self, J1, J2, field=AA, **kwargs): + n1 = J1.dimension() + n2 = J2.dimension() + n = n1+n2 + V = VectorSpace(field, n) + mult_table = [ [ V.zero() for j in range(n) ] + for i in range(n) ] + for i in range(n1): + for j in range(n1): + p = (J1.monomial(i)*J1.monomial(j)).to_vector() + mult_table[i][j] = V(p.list() + [field.zero()]*n2) + + for i in range(n2): + for j in range(n2): + p = (J2.monomial(i)*J2.monomial(j)).to_vector() + mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list()) + + super(DirectSumEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + self.rank.set_cache(J1.rank() + J2.rank())