"""
-Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
-specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
-are used in optimization, and have some additional nice methods beyond
-what can be supported in a general Jordan Algebra.
-
+Representations and constructions for Euclidean Jordan algebras.
+
+A Euclidean Jordan algebra is a Jordan algebra that has some
+additional properties:
+
+ 1. It is finite-dimensional.
+ 2. Its scalar field is the real numbers.
+ 3a. An inner product is defined on it, and...
+ 3b. That inner product is compatible with the Jordan product
+ in the sense that `<x*y,z> = <y,x*z>` for all elements
+ `x,y,z` in the algebra.
+
+Every Euclidean Jordan algebra is formally-real: for any two elements
+`x` and `y` in the algebra, `x^{2} + y^{2} = 0` implies that `x = y =
+0`. Conversely, every finite-dimensional formally-real Jordan algebra
+can be made into a Euclidean Jordan algebra with an appropriate choice
+of inner-product.
+
+Formally-real Jordan algebras were originally studied as a framework
+for quantum mechanics. Today, Euclidean Jordan algebras are crucial in
+symmetric cone optimization, since every symmetric cone arises as the
+cone of squares in some Euclidean Jordan algebra.
+
+It is known that every Euclidean Jordan algebra decomposes into an
+orthogonal direct sum (essentially, a Cartesian product) of simple
+algebras, and that moreover, up to Jordan-algebra isomorphism, there
+are only five families of simple algebras. We provide constructions
+for these simple algebras:
+
+ * :class:`BilinearFormEJA`
+ * :class:`RealSymmetricEJA`
+ * :class:`ComplexHermitianEJA`
+ * :class:`QuaternionHermitianEJA`
+
+Missing from this list is the algebra of three-by-three octononion
+Hermitian matrices, as there is (as of yet) no implementation of the
+octonions in SageMath. In addition to these, we provide two other
+example constructions,
+
+ * :class:`HadamardEJA`
+ * :class:`TrivialEJA`
+
+The Jordan spin algebra is a bilinear form algebra where the bilinear
+form is the identity. The Hadamard EJA is simply a Cartesian product
+of one-dimensional spin algebras. And last but not least, the trivial
+EJA is exactly what you think. Cartesian products of these are also
+supported using the usual ``cartesian_product()`` function; as a
+result, we support (up to isomorphism) all Euclidean Jordan algebras
+that don't involve octonions.
SETUP::
sage: random_eja()
Euclidean Jordan algebra of dimension...
-
"""
from itertools import repeat
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
-from sage.combinat.free_module import CombinatorialFreeModule
+from sage.categories.sets_cat import cartesian_product
+from sage.combinat.free_module import (CombinatorialFreeModule,
+ CombinatorialFreeModule_CartesianProduct)
from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
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_utils import _mat2vec
+from mjo.eja.eja_element import FiniteDimensionalEJAElement
+from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
+from mjo.eja.eja_utils import _all2list, _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 "matrix form," which
+ must be the same form as the arguments to ``jordan_product``
+ and ``inner_product``. In reality, "matrix form" can be either
+ vectors, matrices, or a Cartesian product (ordered tuple)
+ of vectors or matrices. All of these would ideally be vector
+ spaces in sage with no special-casing needed; but in reality
+ we turn vectors into column-matrices and Cartesian products
+ `(a,b)` into column matrices `(a,b)^{T}` after converting
+ `a` and `b` themselves.
+
+ - jordan_product -- function of two ``basis`` elements (in
+ matrix form) that returns their jordan product, also in matrix
+ form; this will be applied to ``basis`` to compute a
+ multiplication table for the algebra.
+
+ - inner_product -- function of two ``basis`` elements (in matrix
+ form) that returns their inner product. This will be applied
+ to ``basis`` to compute an inner-product table (basically a
+ matrix) for this algebra.
"""
+ Element = FiniteDimensionalEJAElement
+
+ def __init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=AA,
+ orthonormalize=True,
+ associative=False,
+ cartesian_product=False,
+ check_field=True,
+ check_axioms=True,
+ prefix='e'):
+
+ # Keep track of whether or not the matrix basis consists of
+ # tuples, since we need special cases for them damned near
+ # everywhere. This is INDEPENDENT of whether or not the
+ # algebra is a cartesian product, since a subalgebra of a
+ # cartesian product will have a basis of tuples, but will not
+ # in general itself be a cartesian product algebra.
+ self._matrix_basis_is_cartesian = False
+ n = len(basis)
+ if n > 0:
+ if hasattr(basis[0], 'cartesian_factors'):
+ self._matrix_basis_is_cartesian = True
+
+ 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.
+ if not cartesian_product:
+ # The field for a cartesian product algebra comes from one
+ # of its factors and is the same for all factors, so
+ # there's no need to "reapply" it on product algebras.
+ if self._matrix_basis_is_cartesian:
+ # OK since if n == 0, the basis does not consist of tuples.
+ P = basis[0].parent()
+ basis = tuple( P(tuple(b_i.change_ring(field) for b_i in b))
+ for b in basis )
+ else:
+ 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()
+ if cartesian_product:
+ category = category.CartesianProducts()
+
+ # Call the superclass constructor so that we can use its from_vector()
+ # method to build our multiplication table.
