X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=1250fbda7981aba5474af0a3d2615c969bc9d1e1;hb=4c8f9aac69d1cb4097b60b10e5b198b6372ec55e;hp=3b13ce10bbf864ceb667f3ba4fc1f8b520327292;hpb=795ac83cd78143e36d47fa267fe6ddf1ca8da111;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 3b13ce1..1250fbd 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -29,14 +29,204 @@ from sage.modules.free_module import FreeModule, VectorSpace from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, PolynomialRing, QuadraticField) -from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement -from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator +from mjo.eja.eja_element import FiniteDimensionalEJAElement +from mjo.eja.eja_operator import FiniteDimensionalEJAOperator from mjo.eja.eja_utils import _mat2vec -class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): +class FiniteDimensionalEJA(CombinatorialFreeModule): r""" - The lowest-level class for representing a Euclidean Jordan algebra. + 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, + 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 + + from sage.structure.element import is_Matrix + basis_is_matrices = False + + degree = 0 + if n > 0: + if is_Matrix(basis[0]): + if basis[0].is_square(): + # TODO: this ugly is_square() hack works around the problem + # of passing to_matrix()ed vectors in as the basis from a + # subalgebra. They aren't REALLY matrices, at least not of + # the type that we assume here... Ugh. + basis_is_matrices = True + from mjo.eja.eja_utils import _vec2mat + vector_basis = tuple( map(_mat2vec,basis) ) + degree = basis[0].nrows()**2 + else: + # convert from column matrices to vectors, yuck + basis = tuple( map(_mat2vec,basis) ) + vector_basis = basis + degree = basis[0].degree() + else: + degree = basis[0].degree() + + # Build an ambient space that fits... + V = VectorSpace(field, degree) + + # We overwrite the name "vector_basis" in a second, but never modify it + # in place, to this effectively makes a copy of it. + deortho_vector_basis = vector_basis + self._deortho_matrix = None + + if orthonormalize: + from mjo.eja.eja_utils import gram_schmidt + if basis_is_matrices: + vector_ip = lambda x,y: inner_product(_vec2mat(x), _vec2mat(y)) + vector_basis = gram_schmidt(vector_basis, vector_ip) + else: + vector_basis = gram_schmidt(vector_basis, inner_product) + + # Normalize the "matrix" basis, too! + basis = vector_basis + + if basis_is_matrices: + basis = tuple( map(_vec2mat,basis) ) + + # Save the matrix "basis" for later... this is the last time we'll + # reference it in this constructor. + if basis_is_matrices: + self._matrix_basis = basis + else: + MS = MatrixSpace(self.base_ring(), degree, 1) + self._matrix_basis = tuple( MS(b) for b in basis ) + + # Now create the vector space for the algebra... + 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) ] + + print("vector_basis:") + print(vector_basis) + # 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 = vector_basis[i] + q_j = vector_basis[j] + + if basis_is_matrices: + q_i = _vec2mat(q_i) + q_j = _vec2mat(q_j) + + elt = jordan_product(q_i, q_j) + ip = inner_product(q_i, q_j) + + if basis_is_matrices: + # do another mat2vec because the multiplication + # table is in terms of vectors + elt = _mat2vec(elt) + + # TODO: the jordan product turns things back into + # matrices here even if they're supposed to be + # vectors. ugh. Can we get rid of vectors all together + # please? + elt = W.coordinate_vector(elt) + 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. @@ -61,190 +251,130 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ return None - def __init__(self, - field, - multiplication_table, - inner_product_table, - prefix='e', - category=None, - matrix_basis=None, - check_field=True, - check_axioms=True): - """ - INPUT: - * field -- the scalar field for this algebra (must be real) + 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. - * multiplication_table -- the multiplication table for this - algebra's implicit basis. Only the lower-triangular portion - of the table is used, since the multiplication is assumed - to be commutative. + 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 ( - ....: FiniteDimensionalEuclideanJordanAlgebra, - ....: 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 - An error is raised if the Jordan product is not commutative:: - - sage: JP = ((1,2),(0,0)) - sage: IP = ((1,0),(0,1)) - sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,JP,IP) - Traceback (most recent call last): - ... - ValueError: Jordan product is not commutative - - An error is raised if the inner-product is not commutative:: + TESTS: - sage: JP = ((1,0),(0,1)) - sage: IP = ((1,2),(0,0)) - sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,JP,IP) - Traceback (most recent call last): - ... - ValueError: inner-product is not commutative + Ensure that this is the usual inner product for the algebras + over `R^n`:: - TESTS: + 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 - The ``field`` we're