X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=081abd303c32d35c9376ad99ecf2732a14ea3e9b;hb=95ae8e7b0ddca840da9631603a2f37cca888468b;hp=d31b5b71a0df190dee14d86f1544cff547f025ef;hpb=44182089cc3e0f0a6aecb6e652ad2a5c42bbb36a;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index d31b5b7..081abd3 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -3,6 +3,17 @@ Euclidean Jordan Algebras. These are formally-real Jordan Algebras; specifically those where u^2 + v^2 = 0 implies that u = v = 0. They are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. + + +SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + +EXAMPLES:: + + sage: random_eja() + Euclidean Jordan algebra of dimension... + """ from itertools import repeat @@ -13,96 +24,325 @@ 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.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace -from sage.rings.all import (ZZ, QQ, RR, RLF, CLF, +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_element import FiniteDimensionalEJAElement +from mjo.eja.eja_operator import FiniteDimensionalEJAOperator from mjo.eja.eja_utils import _mat2vec -class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): - # This is an ugly hack needed to prevent the category framework - # from implementing a coercion from our base ring (e.g. the - # rationals) into the algebra. First of all -- such a coercion is - # nonsense to begin with. But more importantly, it tries to do so - # in the category of rings, and since our algebras aren't - # associative they generally won't be rings. - _no_generic_basering_coercion = True +class FiniteDimensionalEJA(CombinatorialFreeModule): + r""" + A finite-dimensional Euclidean Jordan algebra. + + INPUT: + + - basis -- a tuple of basis elements in their matrix form. + + - jordan_product -- function of two elements (in matrix form) + that returns their jordan product in this algebra; this will + be applied to ``basis`` to compute a multiplication table for + the algebra. + + - inner_product -- function of two elements (in matrix form) that + returns their inner product. This will be applied to ``basis`` to + compute an inner-product table (basically a matrix) for this algebra. + + """ + Element = FiniteDimensionalEJAElement def __init__(self, - field, - mult_table, - rank, - prefix='e', - category=None, - natural_basis=None, - check=True): + basis, + jordan_product, + inner_product, + field=AA, + orthonormalize=True, + associative=False, + check_field=True, + check_axioms=True, + prefix='e'): + + if check_field: + if not field.is_subring(RR): + # Note: this does return true for the real algebraic + # field, the rationals, and any quadratic field where + # we've specified a real embedding. + raise ValueError("scalar field is not real") + + # If the basis given to us wasn't over the field that it's + # supposed to be over, fix that. Or, you know, crash. + basis = tuple( b.change_ring(field) for b in basis ) + + if check_axioms: + # Check commutativity of the Jordan and inner-products. + # This has to be done before we build the multiplication + # and inner-product tables/matrices, because we take + # advantage of symmetry in the process. + if not all( jordan_product(bi,bj) == jordan_product(bj,bi) + for bi in basis + for bj in basis ): + raise ValueError("Jordan product is not commutative") + + if not all( inner_product(bi,bj) == inner_product(bj,bi) + for bi in basis + for bj in basis ): + raise ValueError("inner-product is not commutative") + + + category = MagmaticAlgebras(field).FiniteDimensional() + category = category.WithBasis().Unital() + if associative: + # Element subalgebras can take advantage of this. + category = category.Associative() + + # Call the superclass constructor so that we can use its from_vector() + # method to build our multiplication table. + n = len(basis) + super().__init__(field, + range(n), + prefix=prefix, + category=category, + bracket=False) + + # Now comes all of the hard work. We'll be constructing an + # ambient vector space V that our (vectorized) basis lives in, + # as well as a subspace W of V spanned by those (vectorized) + # basis elements. The W-coordinates are the coefficients that + # we see in things like x = 1*e1 + 2*e2. + vector_basis = basis + + degree = 0 + if n > 0: + # Works on both column and square matrices... + degree = len(basis[0].list()) + + # Build an ambient space that fits our matrix basis when + # written out as "long vectors." + V = VectorSpace(field, degree) + + # The matrix that will hole the orthonormal -> unorthonormal + # coordinate transformation. + self._deortho_matrix = None + + if orthonormalize: + # Save a copy of the un-orthonormalized basis for later. + # Convert it to ambient V (vector) coordinates while we're + # at it, because we'd have to do it later anyway. + deortho_vector_basis = tuple( V(b.list()) for b in basis ) + + from mjo.eja.eja_utils import gram_schmidt + basis = gram_schmidt(basis, inner_product) + + # Save the (possibly orthonormalized) matrix basis for + # later... + self._matrix_basis = basis + + # Now create the vector space for the algebra, which will have + # its own set of non-ambient coordinates (in terms of the + # supplied basis). + vector_basis = tuple( V(b.list()) for b in basis ) + W = V.span_of_basis( vector_basis, check=check_axioms) + + if orthonormalize: + # Now "W" is the vector space of our algebra coordinates. The + # variables "X1", "X2",... refer to the entries of vectors in + # W. Thus to convert back and forth between the orthonormal + # coordinates and the given ones, we need to stick the original + # basis in W. + U = V.span_of_basis( deortho_vector_basis, check=check_axioms) + self._deortho_matrix = matrix( U.coordinate_vector(q) + for q in vector_basis ) + + + # Now we actually compute the multiplication and inner-product + # tables/matrices using the possibly-orthonormalized basis. + self._inner_product_matrix = matrix.zero(field, n) + self._multiplication_table = [ [0 for j in range(i+1)] + for i in range(n) ] + + # Note: the Jordan and inner-products are defined in terms + # of the ambient basis. It's important that their arguments + # are in ambient coordinates as well. + for i in range(n): + for j in range(i+1): + # ortho basis w.r.t. ambient coords + q_i = basis[i] + q_j = basis[j] + + elt = jordan_product(q_i, q_j) + ip = inner_product(q_i, q_j) + + # The jordan product returns a matrixy answer, so we + # have to convert it to the algebra coordinates. + elt = W.coordinate_vector(V(elt.list())) + self._multiplication_table[i][j] = self.from_vector(elt) + self._inner_product_matrix[i,j] = ip + self._inner_product_matrix[j,i] = ip + + self._inner_product_matrix._cache = {'hermitian': True} + self._inner_product_matrix.set_immutable() + + if check_axioms: + if not self._is_jordanian(): + raise ValueError("Jordan identity does not hold") + if not self._inner_product_is_associative(): + raise ValueError("inner product is not associative") + + + def _coerce_map_from_base_ring(self): + """ + Disable the map from the base ring into the algebra. + + Performing a nonsense conversion like this automatically + is counterpedagogical. The fallback is to try the usual + element constructor, which should also fail. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J(1) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + + """ + return None + + + def product_on_basis(self, i, j): + # We only stored the lower-triangular portion of the + # multiplication table. + if j <= i: + return self._multiplication_table[i][j] + else: + return self._multiplication_table[j][i] + + def inner_product(self, x, y): """ + The inner product associated with this Euclidean Jordan algebra. + + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. + SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja) + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: BilinearFormEJA) EXAMPLES: - By definition, Jordan multiplication commutes:: + Our inner product is "associative," which means the following for + a symmetric bilinear form:: sage: set_random_seed() sage: J = random_eja() - sage: x,y = J.random_elements(2) - sage: x*y == y*x + sage: x,y,z = J.random_elements(3) + sage: (x*y).inner_product(z) == y.inner_product(x*z) True TESTS: - The ``field`` we're given must be real:: + Ensure that this is the usual inner product for the algebras + over `R^n`:: - sage: JordanSpinEJA(2,QQbar) - Traceback (most recent call last): - ... - ValueError: field is not real + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + Ensure that this is one-half of the trace inner-product in a + BilinearFormEJA that isn't just the reals (when ``n`` isn't + one). This is in Faraut and Koranyi, and also my "On the + symmetry..." paper:: + + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) + True """ - if check: - if not field.is_subring(RR): - # Note: this does return true for the real algebraic - # field, and any quadratic field where we've specified - # a real embedding. - raise ValueError('field is not real') - - self._rank = rank - self._natural_basis = natural_basis - - if category is None: - category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() - - fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) - fda.