+ CombinatorialFreeModule.__init__(self,
+ field,
+ range(n),
+ prefix=prefix,
+ category=category,
+ bracket=False)
+
+ # Now comes all of the hard work. We'll be constructing an
+ # ambient vector space V that our (vectorized) basis lives in,
+ # as well as a subspace W of V spanned by those (vectorized)
+ # basis elements. The W-coordinates are the coefficients that
+ # we see in things like x = 1*e1 + 2*e2.
+ vector_basis = basis
+
+ degree = 0
+ if n > 0:
+ degree = len(_all2list(basis[0]))
+
+ # Build an ambient space that fits our matrix basis when
+ # written out as "long vectors."
+ V = VectorSpace(field, degree)
+
+ # The matrix that will hole the orthonormal -> unorthonormal
+ # coordinate transformation.
+ self._deortho_matrix = None
+
+ if orthonormalize:
+ # Save a copy of the un-orthonormalized basis for later.
+ # Convert it to ambient V (vector) coordinates while we're
+ # at it, because we'd have to do it later anyway.
+ deortho_vector_basis = tuple( V(_all2list(b)) for b in basis )
+
+ from mjo.eja.eja_utils import gram_schmidt
+ basis = tuple(gram_schmidt(basis, inner_product))
+
+ # Save the (possibly orthonormalized) matrix basis for
+ # later...
+ self._matrix_basis = basis
+
+ # Now create the vector space for the algebra, which will have
+ # its own set of non-ambient coordinates (in terms of the
+ # supplied basis).
+ vector_basis = tuple( V(_all2list(b)) for b in basis )
+ W = V.span_of_basis( vector_basis, check=check_axioms)
+
+ if orthonormalize:
+ # Now "W" is the vector space of our algebra coordinates. The
+ # variables "X1", "X2",... refer to the entries of vectors in
+ # W. Thus to convert back and forth between the orthonormal
+ # coordinates and the given ones, we need to stick the original
+ # basis in W.
+ U = V.span_of_basis( deortho_vector_basis, check=check_axioms)
+ self._deortho_matrix = matrix( U.coordinate_vector(q)
+ for q in vector_basis )
+
+
+ # Now we actually compute the multiplication and inner-product
+ # tables/matrices using the possibly-orthonormalized basis.
+ self._inner_product_matrix = matrix.identity(field, n)
+ self._multiplication_table = [ [0 for j in range(i+1)]
+ for i in range(n) ]
+
+ # Note: the Jordan and inner-products are defined in terms
+ # of the ambient basis. It's important that their arguments
+ # are in ambient coordinates as well.
+ for i in range(n):
+ for j in range(i+1):
+ # ortho basis w.r.t. ambient coords
+ q_i = basis[i]
+ q_j = basis[j]
+
+ # The jordan product returns a matrixy answer, so we
+ # have to convert it to the algebra coordinates.
+ elt = jordan_product(q_i, q_j)
+ elt = W.coordinate_vector(V(_all2list(elt)))
+ self._multiplication_table[i][j] = self.from_vector(elt)
+
+ if not orthonormalize:
+ # If we're orthonormalizing the basis with respect
+ # to an inner-product, then the inner-product
+ # matrix with respect to the resulting basis is
+ # just going to be the identity.
+ ip = inner_product(q_i, q_j)
+ self._inner_product_matrix[i,j] = ip
+ self._inner_product_matrix[j,i] = ip
+
+ self._inner_product_matrix._cache = {'hermitian': True}
+ self._inner_product_matrix.set_immutable()
+
+ if check_axioms:
+ if not self._is_jordanian():
+ raise ValueError("Jordan identity does not hold")
+ if not self._inner_product_is_associative():
+ raise ValueError("inner product is not associative")
+
+
def _coerce_map_from_base_ring(self):
"""
Disable the map from the base ring into the algebra.
"""
return None
- def __init__(self,
- field,
- multiplication_table,
- inner_product_table,
- prefix='e',
- category=None,
- matrix_basis=None,
- check_field=True,
- check_axioms=True):
+
+ def product_on_basis(self, i, j):
+ r"""
+ Returns the Jordan product of the `i` and `j`th basis elements.
+
+ This completely defines the Jordan product on the algebra, and
+ is used direclty by our superclass machinery to implement
+ :meth:`product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: n = J.dimension()
+ sage: ei = J.zero()
+ sage: ej = J.zero()
+ sage: ei_ej = J.zero()*J.zero()
+ sage: if n > 0:
+ ....: i = ZZ.random_element(n)
+ ....: j = ZZ.random_element(n)
+ ....: ei = J.gens()[i]
+ ....: ej = J.gens()[j]
+ ....: ei_ej = J.product_on_basis(i,j)
+ sage: ei*ej == ei_ej
+ True
+
"""
- INPUT:
+ # 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]
- * field -- the scalar field for this algebra (must be real)
+ 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,z = J.random_elements(3)
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
+ TESTS:
+
+ Ensure that this is the usual inner product for the algebras
+ over `R^n`::
+
+ sage: set_random_seed()
+ sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: x*y == y*x
+ sage: actual = x.inner_product(y)
+ sage: expected = x.to_vector().inner_product(y.to_vector())
+ sage: actual == expected
True
- An error is raised if the Jordan product is not commutative::
+ 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: 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
+ 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
- An error is raised if the inner-product is not commutative::
+ """
+ B = self._inner_product_matrix
+ return (B*x.to_vector()).inner_product(y.to_vector())
- 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
- TESTS:
+ def is_associative(self):
+ r"""
+ Return whether or not this algebra's Jordan product is associative.
