X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=081abd303c32d35c9376ad99ecf2732a14ea3e9b;hb=95ae8e7b0ddca840da9631603a2f37cca888468b;hp=bab1002ae7d0e2bc1e0fe1332f28871f55efadf2;hpb=707c1ace788819d3d0542e61dab0134eabea2159;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index bab1002..081abd3 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -29,14 +29,168 @@ 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 + + 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. @@ -61,188 +215,130 @@ 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] - * 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. + 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 ( - ....: 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,QQbar) - Traceback (most recent call last): - ... - ValueError: scalar field is not real + 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): """ @@ -291,6 +387,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x = J.random_element() sage: J(x.to_vector().column()) == x True + """ msg = "not an element of this algebra" if elt == 0: @@ -304,6 +401,12 @@ 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) @@ -313,8 +416,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # 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. + # + # 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() ) + W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()), + check=False) try: coords = W.coordinate_vector(_mat2vec(elt)) @@ -344,74 +451,6 @@ 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, 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): @@ -490,7 +529,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = HadamardEJA(2) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X1, X2... - sage: J = RealSymmetricEJA(3,QQ,orthonormalize=False) + sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... @@ -607,20 +646,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ n = self.dimension() - M = [ [ self.zero() for j in range(n) ] - for i in range(n) ] - for i in range(n): - for j in range(i+1): - M[i][j] = self._multiplication_table[i][j] - M[j][i] = M[i][j] + # Prepend the header row. + M = [["*"] + list(self.gens())] - for i in range(n): - # Prepend the left "header" column entry Can't do this in - # the loop because it messes up the symmetry. - M[i] = [self.monomial(i)] + M[i] + # 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) ] - # Prepend the header row. - M = [["*"] + list(self.gens())] + M return table(M, header_row=True, header_column=True, frame=True) @@ -679,11 +713,7 @@ 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): @@ -701,8 +731,6 @@ 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() @@ -896,14 +924,14 @@ 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 @@ -913,9 +941,9 @@ 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: @@ -1130,208 +1158,66 @@ 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. - SETUP:: - - sage: from mjo.eja.eja_algebra import BilinearFormEJA - - EXAMPLES: - - The supplied basis is orthonormalized by default:: - - 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, - field, - basis, - jordan_product, - inner_product, - orthonormalize=True, - prefix='e', - category=None, - check_field=True, - check_axioms=True): - - 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) - - # If we were asked to orthonormalize, and if the orthonormal - # basis is different from the given one, then we also want to - # compute multiplication and inner-product tables for the - # deorthonormalized basis. These can be used later to - # construct a deorthonormalized copy of this algebra over QQ - # in which several operations are much faster. - self._deortho_multiplication_table = None - self._deortho_inner_product_table = None - - if orthonormalize: - # Compute the deorthonormalized tables before we orthonormalize - # the given basis. - W = V.span_of_basis( vector_basis ) - - if check_axioms: - # If the superclass constructor is going to verify the - # symmetry of this table, it has better at least be - # square... - self._deortho_multiplication_table = [ [0 for j in range(n)] - for i in range(n) ] - self._deortho_inner_product_table = [ [0 for j in range(n)] - for i in range(n) ] - else: - self._deortho_multiplication_table = [ [0 for j in range(i+1)] - for i in range(n) ] - self._deortho_inner_product_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): - # 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) - - elt = W.coordinate_vector(elt) - self._deortho_multiplication_table[i][j] = elt - self._deortho_inner_product_table[i][j] = ip - if check_axioms: - # The tables are square if we're verifying that they - # are commutative. - self._deortho_multiplication_table[j][i] = elt - self._deortho_inner_product_table[j][i] = ip - - if self._deortho_multiplication_table is not None: - self._deortho_multiplication_table = tuple(map(tuple, self._deortho_multiplication_table)) - if self._deortho_inner_product_table is not None: - self._deortho_inner_product_table = tuple(map(tuple, self._deortho_inner_product_table)) - - # 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) - - W = V.span_of_basis( vector_basis ) - - # Normalize the "matrix" basis, too! - basis = vector_basis - - if basis_is_matrices: - basis = tuple( map(_vec2mat,basis) ) - - W = V.span_of_basis( vector_basis ) - - # 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 ) - 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) + SETUP:: - elt = jordan_product(q_i, q_j) - ip = inner_product(q_i, q_j) + sage: from mjo.eja.eja_algebra import BilinearFormEJA - 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 ) + EXAMPLES: - super().__init__(field, - mult_table, - ip_table, - prefix, - category, - basis, # matrix basis - check_field, - check_axioms) + The supplied basis is orthonormalized by default:: + + 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, + 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): @@ -1357,21 +1243,18 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr Algebraic Real Field """ - if self.base_ring() is QQ: + 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. - superclass = super(RationalBasisEuclideanJordanAlgebra, self) - return superclass._charpoly_coefficients() + # 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. - J = FiniteDimensionalEuclideanJordanAlgebra(QQ, - self._deortho_multiplication_table, - self._deortho_inner_product_table) - - # Change back from QQ to our real base ring + # Then, change back from QQ to our real base ring a = ( a_i.change_ring(self.base_ring()) - for a_i in J._charpoly_coefficients() ) + for a_i in self._rational_algebra._charpoly_coefficients() ) # Now convert the coordinate variables back to the # deorthonormalized