__init__(field, - range(len(mult_table)), - prefix=prefix, - category=category) - 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 - ] + B = self._inner_product_matrix + return (B*x.to_vector()).inner_product(y.to_vector()) + + + def _is_commutative(self): + r""" + Whether or not this algebra's multiplication table is commutative. + + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. + """ + return all( self.product_on_basis(i,j) == self.product_on_basis(i,j) + for i in range(self.dimension()) + for j in range(self.dimension()) ) + + def _is_jordanian(self): + r""" + Whether or not this algebra's multiplication table respects the + 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. + """ + + # Used to check whether or not something is zero in an inexact + # ring. This number is sufficient to allow the construction of + # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. + epsilon = 1e-16 + for i in range(self.dimension()): + for j in range(self.dimension()): + for k in range(self.dimension()): + x = self.monomial(i) + y = self.monomial(j) + z = self.monomial(k) + diff = (x*y).inner_product(z) - x.inner_product(y*z) + + if self.base_ring().is_exact(): + if diff != 0: + return False + else: + if diff.abs() > epsilon: + return False + + return True def _element_constructor_(self, elt): """ - Construct an element of this algebra from its natural + Construct an element of this algebra from its vector or matrix representation. This gets called only after the parent element _call_ method @@ -111,7 +351,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: RealSymmetricEJA) EXAMPLES: @@ -130,16 +370,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J(A) Traceback (most recent call last): ... - ArithmeticError: vector is not in free module + ValueError: not an element of this algebra TESTS: Ensure that we can convert any element of the two non-matrix - simple algebras (whose natural representations are their usual - vector representations) back and forth faithfully:: + simple algebras (whose matrix representations are columns) + back and forth faithfully:: sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() + sage: J = HadamardEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True @@ -149,27 +389,46 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ + msg = "not an element of this algebra" if elt == 0: # The superclass implementation of random_element() # needs to be able to coerce "0" into the algebra. return self.zero() + elif elt in self.base_ring(): + # Ensure that no base ring -> algebra coercion is performed + # by this method. There's some stupidity in sage that would + # otherwise propagate to this method; for example, sage thinks + # that the integer 3 belongs to the space of 2-by-2 matrices. + raise ValueError(msg) - natural_basis = self.natural_basis() - basis_space = natural_basis[0].matrix_space() - if elt not in basis_space: - raise ValueError("not a naturally-represented algebra element") + try: + elt = elt.column() + except (AttributeError, TypeError): + # Try to convert a vector into a column-matrix + pass + + if elt not in self.matrix_space(): + raise ValueError(msg) # Thanks for nothing! Matrix spaces aren't vector spaces in - # Sage, so we have to figure out its natural-basis coordinates + # Sage, so we have to figure out its matrix-basis coordinates # ourselves. We use the basis space's ring instead of the # element's ring because the basis space might be an algebraic # closure whereas the base ring of the 3-by-3 identity matrix # could be QQ instead of QQbar. - V = VectorSpace(basis_space.base_ring(), elt.nrows()*elt.ncols()) - W = V.span_of_basis( _mat2vec(s) for s in natural_basis ) - coords = W.coordinate_vector(_mat2vec(elt)) - return self.from_vector(coords) + # + # We pass check=False because the matrix basis is "guaranteed" + # to be linearly independent... right? Ha ha. + V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols()) + W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()), + check=False) + try: + coords = W.coordinate_vector(_mat2vec(elt)) + except ArithmeticError: # vector is not in free module + raise ValueError(msg) + + return self.from_vector(coords) def _repr_(self): """ @@ -183,8 +442,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Ensure that it says what we think it says:: - sage: JordanSpinEJA(2, field=QQ) - Euclidean Jordan algebra of dimension 2 over Rational Field + sage: JordanSpinEJA(2, field=AA) + Euclidean Jordan algebra of dimension 2 over Algebraic Real Field sage: JordanSpinEJA(3, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field @@ -192,171 +451,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): fmt = "Euclidean Jordan algebra of dimension {} over {}" return fmt.format(self.dimension(), self.base_ring()) - def product_on_basis(self, i, j): - return self._multiplication_table[i][j] - - def _a_regular_element(self): - """ - Guess a regular element. Needed to compute the basis for our - characteristic polynomial coefficients. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS: - - Ensure that this hacky method succeeds for every algebra that we - know how to construct:: - - sage: set_random_seed() - sage: J = random_eja() - sage: J._a_regular_element().is_regular() - True - - """ - gs = self.gens() - z = self.sum( (i+1)*gs[i] for i in range(len(gs)) ) - if not z.is_regular(): - raise ValueError("don't know a regular element") - return z - - - @cached_method - def _charpoly_basis_space(self): - """ - Return the vector space spanned by the basis used in our - characteristic polynomial coefficients. This is used not only to - compute those coefficients, but also any time we need to - evaluate the coefficients (like when we compute the trace or - determinant). - """ - z = self._a_regular_element() - # Don't use the parent vector space directly here in case this - # happens to be a subalgebra. In that case, we would be e.g. - # two-dimensional but span_of_basis() would expect three - # coordinates. - V = VectorSpace(self.base_ring(), self.vector_space().dimension()) - basis = [ (z**k).to_vector() for k in range(self.rank()) ] - V1 = V.span_of_basis( basis ) - b = (V1.basis() + V1.complement().basis()) - return V.span_of_basis(b) - - @cached_method - def _charpoly_coeff(self, i): - """ - Return the coefficient polynomial "a_{i}" of this algebra's - general characteristic polynomial. - - Having this be a separate cached method lets us compute and - store the trace/determinant (a_{r-1} and a_{0} respectively) - separate from the entire characteristic polynomial. - """ - (A_of_x, x, xr, detA) = self._charpoly_matrix_system() - R = A_of_x.base_ring() - - if i == self.rank(): - return R.one() - if i > self.rank(): - # Guaranteed by theory - return R.zero() - - # Danger: the in-place modification is done for performance - # reasons (reconstructing a matrix with huge polynomial - # entries is slow), but I don't know how cached_method works, - # so it's highly possible that we're modifying some global - # list variable by reference, here. In other words, you - # probably shouldn't call this method twice on the same - # algebra, at the same time, in two threads - Ai_orig = A_of_x.column(i) - A_of_x.set_column(i,xr) - numerator = A_of_x.det() - A_of_x.set_column(i,Ai_orig) - - # We're relying on the theory here to ensure that each a_i is - # indeed back in R, and the added negative signs are to make - # the whole charpoly expression sum to zero. - return R(-numerator/detA) - - - @cached_method - def _charpoly_matrix_system(self): - """ - Compute the matrix whose entries A_ij are polynomials in - X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector - corresponding to `x^r` and the determinent of the matrix A = - [A_ij]. In other words, all of the fixed (cachable) data needed - to compute the coefficients of the characteristic polynomial. + def characteristic_polynomial_of(self): """ - r = self.rank() - n = self.dimension() - - # Turn my vector space into a module so that "vectors" can - # have multivatiate polynomial entries. - names = tuple('X' + str(i) for i in range(1,n+1)) - R = PolynomialRing(self.base_ring(), names) - - # Using change_ring() on the parent's vector space doesn't work - # here because, in a subalgebra, that vector space has a basis - # and change_ring() tries to bring the basis along with it. And - # that doesn't work unless the new ring is a PID, which it usually - # won't be. - V = FreeModule(R,n) - - # Now let x = (X1,X2,...,Xn) be the vector whose entries are - # indeterminates... - x = V(names) - - # And figure out the "left multiplication by x" matrix in - # that setting. - lmbx_cols = [] - monomial_matrices = [ self.monomial(i).operator().matrix() - for i in range(n) ] # don't recompute these! - for k in range(n): - ek = self.monomial(k).to_vector() - lmbx_cols.append( - sum( x[i]*(monomial_matrices[i]*ek) - for i in range(n) ) ) - Lx = matrix.column(R, lmbx_cols) - - # Now we can compute powers of x "symbolically" - x_powers = [self.one().to_vector(), x] - for d in range(2, r+1): - x_powers.append( Lx*(x_powers[-1]) ) - - idmat = matrix.identity(R, n) - - W = self._charpoly_basis_space() - W = W.change_ring(R.fraction_field()) - - # Starting with the standard coordinates x = (X1,X2,...,Xn) - # and then converting the entries to W-coordinates allows us - # to pass in the standard coordinates to the charpoly and get - # back the right answer. Specifically, with x = (X1,X2,...,Xn), - # we have - # - # W.coordinates(x^2) eval'd at (standard z-coords) - # = - # W-coords of (z^2) - # = - # W-coords of (standard coords of x^2 eval'd at std-coords of z) - # - # We want the middle equivalent thing in our matrix, but use - # the first equivalent thing instead so that we can pass in - # standard coordinates. - x_powers = [ W.coordinate_vector(xp) for xp in x_powers ] - l2 = [idmat.column(k-1) for k in range(r+1, n+1)] - A_of_x = matrix.column(R, n, (x_powers[:r] + l2)) - return (A_of_x, x, x_powers[r], A_of_x.det()) - - - @cached_method - def characteristic_polynomial(self): - """ - 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 @@ -376,7 +478,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Alizadeh, Example 11.11:: sage: J = JordanSpinEJA(3) - sage: p = J.characteristic_polynomial(); p + sage: p = J.characteristic_polynomial_of(); p X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 sage: xvec = J.one().to_vector() sage: p(*xvec) @@ -389,28 +491,51 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): any argument:: sage: J = TrivialEJA() - sage: J.characteristic_polynomial() + sage: J.characteristic_polynomial_of() 1 """ r = self.rank() n = self.dimension() - # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n. - a = [ self._charpoly_coeff(i) for i in range(r+1) ] + # 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) + if r > 0: + R = a[0].parent() + S = PolynomialRing(S, R.variable_names()) + t = S(t) + + return (t**r + sum( a[k]*(t**k) for k in range(r) )) + + def coordinate_polynomial_ring(self): + r""" + The multivariate polynomial ring in which this algebra's + :meth:`characteristic_polynomial_of` lives. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) - return sum( a[k]*(t**k) for k in range(len(a)) ) + EXAMPLES:: + + sage: J = HadamardEJA(2) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2... + sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... + """ + var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) ) + return PolynomialRing(self.base_ring(), var_names) def inner_product(self, x, y): """ @@ -422,7 +547,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: BilinearFormEJA) EXAMPLES: @@ -435,10 +562,34 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: (x*y).inner_product(z) == y.inner_product(x*z) True + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + + Ensure that this is one-half of the trace inner-product in a + BilinearFormEJA that isn't just the reals (when ``n`` isn't + one). This is in Faraut and Koranyi, and also my "On the + symmetry..." paper:: + + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) + True """ - X = x.natural_representation() - Y = y.natural_representation() - return self.natural_inner_product(X,Y) + B = self._inner_product_matrix + return (B*x.to_vector()).inner_product(y.to_vector()) def is_trivial(self): @@ -494,26 +645,46 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): +----++----+----+----+----+ """ - 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 + n = self.dimension() + # Prepend the header row. + M = [["*"] + list(self.gens())] + + # And to each subsequent row, prepend an entry that belongs to + # the left-side "header column." + M += [ [self.monomial(i)] + [ self.product_on_basis(i,j) + for j in range(n) ] + for i in range(n) ] + return table(M, header_row=True, header_column=True, frame=True) - def natural_basis(self): + def matrix_basis(self): """ - Return a more-natural representation of this algebra's basis. + Return an (often more natural) representation of this algebras + basis as an ordered tuple of matrices. + + Every finite-dimensional Euclidean Jordan Algebra is a, up to + Jordan isomorphism, a direct sum of five simple + algebras---four of which comprise Hermitian matrices. And the + last type of algebra can of course be thought of as `n`-by-`1` + column matrices (ambiguusly called column vectors) to avoid + special cases. As a result, matrices (and column vectors) are + a natural representation format for Euclidean Jordan algebra + elements. - Every finite-dimensional Euclidean Jordan Algebra is a direct - sum of five simple algebras, four of which comprise Hermitian - matrices. This method returns the original "natural" basis - for our underlying vector space. (Typically, the natural basis - is used to construct the multiplication table in the first place.) + But, when we construct an algebra from a basis of matrices, + those matrix representations are lost in favor of coordinate + vectors *with respect to* that basis. We could eventually + convert back if we tried hard enough, but having the original + representations handy is valuable enough that we simply store + them and return them from this method. - Note that this will always return a matrix. The standard basis - in `R^n` will be returned as `n`-by-`1` column matrices. + Why implement this for non-matrix algebras? Avoiding special + cases for the :class:`BilinearFormEJA` pays with simplicity in + its own right. But mainly, we would like to be able to assume + that elements of a :class:`DirectSumEJA` can be displayed + nicely, without having to have special classes for direct sums + one of whose components was a matrix algebra. SETUP:: @@ -525,10 +696,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = RealSymmetricEJA(2) sage: J.basis() Finite family {0: e0, 1: e1, 2: e2} - sage: J.natural_basis() + sage: J.matrix_basis() ( - [1 0] [ 0 1/2*sqrt2] [0 0] - [0 0], [1/2*sqrt2 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ) :: @@ -536,43 +707,32 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = JordanSpinEJA(2) sage: J.basis() Finite family {0: e0, 1: e1} - sage: J.natural_basis() + sage: J.matrix_basis() ( [1] [0] [0], [1] ) - """ - if self._natural_basis is None: - M = self.natural_basis_space() - return tuple( M(b.to_vector()) for b in self.basis() ) - else: - return self._natural_basis + return self._matrix_basis - def natural_basis_space(self): - """ - Return the matrix space in which this algebra's natural basis - elements live. + def matrix_space(self): """ - if self._natural_basis is None or len(self._natural_basis) == 0: - return MatrixSpace(self.base_ring(), self.dimension(), 1) - else: - return self._natural_basis[0].matrix_space() - + Return the matrix space in which this algebra's elements live, if + we think of them as matrices (including column vectors of the + appropriate size). - @staticmethod - def natural_inner_product(X,Y): - """ - Compute the inner product of two naturally-represented elements. + 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 matrix + space (empty matrices) can be multiplied. - 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. + Matrix algebras override this with something more useful. """ - return (X.conjugate_transpose()*Y).trace() + if self.is_trivial(): + return MatrixSpace(self.base_ring(), 0) + else: + return self._matrix_basis[0].matrix_space() @cached_method @@ -582,12 +742,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES:: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 @@ -610,6 +770,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: actual == expected True + Ensure that the cached unit element (often precomputed by + hand) agrees with the computed one:: + + sage: set_random_seed() + sage: J = random_eja() + sage: cached = J.one() + sage: J.one.clear_cache() + sage: J.one() == cached + True + """ # We can brute-force compute the matrices of the operators # that correspond to the basis elements of this algebra. @@ -621,19 +791,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # appeal to the "long vectors" isometry. oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ] - # Now we use basis linear algebra to find the coefficients, + # Now we use basic linear algebra to find the coefficients, # of the matrices-as-vectors-linear-combination, which should # work for the original algebra basis too. - A = matrix.column(self.base_ring(), oper_vecs) + A = matrix(self.base_ring(), oper_vecs) # We used the isometry on the left-hand side already, but we # still need to do it for the right-hand side. Recall that we # wanted something that summed to the identity matrix. b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) ) - # Now if there's an identity element in the algebra, this should work. - coeffs = A.solve_right(b) - return self.linear_combination(zip(self.gens(), coeffs)) + # Now if there's an identity element in the algebra, this + # should work. We solve on the left to avoid having to + # transpose the matrix "A". + return self.from_vector(A.solve_left(b)) def peirce_decomposition(self, c): @@ -688,6 +859,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Vector space of degree 6 and dimension 2... sage: J1 Euclidean Jordan algebra of dimension 3... + sage: J0.one().to_matrix() + [0 0 0] + [0 0 0] + [0 0 1] + sage: orig_df = AA.options.display_format + sage: AA.options.display_format = 'radical' + sage: J.from_vector(J5.basis()[0]).to_matrix() + [ 0 0 1/2*sqrt(2)] + [ 0 0 0] + [1/2*sqrt(2) 0 0] + sage: J.from_vector(J5.basis()[1]).to_matrix() + [ 0 0 0] + [ 0 0 1/2*sqrt(2)] + [ 0 1/2*sqrt(2) 0] + sage: AA.options.display_format = orig_df + sage: J1.one().to_matrix() + [1 0 0] + [0 1 0] + [0 0 0] TESTS: @@ -702,9 +892,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J1.superalgebra() == J and J1.dimension() == J.dimension() True - The identity elements in the two subalgebras are the - projections onto their respective subspaces of the - superalgebra's identity element:: + The decomposition is into eigenspaces, and its components are + therefore necessarily orthogonal. Moreover, the identity + elements in the two subalgebras are the projections onto their + respective subspaces of the superalgebra's identity element:: sage: set_random_seed() sage: J = random_eja() @@ -714,6 +905,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): ....: 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() @@ -723,22 +924,26 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): if not c.is_idempotent(): raise ValueError("element is not idempotent: %s" % c) + from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + # Default these to what they should be if they turn out to be # trivial, because eigenspaces_left() won't return eigenvalues # corresponding to trivial spaces (e.g. it returns only the # eigenspace corresponding to lambda=1 if you take the # decomposition relative to the identity element). - trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ()) + trivial = FiniteDimensionalEJASubalgebra(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().left_eigenspaces(): + 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) + subalg = FiniteDimensionalEJASubalgebra(self, + gens, + check_axioms=False) if eigval == 0: J0 = subalg elif eigval == 1: @@ -749,10 +954,61 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return (J0, J5, J1) - def random_elements(self, count): + def random_element(self, thorough=False): + r""" + Return a random element of this algebra. + + Our algebra superclass method only returns a linear + combination of at most two basis elements. We instead + want the vector space "random element" method that + returns a more diverse selection. + + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + element when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + + """ + # For a general base ring... maybe we can trust this to do the + # right thing? Unlikely, but. + V = self.vector_space() + v = V.random_element() + + if