- The ``field`` we're given must be real with ``check_field=True``::
+ SETUP::
- 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: from mjo.eja.eja_algebra import ComplexHermitianEJA
- The multiplication table must be square with ``check_axioms=True``::
+ EXAMPLES::
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()),((1,),))
- Traceback (most recent call last):
- ...
- ValueError: multiplication table is not square
+ sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
+ sage: J.is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.is_associative()
+ True
- 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``::
+ """
+ return "Associative" in self.category().axioms()
- 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.gens()[i]**2)*(self.gens()[i]*self.gens()[j])
+ ==
+ (self.gens()[i])*((self.gens()[i]**2)*self.gens()[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 Jordan or inner-product.
+ """
- 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.gens()[i]
+ y = self.gens()[j]
+ z = self.gens()[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):
"""
...
ValueError: not an element of this algebra
+ Tuples work as well, provided that the matrix basis for the
+ algebra consists of them::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
+ e(0, 1) + e(1, 2)
+
TESTS:
Ensure that we can convert any element of the two non-matrix
sage: J(x.to_vector().column()) == x
True
+ We cannot coerce elements between algebras just because their
+ matrix representations are compatible::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = JordanSpinEJA(3)
+ sage: J2(J1.one())
+ Traceback (most recent call last):
+ ...
+ ValueError: not an element of this algebra
+ sage: J1(J2.zero())
+ Traceback (most recent call last):
+ ...
+ ValueError: not an element of this algebra
+
"""
msg = "not an element of this algebra"
- if elt == 0:
- # The superclass implementation of random_element()
- # needs to be able to coerce "0" into the algebra.
- return self.zero()
- elif elt in self.base_ring():
+ if elt in self.base_ring():
# Ensure that no base ring -> algebra coercion is performed
# by this method. There's some stupidity in sage that would
# otherwise propagate to this method; for example, sage thinks
# that the integer 3 belongs to the space of 2-by-2 matrices.
raise ValueError(msg)
+ try:
+ # Try to convert a vector into a column-matrix...
+ elt = elt.column()
+ except (AttributeError, TypeError):
+ # and ignore failure, because we weren't really expecting
+ # a vector as an argument anyway.
+ pass
+
if elt not in self.matrix_space():
raise ValueError(msg)
# closure whereas the base ring of the 3-by-3 identity matrix
# could be QQ instead of QQbar.
#
+ # And, we also have to handle Cartesian product bases (when
+ # the matrix basis consists of tuples) here. The "good news"
+ # is that we're already converting everything to long vectors,
+ # and that strategy works for tuples as well.
+ #
# We pass check=False because the matrix basis is "guaranteed"
# to be linearly independent... right? Ha ha.
- V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols())
- W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()),
+ elt = _all2list(elt)
+ V = VectorSpace(self.base_ring(), len(elt))
+ W = V.span_of_basis( (V(_all2list(s)) for s in self.matrix_basis()),
check=False)
try:
- coords = W.coordinate_vector(_mat2vec(elt))
+ coords = W.coordinate_vector(V(elt))
except ArithmeticError: # vector is not in free module
raise ValueError(msg)
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):
"""
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.gens()[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)
Why implement this for non-matrix algebras? Avoiding special
cases for the :class:`BilinearFormEJA` pays with simplicity in
its own right. But mainly, we would like to be able to assume
- that elements of a :class:`DirectSumEJA` can be displayed
+ that elements of a :class:`CartesianProductEJA` can be displayed
nicely, without having to have special classes for direct sums
one of whose components was a matrix algebra.
[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):
we think of them as matrices (including column vectors of the
appropriate size).
- Generally this will be an `n`-by-`1` column-vector space,
+ "By default" this will be an `n`-by-`1` column-matrix space,
except when the algebra is trivial. There it's `n`-by-`n`
(where `n` is zero), to ensure that two elements of the matrix
- space (empty matrices) can be multiplied.