ones. @@ -1383,7 +1266,7 @@ 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. @@ -1394,7 +1277,7 @@ class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ConcreteEJA TESTS: @@ -1402,7 +1285,7 @@ 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 @@ -1413,7 +1296,7 @@ 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 @@ -1445,15 +1328,30 @@ 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 real_embed(M): + def dimension_over_reals(): + r""" + The dimension of this matrix's base ring over the reals. + + The reals are dimension one over themselves, obviously; that's + just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi` + have dimension two. Finally, the quaternions have dimension + four over the reals. + + This is used to determine the size of the matrix returned from + :meth:`real_embed`, among other things. + """ + raise NotImplementedError + + @classmethod + def real_embed(cls,M): """ Embed the matrix ``M`` into a space of real matrices. @@ -1466,15 +1364,21 @@ class MatrixEuclideanJordanAlgebra: real_embed(M*N) = real_embed(M)*real_embed(N) """ - raise NotImplementedError + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + return M - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of :meth:`real_embed`. """ - raise NotImplementedError + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero(): + raise ValueError("the matrix 'M' must be a real embedding") + return M @staticmethod def jordan_product(X,Y): @@ -1482,6 +1386,61 @@ class MatrixEuclideanJordanAlgebra: @classmethod def trace_inner_product(cls,X,Y): + r""" + Compute the trace inner-product of two real-embeddings. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA) + + EXAMPLES:: + + This gives the same answer as it would if we computed the trace + from the unembedded (original) matrices:: + + 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 + + :: + + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace().real() + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True + + :: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = J.real_unembed(Xe) + sage: Y = J.real_unembed(Ye) + sage: expected = (X*Y).trace().coefficient_tuple()[0] + sage: actual = J.trace_inner_product(Xe,Ye) + sage: actual == expected + True + + """ Xu = cls.real_unembed(X) Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() @@ -1496,26 +1455,13 @@ class MatrixEuclideanJordanAlgebra: 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(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 @@ -1538,9 +1484,9 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, RDF) + sage: RealSymmetricEJA(2, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field - sage: RealSymmetricEJA(2, RR) + sage: RealSymmetricEJA(2, field=RR) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1581,7 +1527,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, """ @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. @@ -1593,7 +1539,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, 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 @@ -1603,13 +1549,13 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, 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 @@ -1617,27 +1563,39 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, return 4 # Dimension 10 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + return cls(n, **kwargs) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, - basis, + 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) - self.one.set_cache(self(matrix.identity(field,n))) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) -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 + @@ -1645,8 +1603,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: ComplexMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import ComplexMatrixEJA EXAMPLES:: @@ -1656,7 +1613,7 @@ 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] [-----+-----] @@ -1672,16 +1629,15 @@ 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(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() @@ -1695,15 +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:: @@ -1711,7 +1666,7 @@ 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] @@ -1722,36 +1677,42 @@ 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(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() + + # 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]: @@ -1759,42 +1720,10 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): z = submat[0,0] + submat[0,1]*i elements.append(z) - return matrix(F, n/2, elements) - - - @classmethod - def trace_inner_product(cls,X,Y): - """ - Compute a matrix 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`:: + return matrix(F, n/d, elements) - 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 = ComplexHermitianEJA.real_unembed(Xe) - sage: Y = ComplexHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().real() - sage: actual = ComplexHermitianEJA.trace_inner_product(Xe,Ye) - sage: actual == expected - True - """ - return RealMatrixEuclideanJordanAlgebra.trace_inner_product(X,Y)/2 - - -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, @@ -1809,9 +1738,9 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, RDF) + sage: ComplexHermitianEJA(2, field=RDF) Euclidean Jordan algebra of dimension 4 over Real Double Field - sage: ComplexHermitianEJA(2, RR) + sage: ComplexHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -1853,7 +1782,7 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, """ @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. @@ -1872,15 +1801,16 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, 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: # @@ -1906,31 +1836,41 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, return tuple( s.change_ring(field) for s in S ) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA, self).