self.base_ring() is AA: + # The "random element" method of the algebraic reals is + # stupid at the moment, and only returns integers between + # -2 and 2, inclusive: + # + # https://trac.sagemath.org/ticket/30875 + # + # Instead, we implement our own "random vector" method, + # and then coerce that into the algebra. We use the vector + # space degree here instead of the dimension because a + # subalgebra could (for example) be spanned by only two + # vectors, each with five coordinates. We need to + # generate all five coordinates. + if thorough: + v *= QQbar.random_element().real() + else: + v *= QQ.random_element() + + return self.from_vector(V.coordinate_vector(v)) + + def random_elements(self, count, thorough=False): """ Return ``count`` random elements as a tuple. + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + elements when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA @@ -767,23 +1023,69 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in range(count) ) + return tuple( self.random_element(thorough) + for idx in range(count) ) - def rank(self): + @cached_method + def _charpoly_coefficients(self): + r""" + The `r` polynomial coefficients of the "characteristic polynomial + of" function. """ - Return the rank of this EJA. + n = self.dimension() + R = self.coordinate_polynomial_ring() + vars = R.gens() + F = R.fraction_field() + + def L_x_i_j(i,j): + # From a result in my book, these are the entries of the + # basis representation of L_x. + return sum( vars[k]*self.monomial(k).operator().matrix()[i,j] + for k in range(n) ) + + L_x = matrix(F, n, n, L_x_i_j) + + r = None + if self.rank.is_in_cache(): + r = self.rank() + # There's no need to pad the system with redundant + # columns if we *know* they'll be redundant. + n = r + + # Compute an extra power in case the rank is equal to + # the dimension (otherwise, we would stop at x^(r-1)). + x_powers = [ (L_x**k)*self.one().to_vector() + for k in range(n+1) ] + A = matrix.column(F, x_powers[:n]) + AE = A.extended_echelon_form() + E = AE[:,n:] + A_rref = AE[:,:n] + if r is None: + r = A_rref.rank() + b = x_powers[r] + + # The theory says that only the first "r" coefficients are + # nonzero, and they actually live in the original polynomial + # ring and not the fraction field. We negate them because + # in the actual characteristic polynomial, they get moved + # to the other side where x^r lives. + return -A_rref.solve_right(E*b).change_ring(R)[:r] - ALGORITHM: + @cached_method + def rank(self): + r""" + Return the rank of this EJA. - The author knows of no algorithm to compute the rank of an EJA - where only the multiplication table is known. In lieu of one, we - require the rank to be specified when the algebra is created, - and simply pass along that number here. + This is a cached method because we know the rank a priori for + all of the algebras we can construct. Thus we can avoid the + expensive ``_charpoly_coefficients()`` call unless we truly + need to compute the whole characteristic polynomial. SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, ....: RealSymmetricEJA, ....: ComplexHermitianEJA, ....: QuaternionHermitianEJA, @@ -824,8 +1126,18 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: r > 0 or (r == 0 and J.is_trivial()) True + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: + + sage: set_random_seed() # long time + sage: J = random_eja() # long time + sage: caches = J.rank() # long time + sage: J.rank.clear_cache() # long time + sage: J.rank() == cached # long time + True + """ - return self._rank + return len(self._charpoly_coefficients()) def vector_space(self): @@ -846,335 +1158,310 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.zero().to_vector().parent().ambient_vector_space() - Element = FiniteDimensionalEuclideanJordanAlgebraElement - - -class KnownRankEJA(object): - """ - A class for algebras that we actually know we can construct. The - main issue is that, for most of our methods to make sense, we need - to know the rank of our algebra. Thus we can't simply generate a - "random" algebra, or even check that a given basis and product - satisfy the axioms; because even if everything looks OK, we wouldn't - know the rank we need to actuallty build the thing. - - Not really a subclass of FDEJA because doing that causes method - resolution errors, e.g. - - TypeError: Error when calling the metaclass bases - Cannot create a consistent method resolution - order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA - - """ - @staticmethod - def _max_test_case_size(): - """ - Return an integer "size" that is an upper bound on the size of - this algebra when it is used in a random test - case. Unfortunately, the term "size" is quite vague -- when - dealing with `R^n` under either the Hadamard or Jordan spin - product, the "size" refers to the dimension `n`. When dealing - with a matrix algebra (real symmetric or complex/quaternion - Hermitian), it refers to the size of the matrix, which is - far less than the dimension of the underlying vector space. - - We default to five in this class, which is safe in `R^n`. The - matrix algebra subclasses (or any class where the "size" is - interpreted to be far less than the dimension) should override - with a smaller number. - """ - return 5 - - @classmethod - def random_instance(cls, field=QQ, **kwargs): - """ - Return a random instance of this type of algebra. - - Beware, this will crash for "most instances" because the - constructor below looks wrong. - """ - if cls is TrivialEJA: - # The TrivialEJA class doesn't take an "n" argument because - # there's only one. - return cls(field) - - n = ZZ.random_element(cls._max_test_case_size()) + 1 - return cls(n, field, **kwargs) - - -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA): - """ - Return the Euclidean Jordan Algebra corresponding to the set - `R^n` under the Hadamard product. + Element = FiniteDimensionalEJAElement - Note: this is nothing more than the Cartesian product of ``n`` - copies of the spin algebra. Once Cartesian product algebras - are implemented, this can go. +class RationalBasisEJA(FiniteDimensionalEJA): + r""" + New class for algebras whose supplied basis elements have all rational entries. SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import BilinearFormEJA EXAMPLES: - This multiplication table can be verified by hand:: - - sage: J = RealCartesianProductEJA(3) - sage: e0,e1,e2 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - 0 - sage: e0*e2 - 0 - sage: e1*e1 - e1 - sage: e1*e2 - 0 - sage: e2*e2 - e2 + The supplied basis is orthonormalized by default:: - TESTS: - - We can change the generator prefix:: - - sage: RealCartesianProductEJA(3, prefix='r').gens() - (r0, r1, r2) + sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]]) + sage: J = BilinearFormEJA(B) + sage: J.matrix_basis() + ( + [1] [ 0] [ 0] + [0] [1/5] [32/5] + [0], [ 0], [ 5] + ) """ - def __init__(self, n, field=QQ, **kwargs): - V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] - for i in range(n) ] - - fdeja = super(RealCartesianProductEJA, self) - return fdeja.__init__(field, mult_table, rank=n, **kwargs) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. + def __init__(self, + basis, + jordan_product, + inner_product, + field=AA, + check_field=True, + **kwargs): + + if check_field: + # Abuse the check_field parameter to check that the entries of + # out basis (in ambient coordinates) are in the field QQ. + if not all( all(b_i in QQ for b_i in b.list()) for b in basis ): + raise TypeError("basis not rational") + + if field is not QQ: + # There's no point in constructing the extra algebra if this + # one is already rational. + # + # Note: the same Jordan and inner-products work here, + # because they are necessarily defined with respect to + # ambient coordinates and not any particular basis. + self._rational_algebra = FiniteDimensionalEJA( + basis, + jordan_product, + inner_product, + field=QQ, + orthonormalize=False, + check_field=False, + check_axioms=False) + + super().__init__(basis, + jordan_product, + inner_product, + field=field, + check_field=check_field, + **kwargs) + @cached_method + def _charpoly_coefficients(self): + r""" SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA - - TESTS: + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) - Ensure that this is the usual inner product for the algebras - over `R^n`:: + EXAMPLES: - sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.natural_inner_product(X,Y) - True + The base ring of the resulting polynomial coefficients is what + it should be, and not the rationals (unless the algebra was + already over the rationals):: - """ - return x.to_vector().inner_product(y.to_vector()) + sage: J = JordanSpinEJA(3) + sage: J._charpoly_coefficients() + (X1^2 - X2^2 - X3^2, -2*X1) + sage: a0 = J._charpoly_coefficients()[0] + sage: J.base_ring() + Algebraic Real Field + sage: a0.base_ring() + Algebraic Real Field + + """ + if self._rational_algebra is None: + # There's no need to construct *another* algebra over the + # rationals if this one is already over the + # rationals. Likewise, if we never orthonormalized our + # basis, we might as well just use the given one. + return super()._charpoly_coefficients() + + # Do the computation over the rationals. The answer will be + # the same, because all we've done is a change of basis. + # Then, change back from QQ to our real base ring + a = ( a_i.change_ring(self.base_ring()) + for a_i in self._rational_algebra._charpoly_coefficients() ) + + # Now convert the coordinate variables back to the + # deorthonormalized ones. + R = self.coordinate_polynomial_ring() + from sage.modules.free_module_element import vector + X = vector(R, R.gens()) + BX = self._deortho_matrix*X + + subs_dict = { X[i]: BX[i] for i in range(len(X)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) + +class ConcreteEJA(RationalBasisEJA): + r""" + A class for the Euclidean Jordan algebras that we know by name. + + These are the Jordan algebras whose basis, multiplication table, + rank, and so on are known a priori. More to the point, they are + the Euclidean Jordan algebras for which we are able to conjure up + a "random instance." + SETUP:: -def random_eja(field=QQ, nontrivial=False): - """ - Return a "random" finite-dimensional Euclidean Jordan Algebra. + sage: from mjo.eja.eja_algebra import ConcreteEJA - SETUP:: + TESTS: - sage: from mjo.eja.eja_algebra import random_eja + Our basis is normalized with respect to the algebra's inner + product, unless we specify otherwise:: - TESTS:: + sage: set_random_seed() + sage: J = ConcreteEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True - sage: random_eja() - Euclidean Jordan algebra of dimension... + Since our basis is orthonormal with respect to the algebra's inner + product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the + EJA the operator is self-adjoint by the Jordan axiom:: + sage: set_random_seed() + sage: J = ConcreteEJA.random_instance() + sage: x = J.random_element() + sage: x.operator().is_self_adjoint() + True """ - eja_classes = KnownRankEJA.