+ space (empty matrices) can be multiplied. For algebras of
+ matrices, this returns the space in which their
+ real embeddings live.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
+ ....: QuaternionHermitianEJA,
+ ....: TrivialEJA)
+
+ EXAMPLES:
+
+ By default, the matrix representation is just a column-matrix
+ equivalent to the vector representation::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.matrix_space()
+ Full MatrixSpace of 3 by 1 dense matrices over Algebraic
+ Real Field
+
+ The matrix representation in the trivial algebra is
+ zero-by-zero instead of the usual `n`-by-one::
+
+ sage: J = TrivialEJA()
+ sage: J.matrix_space()
+ Full MatrixSpace of 0 by 0 dense matrices over Algebraic
+ Real Field
+
+ The matrix space for complex/quaternion Hermitian matrix EJA
+ is the space in which their real-embeddings live, not the
+ original complex/quaternion matrix space::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J.matrix_space()
+ Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
+ sage: J.matrix_space()
+ Full MatrixSpace of 4 by 4 dense matrices over Rational Field
- Matrix algebras override this with something more useful.
"""
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()
+ return self.matrix_basis()[0].parent()
@cached_method
sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
- EXAMPLES::
+ EXAMPLES:
+
+ We can compute unit element in the Hadamard EJA::
+
+ sage: J = HadamardEJA(5)
+ sage: J.one()
+ e0 + e1 + e2 + e3 + e4
+
+ The unit element in the Hadamard EJA is inherited in the
+ subalgebras generated by its elements::
sage: J = HadamardEJA(5)
sage: J.one()
e0 + e1 + e2 + e3 + e4
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.one()
+ f0
+ sage: A.one().superalgebra_element()
+ e0 + e1 + e2 + e3 + e4
TESTS:
- The identity element acts like the identity::
+ The identity element acts like the identity, regardless of
+ whether or not we orthonormalize::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: J.one()*x == x and x*J.one() == x
+ True
+ sage: A = x.subalgebra_generated_by()
+ sage: y = A.random_element()
+ sage: A.one()*y == y and y*A.one() == y
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: x = J.random_element()
+ sage: J.one()*x == x and x*J.one() == x
+ True
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: y = A.random_element()
+ sage: A.one()*y == y and y*A.one() == y
+ True
+
+ The matrix of the unit element's operator is the identity,
+ regardless of the base field and whether or not we
+ orthonormalize::
sage: set_random_seed()
sage: J = random_eja()
+ sage: actual = J.one().operator().matrix()
+ sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ sage: actual == expected
+ True
sage: x = J.random_element()
- sage: J.one()*x == x and x*J.one() == x
+ sage: A = x.subalgebra_generated_by()
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
True
- The matrix of the unit element's operator is the identity::
+ ::
sage: set_random_seed()
- sage: J = random_eja()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
sage: actual = J.one().operator().matrix()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
sage: actual == expected
True
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
Ensure that the cached unit element (often precomputed by
hand) agrees with the computed one::
sage: J.one() == cached
True
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: cached = J.one()
+ sage: J.one.clear_cache()
+ sage: J.one() == cached
+ True
+
"""
# We can brute-force compute the matrices of the operators
# that correspond to the basis elements of this algebra.
if not c.is_idempotent():
raise ValueError("element is not idempotent: %s" % c)
- from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
-
# 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 = self.subalgebra(())
J0 = trivial # eigenvalue zero
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
J5 = eigspace
else:
gens = tuple( self.from_vector(b) for b in eigspace.basis() )
- subalg = FiniteDimensionalEuclideanJordanSubalgebra(self,
- gens,
- check_axioms=False)
+ subalg = self.subalgebra(gens, check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
r"""
The `r` polynomial coefficients of the "characteristic polynomial
of" function.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS:
+
+ The theory shows that these are all homogeneous polynomials of
+ a known degree::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: all(p.is_homogeneous() for p in J._charpoly_coefficients())
+ True
+
"""
n = self.dimension()
R = self.coordinate_polynomial_ring()
def L_x_i_j(i,j):
# From a result in my book, these are the entries of the
# basis representation of L_x.
- return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
+ return sum( vars[k]*self.gens()[k].operator().matrix()[i,j]
for k in range(n) )
L_x = matrix(F, n, n, L_x_i_j)
# The theory says that only the first "r" coefficients are
# nonzero, and they actually live in the original polynomial
- # ring and not the fraction field. We negate them because
- # in the actual characteristic polynomial, they get moved
- # to the other side where x^r lives.
- return -A_rref.solve_right(E*b).change_ring(R)[:r]
+ # ring and not the fraction field. We negate them because in
+ # the actual characteristic polynomial, they get moved to the
+ # other side where x^r lives. We don't bother to trim A_rref
+ # down to a square matrix and solve the resulting system,
+ # because the upper-left r-by-r portion of A_rref is
+ # guaranteed to be the identity matrix, so e.g.
+ #
+ # A_rref.solve_right(Y)
+ #
+ # would just be returning Y.
+ return (-E*b)[:r].change_ring(R)
@cached_method
def rank(self):
sage: set_random_seed() # long time
sage: J = random_eja() # long time
- sage: caches = J.rank() # long time
+ sage: cached = J.rank() # long time
sage: J.rank.clear_cache() # long time
sage: J.rank() == cached # long time
True
return len(self._charpoly_coefficients())
+ def subalgebra(self, basis, **kwargs):
+ r"""
+ Create a subalgebra of this algebra from the given basis.