given must be real with ``check_field=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: JordanSpinEJA(2, field=QQbar) - Traceback (most recent call last): - ... - ValueError: scalar field is not real - sage: JordanSpinEJA(2, field=QQbar, check_field=False) - Euclidean Jordan algebra of dimension 2 over Algebraic Field + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) + True + """ + B = self._inner_product_matrix + return (B*x.to_vector()).inner_product(y.to_vector()) - The multiplication table must be square with ``check_axioms=True``:: - sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()),((1,),)) - Traceback (most recent call last): - ... - ValueError: multiplication table is not square + def _is_commutative(self): + r""" + Whether or not this algebra's multiplication table is commutative. - The multiplication and inner-product tables must be the same - size (and in particular, the inner-product table must also be - square) with ``check_axioms=True``:: + 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()) ) - sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((1,),),(())) - Traceback (most recent call last): - ... - ValueError: multiplication and inner-product tables are - different sizes - sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((1,),),((1,2),)) - Traceback (most recent call last): - ... - ValueError: multiplication and inner-product tables are - different sizes + 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. """ - 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") + 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)`. - # The multiplication and inner-product tables should be square - # if the user wants us to verify them. And we verify them as - # soon as possible, because we want to exploit their symmetry. - n = len(multiplication_table) - if check_axioms: - if not all( len(l) == n for l in multiplication_table ): - raise ValueError("multiplication table is not square") - - # If the multiplication table is square, we can check if - # the inner-product table is square by comparing it to the - # multiplication table's dimensions. - msg = "multiplication and inner-product tables are different sizes" - if not len(inner_product_table) == n: - raise ValueError(msg) - - if not all( len(l) == n for l in inner_product_table ): - raise ValueError(msg) - - # Check commutativity of the Jordan product (symmetry of - # the multiplication table) and the commutativity of the - # inner-product (symmetry of the inner-product table) - # first if we're going to check them at all.. This has to - # be done before we define product_on_basis(), because - # that method assumes that self._multiplication_table is - # symmetric. And it has to be done before we build - # self._inner_product_matrix, because the process used to - # construct it assumes symmetry as well. - if not all( multiplication_table[j][i] - == multiplication_table[i][j] - for i in range(n) - for j in range(i+1) ): - raise ValueError("Jordan product is not commutative") + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. + """ - if not all( inner_product_table[j][i] - == inner_product_table[i][j] - for i in range(n) - for j in range(i+1) ): - raise ValueError("inner-product is not commutative") + # 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 - self._matrix_basis = matrix_basis - - if category is None: - category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() - - fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) - fda.__init__(field, - range(n), - 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. - # - # Note: we take advantage of symmetry here, and only store - # the lower-triangular portion of the table. - self._multiplication_table = [ [ self.vector_space().zero() - for j in range(i+1) ] - for i in range(n) ] + 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) - for i in range(n): - for j in range(i+1): - elt = self.from_vector(multiplication_table[i][j]) - self._multiplication_table[i][j] = elt - - self._multiplication_table = tuple(map(tuple, self._multiplication_table)) - - # Save our inner product as a matrix, since the efficiency of - # matrix multiplication will usually outweigh the fact that we - # have to store a redundant upper- or lower-triangular part. - # Pre-cache the fact that these are Hermitian (real symmetric, - # in fact) in case some e.g. matrix multiplication routine can - # take advantage of it. - ip_matrix_constructor = lambda i,j: inner_product_table[i][j] if j <= i else inner_product_table[j][i] - self._inner_product_matrix = matrix(field, n, ip_matrix_constructor) - self._inner_product_matrix._cache = {'hermitian': True} - self._inner_product_matrix.set_immutable() + if self.base_ring().is_exact(): + if diff != 0: + return False + else: + if diff.abs() > epsilon: + return False - 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") + return True def _element_constructor_(self, elt): """ @@ -307,6 +437,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # that the integer 3 belongs to