__subclasses__() - if nontrivial: - eja_classes.remove(TrivialEJA) - classname = choice(eja_classes) - return classname.random_instance(field=field) + @staticmethod + def _max_random_instance_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 ambiguous -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is far + less than the dimension of the underlying vector space. + + This method must be implemented in each subclass. + """ + raise NotImplementedError + @classmethod + def random_instance(cls, *args, **kwargs): + """ + Return a random instance of this type of algebra. + This method should be implemented in each subclass. + """ + from sage.misc.prandom import choice + eja_class = choice(cls.__subclasses__()) + # These all bubble up to the RationalBasisEJA superclass + # constructor, so any (kw)args valid there are also valid + # here. + return eja_class.random_instance(*args, **kwargs) -class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): +class MatrixEJA: @staticmethod - def _max_test_case_size(): - # Play it safe, since this will be squared and the underlying - # field can have dimension 4 (quaternions) too. - return 2 + def dimension_over_reals(): + r""" + The dimension of this matrix's base ring over the reals. - def __init__(self, field, basis, rank, 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). + The reals are dimension one over themselves, obviously; that's + just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi` + have dimension two. Finally, the quaternions have dimension + four over the reals. + + This is used to determine the size of the matrix returned from + :meth:`real_embed`, among other things. """ - # Used in this class's fast _charpoly_coeff() override. - self._basis_normalizers = None + raise NotImplementedError - # We're going to loop through this a few times, so now's a good - # time to ensure that it isn't a generator expression. - basis = tuple(basis) + @classmethod + def real_embed(cls,M): + """ + Embed the matrix ``M`` into a space of real matrices. - if rank > 1 and normalize_basis: - # We'll need sqrt(2) to normalize the basis, and this - # winds up in the multiplication table, so the whole - # algebra needs to be over the field extension. - R = PolynomialRing(field, 'z') - z = R.gen() - p = z**2 - 2 - if p.is_irreducible(): - field = 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)) + 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. - Qs = self.multiplication_table_from_matrix_basis(basis) + real_embed(M*N) = real_embed(M)*real_embed(N) - fdeja = super(MatrixEuclideanJordanAlgebra, self) - return fdeja.__init__(field, - Qs, - rank=rank, - natural_basis=basis, - **kwargs) + """ + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + return M - @cached_method - def _charpoly_coeff(self, i): + @classmethod + def real_unembed(cls,M): """ - Override the parent method with something that tries to compute - over a faster (non-extension) field. + The inverse of :meth:`real_embed`. """ - if self._basis_normalizers is None: - # We didn't normalize, so assume that the basis we started - # with had entries in a nice field. - return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i) - else: - basis = ( (b/n) for (b,n) in zip(self.natural_basis(), - self._basis_normalizers) ) - - # Do this over the rationals and convert back at the end. - J = MatrixEuclideanJordanAlgebra(QQ, - basis, - self.rank(), - normalize_basis=False) - (_,x,_,_) = J._charpoly_matrix_system() - p = J._charpoly_coeff(i) - # p might be missing some vars, have to substitute "optionally" - pairs = zip(x.base_ring().gens(), self._basis_normalizers) - substitutions = { v: v*c for (v,c) in pairs } - result = p.subs(substitutions) - - # The result of "subs" can be either a coefficient-ring - # element or a polynomial. Gotta handle both cases. - if result in QQ: - return self.base_ring()(result) - else: - return result.change_ring(self.base_ring()) - + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero(): + raise ValueError("the matrix 'M' must be a real embedding") + return M @staticmethod - def multiplication_table_from_matrix_basis(basis): - """ - At least three of the five simple Euclidean Jordan algebras have the - symmetric multiplication (A,B) |-> (AB + BA)/2, where the - multiplication on the right is matrix multiplication. Given a basis - for the underlying matrix space, this function returns a - multiplication table (obtained by looping through the basis - elements) for an algebra of those matrices. - """ - # In S^2, for example, we nominally have four coordinates even - # though the space is of dimension three only. The vector space V - # is supposed to hold the entire long vector, and the subspace W - # of V will be spanned by the vectors that arise from symmetric - # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. - field = basis[0].base_ring() - dimension = basis[0].nrows() - - V = VectorSpace(field, dimension**2) - W = V.span_of_basis( _mat2vec(s) for s in basis ) - n = len(basis) - mult_table = [[W.zero() for j in 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)) + def jordan_product(X,Y): + return (X*Y + Y*X)/2 - return mult_table + @classmethod + def trace_inner_product(cls,X,Y): + r""" + Compute the trace inner-product of two real-embeddings. + SETUP:: - @staticmethod - def real_embed(M): - """ - Embed the matrix ``M`` into a space of real matrices. + sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA) - 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:: + + This gives the same answer as it would if we computed the trace + from the unembedded (original) matrices:: - real_embed(M*N) = real_embed(M)*real_embed(N) + sage: set_random_seed() + sage: J = RealSymmetricEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace() + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True - """ - raise NotImplementedError + :: + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace().real() + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True - @staticmethod - def real_unembed(M): - """ - The inverse of :meth:`real_embed`. - """ - raise NotImplementedError + :: + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace().coefficient_tuple()[0] + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True - @classmethod - def natural_inner_product(cls,X,Y): + """ Xu = cls.real_unembed(X) Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() - if tr in RLF: - # It's real already. - return tr - - # Otherwise, try the thing that works for complex numbers; and - # if that doesn't work, the thing that works for quaternions. try: - return tr.vector()[0] # real part, imag part is index 1 + # Works in QQ, AA, RDF, et cetera. + return tr.real() except AttributeError: - # A quaternions doesn't have a vector() method, but does + # 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] -class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): - @staticmethod - def real_embed(M): - """ - The identity function, for embedding real matrices into real - matrices. - """ - return M - +class RealMatrixEJA(MatrixEJA): @staticmethod - def real_unembed(M): - """ - The identity function, for unembedding real matrices from real - matrices. - """ - return M + def dimension_over_reals(): + return 1 -class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): +class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1197,9 +1484,9 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, AA) - Euclidean Jordan algebra of dimension 3 over Algebraic Real Field - sage: RealSymmetricEJA(2, RR) + sage: RealSymmetricEJA(2, field=RDF) + Euclidean Jordan algebra of dimension 3 over Real Double Field + sage: RealSymmetricEJA(2, field=RR) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1208,7 +1495,7 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): The dimension of this algebra is `(n^2 + n) / 2`:: sage: set_random_seed() - sage: n_max = RealSymmetricEJA._max_test_case_size() + sage: n_max = RealSymmetricEJA._max_random_instance_size() sage: n = ZZ.random_element(1, n_max) sage: J = RealSymmetricEJA(n) sage: J.dimension() == (n^2 + n)/2 @@ -1219,9 +1506,9 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: set_random_seed() sage: J = RealSymmetricEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1233,28 +1520,14 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: RealSymmetricEJA(3, prefix='q').gens() (q0, q1, q2, q3, q4, q5) - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: - - sage: set_random_seed() - sage: J = RealSymmetricEJA.random_instance() - sage: all( b.norm() == 1 for b in J.gens() ) - True - - Since our natural basis is normalized with respect to the natural - inner product, and since we know that this algebra is an EJA, any - left-multiplication operator's matrix will be symmetric because - natural->EJA basis representation is an isometry and within the EJA - the operator is self-adjoint by the Jordan axiom:: + We can construct the (trivial) algebra of rank zero:: - sage: set_random_seed() - sage: x = RealSymmetricEJA.random_instance().random_element() - sage: x.operator().matrix().is_symmetric() - True + sage: RealSymmetricEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Return a basis for the space of real symmetric n-by-n matrices. @@ -1266,7 +1539,7 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = RealSymmetricEJA._denormalized_basis(n,QQ) + sage: B = RealSymmetricEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B) True @@ -1276,28 +1549,53 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): S = [] for i in range(n): for j in range(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) + Eij = matrix(ZZ, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij else: Sij = Eij + Eij.transpose() S.append(Sij) - return S + return tuple(S) @staticmethod - def _max_test_case_size(): + def _max_random_instance_size(): return 4 # Dimension 10 + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + def __init__(self, n, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + **kwargs) + + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) - def __init__(self, n, field=QQ, **kwargs): - basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs) -class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): +class ComplexMatrixEJA(MatrixEJA): @staticmethod - def real_embed(M): + def dimension_over_reals(): + return 2 + + @classmethod + def real_embed(cls,M): """ Embed the n-by-n complex matrix ``M`` into the space of real matrices of size 2n-by-2n via the map the sends each entry `z = a + @@ -1305,18 +1603,17 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) sage: x4 = F(6) sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) - sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M) + sage: ComplexMatrixEJA.real_embed(M) [ 4 -2| 1 2] [ 2 4|-2 1] [-----+-----] @@ -1328,21 +1625,19 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() - sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size() - sage: n = ZZ.random_element(n_max) - sage: F = QuadraticField(-1, 'i') + sage: n = ZZ.random_element(3) + sage: F = QuadraticField(-1, 'I') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) - sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X) - sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y) - sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y) + sage: Xe = ComplexMatrixEJA.real_embed(X) + sage: Ye = ComplexMatrixEJA.real_embed(Y) + sage: XYe = ComplexMatrixEJA.real_embed(X*Y) sage: Xe*Ye == XYe True """ + super(ComplexMatrixEJA,cls).real_embed(M) 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() @@ -1356,15 +1651,14 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return matrix.block(field, n, blocks) - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of _embed_complex_matrix(). SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: @@ -1372,42 +1666,53 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [-2, 1, -4, 3], ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) - sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + sage: ComplexMatrixEJA.real_unembed(A) + [ 2*I + 1 4*I + 3] + [ 10*I + 9 12*I + 11] TESTS: Unembedding is the inverse of embedding:: sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: M = random_matrix(F, 3) - sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) - sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + sage: Me = ComplexMatrixEJA.real_embed(M) + sage: ComplexMatrixEJA.real_unembed(Me) == M True """ + super(ComplexMatrixEJA,cls).real_unembed(M) n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(2).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") + d = cls.dimension_over_reals() # If "M" was normalized, its base ring might have roots # adjoined and they can stick around after unembedding. field = M.base_ring() R = PolynomialRing(field, 'z') z = R.gen() - F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + + # Sage doesn't know how to adjoin the complex "i" (the root of + # x^2 + 1) to a field in a general way. Here, we just enumerate + # all of the cases that I have cared to support so far. + if field is AA: + # Sage doesn't know how to embed AA into QQbar, i.e. how + # to adjoin sqrt(-1) to AA. + F = QQbar + elif not field.is_exact(): + # RDF or RR + F = field.complex_field() + else: + # Works for QQ and... maybe some other fields. + 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] + for k in range(n/d): + for j in range(n/d): + submat = M[d*k:d*k+d,d*j:d*j+d] if submat[0,0] != submat[1,1]: raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: @@ -1415,41 +1720,10 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): z = submat[0,0] + submat[0,1]*i elements.append(z) - return matrix(F, n/2, elements) - - - @classmethod - def natural_inner_product(cls,X,Y): - """ - Compute a natural inner product in this algebra directly from - its real embedding. - - SETUP:: - - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA - - TESTS: - - This gives the same answer as the slow, default method implemented - in :class:`MatrixEuclideanJordanAlgebra`:: - - sage: set_random_seed() - sage: J = ComplexHermitianEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: Xe = x.natural_representation() - sage: Ye = y.natural_representation() - sage: X = ComplexHermitianEJA.real_unembed(Xe) - sage: Y = ComplexHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().vector()[0] - sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) - sage: actual == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 + return matrix(F, n/d, elements) -class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): +class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1464,9 +1738,9 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, AA) - Euclidean Jordan algebra of dimension 4 over Algebraic Real Field - sage: ComplexHermitianEJA(2, RR) + sage: ComplexHermitianEJA(2, field=RDF) + Euclidean Jordan algebra of dimension 4 over Real Double Field + sage: ComplexHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -1475,7 +1749,7 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): The dimension of this algebra is `n^2`:: sage: set_random_seed() - sage: n_max = ComplexHermitianEJA._max_test_case_size() + sage: n_max = ComplexHermitianEJA._max_random_instance_size() sage: n = ZZ.random_element(1, n_max) sage: J = ComplexHermitianEJA(n) sage: J.dimension() == n^2 @@ -1486,9 +1760,9 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: set_random_seed() sage: J = ComplexHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1500,29 +1774,15 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: ComplexHermitianEJA(2, prefix='z').gens() (z0, z1, z2, z3) - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: + We can construct the (trivial) algebra of rank zero:: - sage: set_random_seed() - sage: J = ComplexHermitianEJA.random_instance() - sage: all( b.norm() == 1 for b in J.gens() ) - True - - Since our natural basis is normalized with respect to the natural - inner product, and since we know that this algebra is an EJA, any - left-multiplication operator's matrix will be symmetric because - natural->EJA basis representation is an isometry and within the EJA - the operator is self-adjoint by the Jordan axiom:: - - sage: set_random_seed() - sage: x = ComplexHermitianEJA.random_instance().random_element() - sage: x.operator().matrix().is_symmetric() - True + sage: ComplexHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Returns a basis for the space of complex Hermitian n-by-n matrices. @@ -1541,15 +1801,16 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: field = QuadraticField(2, 'sqrt2') - sage: B = ComplexHermitianEJA._denormalized_basis(n, field) + sage: B = ComplexHermitianEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B) True """ + field = ZZ R = PolynomialRing(field, 'z') z = R.gen() F = field.extension(z**2 + 1, 'I') - I = F.gen() + I = F.gen(1) # This is like the symmetric case, but we need to be careful: # @@ -1572,17 +1833,44 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): # 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 ) + return tuple( s.change_ring(field) for s in S ) + + def __init__(self, n, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - def __init__(self, n, field=QQ, **kwargs): - basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs) + super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + **kwargs) + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) + + @staticmethod + def _max_random_instance_size(): + return 3 # Dimension 9 + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) -class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): +class QuaternionMatrixEJA(MatrixEJA): @staticmethod - def real_embed(M): + def dimension_over_reals(): + return 4 + + @classmethod + def real_embed(cls,M): """ Embed the n-by-n quaternion matrix ``M`` into the space of real matrices of size 4n-by-4n by first sending each quaternion entry `z @@ -1592,8 +1880,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -1601,7 +1888,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: i,j,k = Q.gens() sage: x = 1 + 2*i + 3*j + 4*k sage: M = matrix(Q, 1, [[x]]) - sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M) + sage: QuaternionMatrixEJA.real_embed(M) [ 1 2 3 4] [-2 1 -4 3] [-3 4 1 -2] @@ -1610,24 +1897,22 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() - sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size() - sage: n = ZZ.random_element(n_max) + sage: n = ZZ.random_element(2) sage: Q = QuaternionAlgebra(QQ,-1,-1) sage: X = random_matrix(Q, n) sage: Y = random_matrix(Q, n) - sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X) - sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y) - sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y) + sage: Xe = QuaternionMatrixEJA.real_embed(X) + sage: Ye = QuaternionMatrixEJA.real_embed(Y) + sage: XYe = QuaternionMatrixEJA.real_embed(X*Y) sage: Xe*Ye == XYe True """ + super(QuaternionMatrixEJA,cls).real_embed(M) quaternions = M.base_ring() n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - F = QuadraticField(-1, 'i') + F = QuadraticField(-1, 'I') i = F.gen() blocks = [] @@ -1639,7 +1924,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): d = t[3] cplxM = matrix(F, 2, [[ a + b*i, c + d*i], [-c + d*i, a - b*i]]) - realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM) + realM = ComplexMatrixEJA.real_embed(cplxM) blocks.append(realM) # We should have real entries by now, so use the realest field @@ -1648,15 +1933,14 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of _embed_quaternion_matrix(). SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -1664,7 +1948,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [-2, 1, -4, 3], ....: [-3, 4, 1, -2], ....: [-4, -3, 2, 1]]) - sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M) + sage: QuaternionMatrixEJA.real_unembed(M) [1 + 2*i + 3*j + 4*k] TESTS: @@ -1674,16 +1958,14 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: Q = QuaternionAlgebra(QQ, -1, -1) sage: M = random_matrix(Q, 3) - sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M) - sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + sage: Me = QuaternionMatrixEJA.real_embed(M) + sage: QuaternionMatrixEJA.real_unembed(Me) == M True """ + super(QuaternionMatrixEJA,cls).real_unembed(M) n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(4).is_zero(): - raise ValueError("the matrix 'M' must be a quaternion embedding") + d = cls.dimension_over_reals() # Use the base ring of the matrix to ensure that its entries can be # multiplied by elements of the quaternion algebra. @@ -1695,57 +1977,25 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): # 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] ) + for l in range(n/d): + for m in range(n/d): + submat = ComplexMatrixEJA.real_unembed( + M[d*l:d*l+d,d*m:d*m+d] ) if submat[0,0] != submat[1,1].conjugate(): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): raise ValueError('bad off-diagonal submatrix') - z = submat[0,0].vector()[0] # real part - z += submat[0,0].vector()[1]*i # imag part - z += submat[0,1].vector()[0]*j # real part - z += submat[0,1].vector()[1]*k # imag part + z = submat[0,0].real() + z += submat[0,0].imag()*i + z += submat[0,1].real()*j + z += submat[0,1].imag()*k elements.append(z) - return matrix(Q, n/4, elements) - - - @classmethod - def natural_inner_product(cls,X,Y): - """ - Compute a natural inner product in this algebra directly from - its real embedding. - - SETUP:: - - sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - - TESTS: - - This gives the same answer as the slow, default method implemented - in :class:`MatrixEuclideanJordanAlgebra`:: - - sage: set_random_seed() - sage: J = QuaternionHermitianEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: Xe = x.natural_representation() - sage: Ye = y.natural_representation() - sage: X = QuaternionHermitianEJA.real_unembed(Xe) - sage: Y = QuaternionHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().coefficient_tuple()[0] - sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye) - sage: actual == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 + return matrix(Q, n/d, elements) -class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, - KnownRankEJA): - """ +class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): + r""" The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the real-part-of-trace inner product. It has dimension `2n^2 - n` over @@ -1759,9 +2009,9 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, AA) - Euclidean Jordan algebra of dimension 6 over Algebraic Real Field - sage: QuaternionHermitianEJA(2, RR) + sage: QuaternionHermitianEJA(2, field=RDF) + Euclidean Jordan algebra of dimension 6 over Real Double Field + sage: QuaternionHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -1770,7 +2020,7 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, The dimension of this algebra is `2*n^2 - n`:: sage: set_random_seed() - sage: n_max = QuaternionHermitianEJA._max_test_case_size() + sage: n_max = QuaternionHermitianEJA._max_random_instance_size() sage: n = ZZ.random_element(1, n_max) sage: J = QuaternionHermitianEJA(n) sage: J.dimension() == 2*(n^2) - n @@ -1781,9 +2031,9 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = QuaternionHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1795,28 +2045,14 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: QuaternionHermitianEJA(2, prefix='a').gens() (a0, a1, a2, a3, a4, a5) - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: - - 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:: + We can construct the (trivial) algebra of rank zero:: - sage: set_random_seed() - sage: x = QuaternionHermitianEJA.random_instance().random_element() - sage: x.operator().matrix().is_symmetric() - True + sage: QuaternionHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. @@ -1834,11 +2070,12 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ) + sage: B = QuaternionHermitianEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B ) True """ + field = ZZ Q = QuaternionAlgebra(QQ,-1,-1) I,J,K = Q.gens() @@ -1868,19 +2105,284 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, # 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 ) + return tuple( s.change_ring(field) for s in S ) + + + def __init__(self, n, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + **kwargs) + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) + + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum rank of a random QuaternionHermitianEJA. + """ + return 2 # Dimension 6 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class HadamardEJA(ConcreteEJA): + """ + Return the Euclidean Jordan Algebra corresponding to the set + `R^n` under the Hadamard product. + + Note: this is nothing more than the Cartesian product of ``n`` + copies of the spin algebra. Once Cartesian product algebras + are implemented, this can go. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + EXAMPLES: + + This multiplication table can be verified by hand:: + + sage: J = HadamardEJA(3) + sage: e0,e1,e2 = J.gens() + sage: e0*e0 + e0 + sage: e0*e1 + 0 + sage: e0*e2 + 0 + sage: e1*e1 + e1 + sage: e1*e2 + 0 + sage: e2*e2 + e2 + + TESTS: + + We can change the generator prefix:: + + sage: HadamardEJA(3, prefix='r').gens() + (r0, r1, r2) + + """ + def __init__(self, n, **kwargs): + if n == 0: + jordan_product = lambda x,y: x + inner_product = lambda x,y: x + else: + def jordan_product(x,y): + P = x.parent() + return P( xi*yi for (xi,yi) in zip(x,y) ) + + def inner_product(x,y): + return (x.T*y)[0,0] + + # New defaults for keyword arguments. Don't orthonormalize + # because our basis is already orthonormal with respect to our + # inner-product. Don't check the axioms, because we know this + # is a valid EJA... but do double-check if the user passes + # check_axioms=True. Note: we DON'T override the "check_field" + # default here, because the user can pass in a field! + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() ) + super().__init__(column_basis, jordan_product, inner_product, **kwargs) + self.rank.set_cache(n) + + if n == 0: + self.one.set_cache( self.zero() ) + else: + self.one.set_cache( sum(self.gens()) ) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random HadamardEJA. + """ + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class BilinearFormEJA(ConcreteEJA): + r""" + The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` + with the half-trace inner product and jordan product ``x*y = + (,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is + a symmetric positive-definite "bilinear form" matrix. Its + dimension is the size of `B`, and it has rank two in dimensions + larger than two. It reduces to the ``JordanSpinEJA`` when `B` is + the identity matrix of order ``n``. + + We insist that the one-by-one upper-left identity block of `B` be + passed in as well so that we can be passed a matrix of size zero + to construct a trivial algebra. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) + + EXAMPLES: + + When no bilinear form is specified, the identity matrix is used, + and the resulting algebra is the Jordan spin algebra:: + + sage: B = matrix.identity(AA,3) + sage: J0 = BilinearFormEJA(B) + sage: J1 = JordanSpinEJA(3) + sage: J0.multiplication_table() == J0.multiplication_table() + True + + An error is raised if the matrix `B` does not correspond to a + positive-definite bilinear form:: + + sage: B = matrix.random(QQ,2,3) + sage: J = BilinearFormEJA(B) + Traceback (most recent call last): + ... + ValueError: bilinear form is not positive-definite + sage: B = matrix.zero(QQ,3) + sage: J = BilinearFormEJA(B) + Traceback (most recent call last): + ... + ValueError: bilinear form is not positive-definite + + TESTS: + + We can create a zero-dimensional algebra:: + + sage: B = matrix.identity(AA,0) + sage: J = BilinearFormEJA(B) + sage: J.basis() + Finite family {} + + We can check the multiplication condition given in the Jordan, von + Neumann, and Wigner paper (and also discussed on my "On the + symmetry..." paper). Note that this relies heavily on the standard + choice of basis, as does anything utilizing the bilinear form + matrix. We opt not to orthonormalize the basis, because if we + did, we would have to normalize the `s_{i}` in a similar manner:: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B11 = matrix.identity(QQ,1) + sage: B22 = M.transpose()*M + sage: B = block_matrix(2,2,[ [B11,0 ], + ....: [0, B22 ] ]) + sage: J = BilinearFormEJA(B, orthonormalize=False) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() ) + ....: for ei in eis ] + sage: actual = [ sis[i]*sis[j] + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: expected = [ J.one() if i == j else J.zero() + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: actual == expected + True + """ + def __init__(self, B, **kwargs): + if not B.is_positive_definite(): + raise ValueError("bilinear form is not positive-definite") + + def inner_product(x,y): + return (y.T*B*x)[0,0] + + def jordan_product(x,y): + P = x.parent() + x0 = x[0,0] + xbar = x[1:,0] + y0 = y[0,0] + ybar = y[1:,0] + z0 = (y.T*x)[0,0] + zbar = y0*xbar + x0*ybar + return P([0] + zbar.list()) + + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + n = B.nrows() + column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() ) + super(BilinearFormEJA, self).