+ """
+ from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
+ return FiniteDimensionalEJASubalgebra(self, basis, **kwargs)
+
+
def vector_space(self):
"""
Return the vector space that underlies this algebra.
return self.zero().to_vector().parent().ambient_vector_space()
- Element = FiniteDimensionalEuclideanJordanAlgebraElement
-class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class RationalBasisEJA(FiniteDimensionalEJA):
r"""
New class for algebras whose supplied basis elements have all rational entries.
jordan_product,
inner_product,
field=AA,
- orthonormalize=True,
- prefix='e',
- category=None,
check_field=True,
- check_axioms=True):
+ **kwargs):
if check_field:
# Abuse the check_field parameter to check that the entries of
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.
# 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
-
- 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,
+ **kwargs)
@cached_method
def _charpoly_coefficients(self):
# 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.
a = ( a_i.change_ring(self.base_ring())
for a_i in self._rational_algebra._charpoly_coefficients() )
- # Now convert the coordinate variables back to the
+ if self._deortho_matrix is None:
+ # This can happen if our base ring was, say, AA and we
+ # chose not to (or didn't need to) orthonormalize. It's
+ # still faster to do the computations over QQ even if
+ # the numbers in the boxes stay the same.
+ return tuple(a)
+
+ # Otherwise, convert the coordinate variables back to the
# deorthonormalized ones.
R = self.coordinate_polynomial_ring()
from sage.modules.free_module_element import vector
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.
SETUP::
- sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ConcreteEJA
TESTS:
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
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
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"""
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
**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):
+ # A manual dictionary-cache for the complex_extension() method,
+ # since apparently @classmethods can't also be @cached_methods.
+ _complex_extension = {}
+
+ @classmethod
+ def complex_extension(cls,field):
+ r"""
+ The complex field that we embed/unembed, as an extension
+ of the given ``field``.
+ """
+ if field in cls._complex_extension:
+ return cls._complex_extension[field]
+
+ # Sage doesn't know how to adjoin the complex "i" (the root of
+ # x^2 + 1) to a field in a general way. Here, we just enumerate
+ # all of the cases that I have cared to support so far.
+ if field is AA:
+ # Sage doesn't know how to embed AA into QQbar, i.e. how
+ # to adjoin sqrt(-1) to AA.
+ F = QQbar
+ elif not field.is_exact():
+ # RDF or RR
+ F = field.complex_field()
+ else:
+ # Works for QQ and... maybe some other fields.
+ R = PolynomialRing(field, 'z')
+ z = R.gen()
+ F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+
+ cls._complex_extension[field] = F
+ return F
+
@staticmethod
def dimension_over_reals():
return 2
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: ComplexMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
EXAMPLES::
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]
[-----+-----]
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...
blocks = []
for z in M.list():
- a = z.list()[0] # real part, I guess
- b = z.list()[1] # imag part, I guess
- blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
+ a = z.real()
+ b = z.imag()
+ blocks.append(matrix(field, 2, [ [ a, b],
+ [-b, a] ]))
return matrix.block(field, n, blocks)
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: ComplexMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
EXAMPLES::
....: [-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]
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()
-
- # 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())
+ F = cls.complex_extension(M.base_ring())
i = F.gen()
# Go top-left to bottom-right (reading order), converting every
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,
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: field = QuadraticField(2, 'sqrt2')
sage: B = ComplexHermitianEJA._denormalized_basis(n)
sage: all( M.is_symmetric() for M in B)
True
# * The diagonal will (as a result) be real.
#
S = []
+ Eij = matrix.zero(F,n)
for i in range(n):
for j in range(i+1):
- Eij = matrix(F, n, lambda k,l: k==i and l==j)
+ # "build" E_ij
+ Eij[i,j] = 1
if i == j:
Sij = cls.real_embed(Eij)
S.append(Sij)
else:
# The second one has a minus because it's conjugated.
- Sij_real = cls.real_embed(Eij + Eij.transpose())
+ Eij[j,i] = 1 # Eij = Eij + Eij.transpose()
+ Sij_real = cls.real_embed(Eij)
S.append(Sij_real)
- Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose())
+ # Eij = I*Eij - I*Eij.transpose()
+ Eij[i,j] = I
+ Eij[j,i] = -I
+ Sij_imag = cls.real_embed(Eij)
S.append(Sij_imag)
+ Eij[j,i] = 0
+ # "erase" E_ij
+ Eij[i,j] = 0
# Since we embedded these, we can drop back to the "field" that we
# started with instead of the complex extension "F".
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))
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
+class QuaternionMatrixEJA(MatrixEJA):
+
+ # A manual dictionary-cache for the quaternion_extension() method,
+ # since apparently @classmethods can't also be @cached_methods.
+ _quaternion_extension = {}
+
+ @classmethod
+ def quaternion_extension(cls,field):
+ r"""
+ The quaternion field that we embed/unembed, as an extension
+ of the given ``field``.