the space of 2-by-2 matrices. raise ValueError(msg) + 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) @@ -351,74 +487,6 @@ 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): - # 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 _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 @cached_method def characteristic_polynomial_of(self): @@ -681,11 +749,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): [0], [1] ) """ - if self._matrix_basis is None: - M = self.matrix_space() - return tuple( M(b.to_vector()) for b in self.basis() ) - else: - return self._matrix_basis + return self._matrix_basis def matrix_space(self): @@ -703,8 +767,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ if self.is_trivial(): return MatrixSpace(self.base_ring(), 0) - elif self._matrix_basis is None or len(self._matrix_basis) == 0: - return MatrixSpace(self.base_ring(), self.dimension(), 1) else: return self._matrix_basis[0].matrix_space() @@ -898,14 +960,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): if not c.is_idempotent(): raise ValueError("element is not idempotent: %s" % c) - from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra + 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 @@ -915,9 +977,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): J5 = eigspace else: gens = tuple( self.from_vector(b) for b in eigspace.basis() ) - subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, - gens, - check_axioms=False) + subalg = FiniteDimensionalEJASubalgebra(self, + gens, + check_axioms=False) if eigval == 0: J0 = subalg elif eigval == 1: @@ -1132,9 +1194,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.zero().to_vector().parent().ambient_vector_space() - Element = FiniteDimensionalEuclideanJordanAlgebraElement + Element = FiniteDimensionalEJAElement -class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): +class RationalBasisEJA(FiniteDimensionalEJA): r""" New class for algebras whose supplied basis elements have all rational entries. @@ -1162,10 +1224,9 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr inner_product, field=AA, orthonormalize=True, - prefix='e', - category=None, check_field=True, - check_axioms=True): + check_axioms=True, + **kwargs): if check_field: # Abuse the check_field parameter to check that the entries of @@ -1173,32 +1234,6 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr if not all( all(b_i in QQ for b_i in b.list()) for b in basis ): raise TypeError("basis not rational") - # Temporary(?) hack to ensure that the matrix and vector bases - # are over the same ring. - basis = tuple( b.change_ring(field) for b in basis ) - - n = len(basis) - vector_basis = basis - - from sage.structure.element import is_Matrix - basis_is_matrices = False - - degree = 0 - if n > 0: - if is_Matrix(basis[0]): - basis_is_matrices = True - from mjo.eja.eja_utils import _vec2mat - vector_basis = tuple( map(_mat2vec,basis) ) - degree = basis[0].nrows()**2 - else: - degree = basis[0].degree() - - V = VectorSpace(field, degree) - - # Save a copy of an algebra with the original, rational basis - # and over QQ where computations are fast. - self._rational_algebra = None - if field is not QQ: # There's no point in constructing the extra algebra if this # one is already rational. @@ -1206,128 +1241,23 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr # 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 = RationalBasisEuclideanJordanAlgebra( + self._rational_algebra = FiniteDimensionalEJA( basis, jordan_product, inner_product, field=QQ, orthonormalize=False, - prefix=prefix, - category=category, check_field=False, - check_axioms=False) - - if orthonormalize: - # Compute the deorthonormalized tables before we orthonormalize - # the given basis. The "check" parameter here guarantees that - # the basis is linearly-independent. - W = V.span_of_basis( vector_basis, check=check_axioms) - - # 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): - # given basis w.r.t. ambient coords - q_i = vector_basis[i] - q_j = vector_basis[j] - - if basis_is_matrices: - q_i = _vec2mat(q_i) - q_j = _vec2mat(q_j) - - elt = jordan_product(q_i, q_j) - ip = inner_product(q_i, q_j) - - if basis_is_matrices: - # do another mat2vec because the multiplication - # table is in terms of vectors - elt = _mat2vec(elt) - - # We overwrite the name "vector_basis" in a second, but never modify it - # in place, to this effectively makes a copy of it. - deortho_vector_basis = vector_basis - self._deortho_matrix = None - - if orthonormalize: - from mjo.eja.eja_utils import gram_schmidt - if basis_is_matrices: - vector_ip = lambda x,y: inner_product(_vec2mat(x), _vec2mat(y)) - vector_basis = gram_schmidt(vector_basis, vector_ip) - else: - vector_basis = gram_schmidt(vector_basis, inner_product) - - # Normalize the "matrix" basis, too! - basis = vector_basis + check_axioms=False, + **kwargs) - if basis_is_matrices: - basis = tuple( map(_vec2mat,basis) ) - - W = V.span_of_basis( vector_basis, check=check_axioms) - - # 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 ) - - # If the superclass constructor is going to verify the - # symmetry of this table, it has better at least be - # square... - if check_axioms: - mult_table = [ [0 for j in range(n)] for i in range(n) ] - ip_table = [ [0 for j in range(n)] for i in range(n) ] - else: - mult_table = [ [0 for j in range(i+1)] for i in range(n) ] - ip_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 = vector_basis[i] - q_j = vector_basis[j] - - if basis_is_matrices: - q_i = _vec2mat(q_i) - q_j = _vec2mat(q_j) - - elt = jordan_product(q_i, q_j) - ip = inner_product(q_i, q_j) - - if basis_is_matrices: - # do another mat2vec because the multiplication - # table is in terms of vectors - elt = _mat2vec(elt) - - elt = W.coordinate_vector(elt) - mult_table[i][j] = elt - ip_table[i][j] = ip - if check_axioms: - # The tables are square if we're verifying that they - # are commutative. - mult_table[j][i] = elt - ip_table[j][i] = ip - - if basis_is_matrices: - for m in basis: - m.set_immutable() - else: - basis = tuple( x.column() for x in basis ) - - super().__init__(field, - mult_table, - ip_table, - prefix, - category, - basis, # matrix basis - check_field, - check_axioms) + super().__init__(basis, + jordan_product, + inner_product, + field=field, + check_field=check_field, + check_axioms=check_axioms, + **kwargs) @cached_method def _charpoly_coefficients(self): @@ -1358,8 +1288,7 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr # 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. - superclass = super(RationalBasisEuclideanJordanAlgebra, self) - return superclass._charpoly_coefficients() + 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. @@ -1377,7 +1306,7 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr 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 ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): +class ConcreteEJA(RationalBasisEJA): r""" A class for the Euclidean Jordan algebras that we know by name. @@ -1388,7 +1317,7 @@ class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ConcreteEJA TESTS: @@ -1396,7 +1325,7 @@ class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): product, unless we specify otherwise:: sage: set_random_seed() - sage: J = ConcreteEuclideanJordanAlgebra.random_instance() + sage: J = ConcreteEJA.random_instance() sage: all( b.norm() == 1 for b in J.gens() ) True @@ -1407,7 +1336,7 @@ class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): EJA the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() - sage: J = ConcreteEuclideanJordanAlgebra.random_instance() + sage: J = ConcreteEJA.random_instance() sage: x = J.random_element() sage: x.operator().is_self_adjoint() True @@ -1439,13 +1368,13 @@ class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): from sage.misc.prandom import choice eja_class = choice(cls.__subclasses__()) - # These all bubble up to the RationalBasisEuclideanJordanAlgebra - # superclass constructor, so any (kw)args valid there are also - # valid here. + # 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: +class MatrixEJA: @staticmethod def dimension_over_reals(): r""" @@ -1566,14 +1495,13 @@ class MatrixEuclideanJordanAlgebra: return tr.coefficient_tuple()[0] -class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): +class RealMatrixEJA(MatrixEJA): @staticmethod def dimension_over_reals(): return 1 -class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, - RealMatrixEuclideanJordanAlgebra): +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 @@ -1693,15 +1621,15 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, **kwargs) # TODO: this could be factored out somehow, but is left here - # because the MatrixEuclideanJordanAlgebra is not presently - # a subclass of the FDEJA class that defines rank() and one(). + # 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)) -class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): +class ComplexMatrixEJA(MatrixEJA): @staticmethod def dimension_over_reals(): return 2 @@ -1715,8 +1643,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: @@ -1726,7 +1653,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): 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] [-----+-----] @@ -1742,14 +1669,14 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): 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(ComplexMatrixEuclideanJordanAlgebra,cls).real_embed(M) + super(ComplexMatrixEJA,cls).real_embed(M) n = M.nrows() # We don't need any adjoined elements... @@ -1771,8 +1698,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: @@ -1780,7 +1706,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [-2, 1, -4, 3], ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) - sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A) + sage: ComplexMatrixEJA.real_unembed(A) [ 2*I + 1 4*I + 3] [ 10*I + 9 12*I + 11] @@ -1791,12 +1717,12 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: F = QuadraticField(-1, 'I') sage: M = random_matrix(F, 3) - sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) - sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + sage: Me = ComplexMatrixEJA.real_embed(M) + sage: ComplexMatrixEJA.real_unembed(Me) == M True """ - super(ComplexMatrixEuclideanJordanAlgebra,cls).real_unembed(M) + super(ComplexMatrixEJA,cls).real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() @@ -1837,8 +1763,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return matrix(F, n/d, elements) -class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, - ComplexMatrixEuclideanJordanAlgebra): +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, @@ -1961,8 +1886,8 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, self.trace_inner_product, **kwargs) # TODO: this could be factored out somehow, but is left here - # because the MatrixEuclideanJordanAlgebra is not presently - # a subclass of the FDEJA class that defines rank() and one(). + # 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)) @@ -1979,7 +1904,7 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, n = ZZ.random_element(cls._max_random_instance_size() + 1) return cls(n, **kwargs) -class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): +class QuaternionMatrixEJA(MatrixEJA): @staticmethod def dimension_over_reals(): return 4 @@ -1995,8 +1920,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -2004,7 +1928,7 @@ 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] @@ -2017,14 +1941,14 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): 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(QuaternionMatrixEuclideanJordanAlgebra,cls).real_embed(M) + super(QuaternionMatrixEJA,cls).real_embed(M) quaternions = M.base_ring() n = M.nrows() @@ -2040,7 +1964,7 @@ 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 @@ -2056,8 +1980,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -2065,7 +1988,7 @@ 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: @@ -2075,12 +1998,12 @@ 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(QuaternionMatrixEuclideanJordanAlgebra,cls).real_unembed(M) + super(QuaternionMatrixEJA,cls).real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() @@ -2096,7 +2019,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): elements = [] for l in range(n/d): for m in range(n/d): - submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed( + 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') @@ -2111,8 +2034,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return matrix(Q, n/d, elements) -class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, - QuaternionMatrixEuclideanJordanAlgebra): +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 @@ -2236,8 +2158,8 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, self.trace_inner_product, **kwargs) # TODO: this could be factored out somehow, but is left here - # because the MatrixEuclideanJordanAlgebra is not presently - # a subclass of the FDEJA class that defines rank() and one(). + # 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)) @@ -2259,7 +2181,7 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return cls(n, **kwargs) -class HadamardEJA(ConcreteEuclideanJordanAlgebra): +class HadamardEJA(ConcreteEJA): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. @@ -2344,7 +2266,7 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): return cls(n, **kwargs) -class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): +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 = @@ -2576,7 +2498,7 @@ class JordanSpinEJA(BilinearFormEJA): return cls(n, **kwargs) -class TrivialEJA(ConcreteEuclideanJordanAlgebra): +class TrivialEJA(ConcreteEJA): """ The trivial Euclidean Jordan algebra consisting of only a zero element. @@ -2629,7 +2551,7 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): # inappropriate for us. return cls(**kwargs) -class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): +class DirectSumEJA(ConcreteEJA): r""" The external (orthogonal) direct sum of two other Euclidean Jordan algebras. Essentially the Cartesian product of its two factors. @@ -2757,8 +2679,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): # 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 = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J1,P1) - pi_right = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J2,P2) + pi_left = FiniteDimensionalEJAOperator(self,J1,P1) + pi_right = FiniteDimensionalEJAOperator(self,J2,P2) return (pi_left, pi_right) def inclusions(self): @@ -2824,8 +2746,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): # 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 = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,self,I1) - iota_right = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,self,I2) + iota_left = FiniteDimensionalEJAOperator(J1,self,I1) + iota_right = FiniteDimensionalEJAOperator(J2,self,I2) return (iota_left, iota_right) def inner_product(self, x, y): @@ -2869,4 +2791,4 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): -random_eja = ConcreteEuclideanJordanAlgebra.random_instance +random_eja = ConcreteEJA.random_instance