__init__(column_basis, + jordan_product, + inner_product, + **kwargs) + + # The rank of this algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded + # by the ambient dimension). + self.rank.set_cache(min(n,2)) + + if n == 0: + self.one.set_cache( self.zero() ) + else: + self.one.set_cache( self.monomial(0) ) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random BilinearFormEJA. + """ + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + if n.is_zero(): + B = matrix.identity(ZZ, n) + return cls(B, **kwargs) + B11 = matrix.identity(ZZ, 1) + M = matrix.random(ZZ, n-1) + I = matrix.identity(ZZ, n-1) + alpha = ZZ.zero() + while alpha.is_zero(): + alpha = ZZ.random_element().abs() + B22 = M.transpose()*M + alpha*I - def __init__(self, n, field=QQ, **kwargs): - basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) + from sage.matrix.special import block_matrix + B = block_matrix(2,2, [ [B11, ZZ(0) ], + [ZZ(0), B22 ] ]) + return cls(B, **kwargs) -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): + +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:: @@ -1913,56 +2415,51 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: JordanSpinEJA(2, prefix='B').gens() (B0, B1) - """ - def __init__(self, n, field=QQ, **kwargs): - 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:] - # z = x*y - z0 = x.inner_product(y) - zbar = y0*xbar + x0*ybar - z = V([z0] + zbar.list()) - mult_table[i][j] = z - - # The rank of the spin algebra is two, unless we're in a - # one-dimensional ambient space (because the rank is bounded by - # the ambient dimension). - fdeja = super(JordanSpinEJA, self) - return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. - - SETUP:: - - sage: from mjo.eja.eja_algebra import JordanSpinEJA - - TESTS: + TESTS: - Ensure that this is the usual inner product for the algebras - over `R^n`:: + 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) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected True + """ + def __init__(self, n, **kwargs): + # This is a special case of the BilinearFormEJA with the + # identity matrix as its bilinear form. + B = matrix.identity(ZZ, n) + + # Don't orthonormalize because our basis is already + # orthonormal with respect to our inner-product. + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + + # But also don't pass check_field=False here, because the user + # can pass in a field! + super(JordanSpinEJA, self).__init__(B, **kwargs) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random JordanSpinEJA. + """ + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + + Needed here to override the implementation for ``BilinearFormEJA``. """ - return x.to_vector().inner_product(y.to_vector()) + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) -class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class TrivialEJA(ConcreteEJA): """ The trivial Euclidean Jordan algebra consisting of only a zero element. @@ -1986,14 +2483,273 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: J.one().norm() 0 sage: J.one().subalgebra_generated_by() - Euclidean Jordan algebra of dimension 0 over Rational Field + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field sage: J.rank() 0 """ - def __init__(self, field=QQ, **kwargs): - mult_table = [] - fdeja = super(TrivialEJA, self) + def __init__(self, **kwargs): + jordan_product = lambda x,y: x + inner_product = lambda x,y: 0 + basis = () + + # New defaults for keyword arguments + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(TrivialEJA, self).__init__(basis, + jordan_product, + inner_product, + **kwargs) # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. - return fdeja.__init__(field, mult_table, rank=0, **kwargs) + self.rank.set_cache(0) + self.one.set_cache( self.zero() ) + + @classmethod + def random_instance(cls, **kwargs): + # We don't take a "size" argument so the superclass method is + # inappropriate for us. + return cls(**kwargs) + +# class DirectSumEJA(ConcreteEJA): +# 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 (random_eja, +# ....: 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 + +# TESTS: + +# The external direct sum construction is only valid when the two factors +# have the same base ring; an error is raised otherwise:: + +# sage: set_random_seed() +# sage: J1 = random_eja(field=AA) +# sage: J2 = random_eja(field=QQ,orthonormalize=False) +# sage: J = DirectSumEJA(J1,J2) +# Traceback (most recent call last): +# ... +# ValueError: algebras must share the same base field + +# """ +# def __init__(self, J1, J2, **kwargs): +# if J1.base_ring() != J2.base_ring(): +# raise ValueError("algebras must share the same base field") +# field = J1.base_ring() + +# self._factors = (J1, J2) +# n1 = J1.dimension() +# n2 = J2.dimension() +# n = n1+n2 +# V = VectorSpace(field, n) +# mult_table = [ [ V.zero() for j in range(i+1) ] +# for i in range(n) ] +# for i in range(n1): +# for j in range(i+1): +# 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(i+1): +# p = (J2.monomial(i)*J2.monomial(j)).to_vector() +# mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list()) + +# # TODO: build the IP table here from the two constituent IP +# # matrices (it'll be block diagonal, I think). +# ip_table = [ [ field.zero() for j in range(i+1) ] +# for i in range(n) ] +# super(DirectSumEJA, self).__init__(field, +# mult_table, +# ip_table, +# check_axioms=False, +# **kwargs) +# self.rank.set_cache(J1.rank() + J2.rank()) + + +# def factors(self): +# r""" +# Return the pair of this algebra's factors. + +# SETUP:: + +# sage: from mjo.eja.eja_algebra import (HadamardEJA, +# ....: JordanSpinEJA, +# ....: DirectSumEJA) + +# EXAMPLES:: + +# sage: J1 = HadamardEJA(2, field=QQ) +# sage: J2 = JordanSpinEJA(3, field=QQ) +# sage: J = DirectSumEJA(J1,J2) +# sage: J.factors() +# (Euclidean Jordan algebra of dimension 2 over Rational Field, +# Euclidean Jordan algebra of dimension 3 over Rational Field) + +# """ +# return self._factors + +# def projections(self): +# r""" +# Return a pair of projections onto this algebra's factors. + +# SETUP:: + +# sage: from mjo.eja.eja_algebra import (JordanSpinEJA, +# ....: ComplexHermitianEJA, +# ....: DirectSumEJA) + +# EXAMPLES:: + +# sage: J1 = JordanSpinEJA(2) +# sage: J2 = ComplexHermitianEJA(2) +# sage: J = DirectSumEJA(J1,J2) +# sage: (pi_left, pi_right) = J.projections() +# sage: J.one().to_vector() +# (1, 0, 1, 0, 0, 1) +# sage: pi_left(J.one()).to_vector() +# (1, 0) +# sage: pi_right(J.one()).to_vector() +# (1, 0, 0, 1) + +# """ +# (J1,J2) = self.factors() +# m = J1.dimension() +# n = J2.dimension() +# V_basis = self.vector_space().basis() +# # Need to specify the dimensions explicitly so that we don't +# # wind up with a zero-by-zero matrix when we want e.g. a +# # zero-by-two matrix (important for composing things). +# P1 = matrix(self.base_ring(), m, m+n, V_basis[:m]) +# P2 = matrix(self.base_ring(), n, m+n, V_basis[m:]) +# pi_left = FiniteDimensionalEJAOperator(self,J1,P1) +# pi_right = FiniteDimensionalEJAOperator(self,J2,P2) +# return (pi_left, pi_right) + +# def inclusions(self): +# r""" +# Return the pair of inclusion maps from our factors into us. + +# SETUP:: + +# sage: from mjo.eja.eja_algebra import (random_eja, +# ....: JordanSpinEJA, +# ....: RealSymmetricEJA, +# ....: DirectSumEJA) + +# EXAMPLES:: + +# sage: J1 = JordanSpinEJA(3) +# sage: J2 = RealSymmetricEJA(2) +# sage: J = DirectSumEJA(J1,J2) +# sage: (iota_left, iota_right) = J.inclusions() +# sage: iota_left(J1.zero()) == J.zero() +# True +# sage: iota_right(J2.zero()) == J.zero() +# True +# sage: J1.one().to_vector() +# (1, 0, 0) +# sage: iota_left(J1.one()).to_vector() +# (1, 0, 0, 0, 0, 0) +# sage: J2.one().to_vector() +# (1, 0, 1) +# sage: iota_right(J2.one()).to_vector() +# (0, 0, 0, 1, 0, 1) +# sage: J.one().to_vector() +# (1, 0, 0, 1, 0, 1) + +# TESTS: + +# Composing a projection with the corresponding inclusion should +# produce the identity map, and mismatching them should produce +# the zero map:: + +# sage: set_random_seed() +# sage: J1 = random_eja() +# sage: J2 = random_eja() +# sage: J = DirectSumEJA(J1,J2) +# sage: (iota_left, iota_right) = J.inclusions() +# sage: (pi_left, pi_right) = J.projections() +# sage: pi_left*iota_left == J1.one().operator() +# True +# sage: pi_right*iota_right == J2.one().operator() +# True +# sage: (pi_left*iota_right).is_zero() +# True +# sage: (pi_right*iota_left).is_zero() +# True + +# """ +# (J1,J2) = self.factors() +# m = J1.dimension() +# n = J2.dimension() +# V_basis = self.vector_space().basis() +# # Need to specify the dimensions explicitly so that we don't +# # wind up with a zero-by-zero matrix when we want e.g. a +# # two-by-zero matrix (important for composing things). +# I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m]) +# I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:]) +# iota_left = FiniteDimensionalEJAOperator(J1,self,I1) +# iota_right = FiniteDimensionalEJAOperator(J2,self,I2) +# return (iota_left, iota_right) + +# def inner_product(self, x, y): +# r""" +# The standard Cartesian inner-product. + +# We project ``x`` and ``y`` onto our factors, and add up the +# inner-products from the subalgebras. + +# SETUP:: + + +# sage: from mjo.eja.eja_algebra import (HadamardEJA, +# ....: QuaternionHermitianEJA, +# ....: DirectSumEJA) + +# EXAMPLE:: + +# sage: J1 = HadamardEJA(3,field=QQ) +# sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False) +# sage: J = DirectSumEJA(J1,J2) +# sage: x1 = J1.one() +# sage: x2 = x1 +# sage: y1 = J2.one() +# sage: y2 = y1 +# sage: x1.inner_product(x2) +# 3 +# sage: y1.inner_product(y2) +# 2 +# sage: J.one().inner_product(J.one()) +# 5 + +# """ +# (pi_left, pi_right) = self.projections() +# x1 = pi_left(x) +# x2 = pi_right(x) +# y1 = pi_left(y) +# y2 = pi_right(y) + +# return (x1.inner_product(y1) + x2.inner_product(y2)) + + + +random_eja = ConcreteEJA.random_instance