+ """
+ if field in cls._quaternion_extension:
+ return cls._quaternion_extension[field]
+
+ Q = QuaternionAlgebra(field,-1,-1)
+
+ cls._quaternion_extension[field] = Q
+ return Q
+
@staticmethod
def dimension_over_reals():
return 4
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: QuaternionMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
EXAMPLES::
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]
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()
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
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: QuaternionMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
EXAMPLES::
....: [-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:
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()
# Use the base ring of the matrix to ensure that its entries can be
# multiplied by elements of the quaternion algebra.
- field = M.base_ring()
- Q = QuaternionAlgebra(field,-1,-1)
+ Q = cls.quaternion_extension(M.base_ring())
i,j,k = Q.gens()
# Go top-left to bottom-right (reading order), converting every
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')
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
# * The diagonal will (as a result) be real.
#
S = []
+ Eij = matrix.zero(Q,n)
for i in range(n):
for j in range(i+1):
- Eij = matrix(Q, n, lambda k,l: k==i and l==j)
+ # "build" E_ij
+ Eij[i,j] = 1
if i == j:
Sij = cls.real_embed(Eij)
S.append(Sij)
else:
# The second, third, and fourth ones have a minus
# because they're conjugated.
- Sij_real = cls.real_embed(Eij + Eij.transpose())
+ # Eij = Eij + Eij.transpose()
+ Eij[j,i] = 1
+ Sij_real = cls.real_embed(Eij)
S.append(Sij_real)
- Sij_I = cls.real_embed(I*Eij - I*Eij.transpose())
+ # Eij = I*(Eij - Eij.transpose())
+ Eij[i,j] = I
+ Eij[j,i] = -I
+ Sij_I = cls.real_embed(Eij)
S.append(Sij_I)
- Sij_J = cls.real_embed(J*Eij - J*Eij.transpose())
+ # Eij = J*(Eij - Eij.transpose())
+ Eij[i,j] = J
+ Eij[j,i] = -J
+ Sij_J = cls.real_embed(Eij)
S.append(Sij_J)
- Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
+ # Eij = K*(Eij - Eij.transpose())
+ Eij[i,j] = K
+ Eij[j,i] = -K
+ Sij_K = cls.real_embed(Eij)
S.append(Sij_K)
+ Eij[j,i] = 0
+ # "erase" E_ij
+ Eij[i,j] = 0
# Since we embedded these, we can drop back to the "field" that we
# started with instead of the quaternion algebra "Q".
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))
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.
"""
def __init__(self, n, **kwargs):
- def jordan_product(x,y):
- P = x.parent()
- return P(tuple( xi*yi for (xi,yi) in zip(x,y) ))
- def inner_product(x,y):
- return x.inner_product(y)
+ 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
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
-
- standard_basis = FreeModule(ZZ, n).basis()
- super(HadamardEJA, self).__init__(standard_basis,
- jordan_product,
- inner_product,
- **kwargs)
+ column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ **kwargs)
self.rank.set_cache(n)
if n == 0:
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 =
....: for j in range(n-1) ]
sage: actual == expected
True
+
"""
def __init__(self, B, **kwargs):
- if not B.is_positive_definite():
- raise ValueError("bilinear form is not positive-definite")
+ # The matrix "B" is supplied by the user in most cases,
+ # so it makes sense to check whether or not its positive-
+ # definite unless we are specifically asked not to...
+ if ("check_axioms" not in kwargs) or kwargs["check_axioms"]:
+ if not B.is_positive_definite():
+ raise ValueError("bilinear form is not positive-definite")
+
+ # However, all of the other data for this EJA is computed
+ # by us in manner that guarantees the axioms are
+ # satisfied. So, again, unless we are specifically asked to
+ # verify things, we'll skip the rest of the checks.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
def inner_product(x,y):
- return (B*x).inner_product(y)
+ return (y.T*B*x)[0,0]
def jordan_product(x,y):
P = x.parent()
- x0 = x[0]
- xbar = x[1:]
- y0 = y[0]
- ybar = y[1:]
- z0 = inner_product(x,y)
+ x0 = x[0,0]
+ xbar = x[1:,0]
+ y0 = y[0,0]
+ ybar = y[1:,0]
+ z0 = inner_product(y,x)
zbar = y0*xbar + x0*ybar
- return P((z0,) + tuple(zbar))
-
- # 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
+ return P([z0] + zbar.list())
n = B.nrows()
- standard_basis = FreeModule(ZZ, n).basis()
- super(BilinearFormEJA, self).__init__(standard_basis,
+ column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
+ super(BilinearFormEJA, self).__init__(column_basis,
jordan_product,
inner_product,
**kwargs)
return cls(n, **kwargs)
-class TrivialEJA(ConcreteEuclideanJordanAlgebra):
+class TrivialEJA(ConcreteEJA):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
# inappropriate for us.
return cls(**kwargs)
-class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+
+class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct,
+ FiniteDimensionalEJA):
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.
+ The external (orthogonal) direct sum of two or more Euclidean
+ Jordan algebras. Every Euclidean Jordan algebra decomposes into an
+ orthogonal direct sum of simple Euclidean Jordan algebras which is
+ then isometric to a Cartesian product, so no generality is lost by
+ providing only this construction.