__init__(field, - basis, - self.jordan_product, - self.trace_inner_product, - **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(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) - # TODO: pre-cache the identity! + 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, field=AA, **kwargs): + 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, field, **kwargs) + 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 @@ -1940,8 +1880,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import \ - ....: QuaternionMatrixEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA EXAMPLES:: @@ -1949,7 +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] @@ -1962,17 +1901,16 @@ 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(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') i = F.gen() @@ -1986,7 +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 @@ -1995,15 +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:: @@ -2011,7 +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: @@ -2021,16 +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. @@ -2042,10 +1977,10 @@ 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(): @@ -2056,42 +1991,10 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): z += submat[0,1].imag()*k elements.append(z) - return matrix(Q, n/4, elements) - - - @classmethod - def trace_inner_product(cls,X,Y): - """ - Compute a matrix 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.to_matrix() - sage: Ye = y.to_matrix() - sage: X = QuaternionHermitianEJA.real_unembed(Xe) - sage: Y = QuaternionHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().coefficient_tuple()[0] - sage: actual = QuaternionHermitianEJA.trace_inner_product(Xe,Ye) - sage: actual == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.trace_inner_product(X,Y)/4 + 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 @@ -2106,9 +2009,9 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, RDF) + sage: QuaternionHermitianEJA(2, field=RDF) Euclidean Jordan algebra of dimension 6 over Real Double Field - sage: QuaternionHermitianEJA(2, RR) + sage: QuaternionHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -2149,7 +2052,7 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, """ @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. @@ -2167,11 +2070,12 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, 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() @@ -2204,15 +2108,22 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return tuple( s.change_ring(field) for s in S ) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA, self).__init__(field, - basis, + 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) - # TODO: cache one()! + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) + @staticmethod def _max_random_instance_size(): @@ -2222,15 +2133,15 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return 2 # Dimension 6 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + 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. @@ -2270,20 +2181,29 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): (r0, r1, r2) """ - def __init__(self, n, field=AA, **kwargs): - V = VectorSpace(field, n) - basis = V.basis() - - def jordan_product(x,y): - return V([ xi*yi for (xi,yi) in zip(x,y) ]) - def inner_product(x,y): - return x.inner_product(y) - - super(HadamardEJA, self).__init__(field, - basis, - jordan_product, - inner_product, - **kwargs) + 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: @@ -2299,15 +2219,15 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + 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 = @@ -2388,27 +2308,30 @@ class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): sage: actual == expected True """ - def __init__(self, B, field=AA, **kwargs): + def __init__(self, B, **kwargs): if not B.is_positive_definite(): raise ValueError("bilinear form is not positive-definite") - n = B.nrows() - V = VectorSpace(field, n) - def inner_product(x,y): - return (B*x).inner_product(y) + return (y.T*B*x)[0,0] def jordan_product(x,y): - x0 = x[0] - xbar = x[1:] - y0 = y[0] - ybar = y[1:] - z0 = inner_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 V([z0] + zbar.list()) + return P([0] + zbar.list()) - super(BilinearFormEJA, self).__init__(field, - V.basis(), + # 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) @@ -2431,28 +2354,28 @@ class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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(field, n) - return cls(B, field, **kwargs) + B = matrix.identity(ZZ, n) + return cls(B, **kwargs) - B11 = matrix.identity(field,1) - M = matrix.random(field, n-1) - I = matrix.identity(field, n-1) - alpha = field.zero() + 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 = field.random_element().abs() + alpha = ZZ.random_element().abs() B22 = M.transpose()*M + alpha*I from sage.matrix.special import block_matrix B = block_matrix(2,2, [ [B11, ZZ(0) ], [ZZ(0), B22 ] ]) - return cls(B, field, **kwargs) + return cls(B, **kwargs) class JordanSpinEJA(BilinearFormEJA): @@ -2505,11 +2428,18 @@ class JordanSpinEJA(BilinearFormEJA): True """ - def __init__(self, n, field=AA, **kwargs): - # This is a special case of the BilinearFormEJA with the identity - # matrix as its bilinear form. - B = matrix.identity(field, n) - super(JordanSpinEJA, self).__init__(B, field, **kwargs) + 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(): @@ -2519,17 +2449,17 @@ class JordanSpinEJA(BilinearFormEJA): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. Needed here to override the implementation for ``BilinearFormEJA``. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) -class TrivialEJA(ConcreteEuclideanJordanAlgebra): +class TrivialEJA(ConcreteEJA): """ The trivial Euclidean Jordan algebra consisting of only a zero element. @@ -2558,12 +2488,16 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): 0 """ - def __init__(self, field=AA, **kwargs): + def __init__(self, **kwargs): jordan_product = lambda x,y: x - inner_product = lambda x,y: field(0) + inner_product = lambda x,y: 0 basis = () - super(TrivialEJA, self).__init__(field, - 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) @@ -2573,249 +2507,249 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): self.one.set_cache( self.zero() ) @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): # We don't take a "size" argument so the superclass method is # inappropriate for us. - return cls(field, **kwargs) - -class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): - r""" - The external (orthogonal) direct sum of two other Euclidean Jordan - algebras. Essentially the Cartesian product of its two factors. - Every Euclidean Jordan algebra decomposes into an orthogonal - direct sum of simple Euclidean Jordan algebras, so no generality - is lost by providing only this construction. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (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(AA) - sage: J2 = random_eja(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,QQ) - sage: J2 = JordanSpinEJA(3,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 = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J1,P1) - pi_right = FiniteDimensionalEuclideanJordanAlgebraOperator(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 = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,self,I1) - iota_right = FiniteDimensionalEuclideanJordanAlgebraOperator(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,QQ) - sage: J2 = QuaternionHermitianEJA(2,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 = ConcreteEuclideanJordanAlgebra.random_instance + 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