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: CartesianProductEJA,
....: HadamardEJA,
- ....: RealSymmetricEJA,
- ....: DirectSumEJA)
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The Jordan product is inherited from our factors and implemented by
+ our CombinatorialFreeModule Cartesian product superclass::
+ sage: set_random_seed()
sage: J1 = HadamardEJA(2)
- sage: J2 = RealSymmetricEJA(3)
- sage: J = DirectSumEJA(J1,J2)
- sage: J.dimension()
- 8
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: x,y = J.random_elements(2)
+ sage: x*y in J
+ True
+
+ The ability to retrieve the original factors is implemented by our
+ CombinatorialFreeModule Cartesian product superclass::
+
+ sage: J1 = HadamardEJA(2, field=QQ)
+ sage: J2 = JordanSpinEJA(3, field=QQ)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_factors()
+ (Euclidean Jordan algebra of dimension 2 over Rational Field,
+ Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+ You can provide more than two factors::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = JordanSpinEJA(3)
+ sage: J3 = RealSymmetricEJA(3)
+ sage: cartesian_product([J1,J2,J3])
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real
+ Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic
+ Real Field (+) Euclidean Jordan algebra of dimension 6 over
+ Algebraic Real Field
+
+ Rank is additive on a Cartesian product::
+
+ sage: J1 = HadamardEJA(1)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
sage: J.rank()
- 5
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The same rank computation works over the rationals, with whatever
+ basis you like::
+
+ sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False)
+ sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The product algebra will be associative if and only if all of its
+ components are associative::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J1.is_associative()
+ True
+ sage: J2 = HadamardEJA(3)
+ sage: J2.is_associative()
+ True
+ sage: J3 = RealSymmetricEJA(3)
+ sage: J3.is_associative()
+ False
+ sage: CP1 = cartesian_product([J1,J2])
+ sage: CP1.is_associative()
+ True
+ sage: CP2 = cartesian_product([J1,J3])
+ sage: CP2.is_associative()
+ False
TESTS:
- The external direct sum construction is only valid when the two factors
- have the same base ring; an error is raised otherwise::
+ All factors must share the same base field::
- sage: set_random_seed()
- sage: J1 = random_eja(field=AA)
- sage: J2 = random_eja(field=QQ,orthonormalize=False)
- sage: J = DirectSumEJA(J1,J2)
+ sage: J1 = HadamardEJA(2, field=QQ)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: CartesianProductEJA((J1,J2))
Traceback (most recent call last):
...
- ValueError: algebras must share the same base field
+ ValueError: all factors must share the same base field
+
+ The cached unit element is the same one that would be computed::
+
+ sage: set_random_seed() # long time
+ sage: J1 = random_eja() # long time
+ sage: J2 = random_eja() # long time
+ sage: J = cartesian_product([J1,J2]) # long time
+ sage: actual = J.one() # long time
+ sage: J.one.clear_cache() # long time
+ sage: expected = J.one() # long time
+ sage: actual == expected # long time
+ True
"""
- 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)
+ Element = FiniteDimensionalEJAElement
+
+
+ def __init__(self, algebras, **kwargs):
+ CombinatorialFreeModule_CartesianProduct.__init__(self,
+ algebras,
+ **kwargs)
+ field = algebras[0].base_ring()
+ if not all( J.base_ring() == field for J in algebras ):
+ raise ValueError("all factors must share the same base field")
+
+ associative = all( m.is_associative() for m in algebras )
+
+ # The definition of matrix_space() and self.basis() relies
+ # only on the stuff in the CFM_CartesianProduct class, which
+ # we've already initialized.
+ Js = self.cartesian_factors()
+ m = len(Js)
+ MS = self.matrix_space()
+ basis = tuple(
+ MS(tuple( self.cartesian_projection(i)(b).to_matrix()
+ for i in range(m) ))
+ for b in self.basis()
+ )
- 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):
+ # Define jordan/inner products that operate on that matrix_basis.
+ def jordan_product(x,y):
+ return MS(tuple(
+ (Js[i](x[i])*Js[i](y[i])).to_matrix() for i in range(m)
+ ))
+
+ def inner_product(x, y):
+ return sum(
+ Js[i](x[i]).inner_product(Js[i](y[i])) for i in range(m)
+ )
+
+ # There's no need to check the field since it already came
+ # from an EJA. Likewise the axioms are guaranteed to be
+ # satisfied, unless the guy writing this class sucks.
+ #
+ # If you want the basis to be orthonormalized, orthonormalize
+ # the factors.
+ FiniteDimensionalEJA.__init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ orthonormalize=False,
+ associative=associative,
+ cartesian_product=True,
+ check_field=False,
+ check_axioms=False)
+
+ ones = tuple(J.one() for J in algebras)
+ self.one.set_cache(self._cartesian_product_of_elements(ones))
+ self.rank.set_cache(sum(J.rank() for J in algebras))
+
+ def matrix_space(self):
r"""
- Return the pair of this algebra's factors.
+ Return the space that our matrix basis lives in as a Cartesian
+ product.
SETUP::
sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: JordanSpinEJA,
- ....: DirectSumEJA)
+ ....: RealSymmetricEJA)
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)
+ sage: J1 = HadamardEJA(1)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.matrix_space()
+ The Cartesian product of (Full MatrixSpace of 1 by 1 dense
+ matrices over Algebraic Real Field, Full MatrixSpace of 2
+ by 2 dense matrices over Algebraic Real Field)
"""
- return self._factors
+ from sage.categories.cartesian_product import cartesian_product
+ return cartesian_product( [J.matrix_space()
+ for J in self.cartesian_factors()] )
- def projections(self):
+ @cached_method
+ def cartesian_projection(self, i):
r"""
- Return a pair of projections onto this algebra's factors.
-
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: ComplexHermitianEJA,
- ....: DirectSumEJA)
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
+ ....: HadamardEJA,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The projection morphisms are Euclidean Jordan algebra
+ operators::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_projection(0)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [1 0 0 0 0]
+ [0 1 0 0 0]
+ Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field (+) Euclidean Jordan algebra of dimension 3 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field
+ sage: J.cartesian_projection(1)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [0 0 1 0 0]
+ [0 0 0 1 0]
+ [0 0 0 0 1]
+ Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field (+) Euclidean Jordan algebra of dimension 3 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic
+ Real Field
+
+ The projections work the way you'd expect on the vector
+ representation of an element::
sage: J1 = JordanSpinEJA(2)
sage: J2 = ComplexHermitianEJA(2)
- sage: J = DirectSumEJA(J1,J2)
- sage: (pi_left, pi_right) = J.projections()
- sage: J.one().to_vector()
- (1, 0, 1, 0, 0, 1)
+ sage: J = cartesian_product([J1,J2])
+ sage: pi_left = J.cartesian_projection(0)
+ sage: pi_right = J.cartesian_projection(1)
sage: pi_left(J.one()).to_vector()
(1, 0)
sage: pi_right(J.one()).to_vector()
(1, 0, 0, 1)
+ sage: J.one().to_vector()
+ (1, 0, 1, 0, 0, 1)
+
+ TESTS:
+
+ The answer never changes::
+
+ sage: set_random_seed()
+ sage: J1 = random_eja()
+ sage: J2 = random_eja()
+ sage: J = cartesian_product([J1,J2])
+ sage: P0 = J.cartesian_projection(0)
+ sage: P1 = J.cartesian_projection(0)
+ sage: P0 == P1
+ True
"""
- (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.
+ Ji = self.cartesian_factors()[i]
+ # Requires the fix on Trac 31421/31422 to work!
+ Pi = super().cartesian_projection(i)
+ return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix())
+ @cached_method
+ def cartesian_embedding(self, i):
+ r"""
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
....: JordanSpinEJA,
- ....: RealSymmetricEJA,
- ....: DirectSumEJA)
+ ....: HadamardEJA,
+ ....: RealSymmetricEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The embedding morphisms are Euclidean Jordan algebra
+ operators::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_embedding(0)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [1 0]
+ [0 1]
+ [0 0]
+ [0 0]
+ [0 0]
+ Domain: Euclidean Jordan algebra of dimension 2 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 2 over
+ Algebraic Real Field (+) Euclidean Jordan algebra of
+ dimension 3 over Algebraic Real Field
+ sage: J.cartesian_embedding(1)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [0 0 0]
+ [0 0 0]
+ [1 0 0]
+ [0 1 0]
+ [0 0 1]
+ Domain: Euclidean Jordan algebra of dimension 3 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 2 over
+ Algebraic Real Field (+) Euclidean Jordan algebra of
+ dimension 3 over Algebraic Real Field
+
+ The embeddings work the way you'd expect on the vector
+ representation of an element::
sage: J1 = JordanSpinEJA(3)
sage: J2 = RealSymmetricEJA(2)
- sage: J = DirectSumEJA(J1,J2)
- sage: (iota_left, iota_right) = J.inclusions()
+ sage: J = cartesian_product([J1,J2])
+ sage: iota_left = J.cartesian_embedding(0)
+ sage: iota_right = J.cartesian_embedding(1)
sage: iota_left(J1.zero()) == J.zero()
True
sage: iota_right(J2.zero()) == J.zero()
TESTS:
+ The answer never changes::
+
+ sage: set_random_seed()
+ sage: J1 = random_eja()
+ sage: J2 = random_eja()
+ sage: J = cartesian_product([J1,J2])
+ sage: E0 = J.cartesian_embedding(0)
+ sage: E1 = J.cartesian_embedding(0)
+ sage: E0 == E1
+ True
+
Composing a projection with the corresponding inclusion should
produce the identity map, and mismatching them should produce
the zero map::
sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
- sage: J = DirectSumEJA(J1,J2)
- sage: (iota_left, iota_right) = J.inclusions()
- sage: (pi_left, pi_right) = J.projections()
+ sage: J = cartesian_product([J1,J2])
+ sage: iota_left = J.cartesian_embedding(0)
+ sage: iota_right = J.cartesian_embedding(1)
+ sage: pi_left = J.cartesian_projection(0)
+ sage: pi_right = J.cartesian_projection(1)
sage: pi_left*iota_left == J1.one().operator()
True
sage: pi_right*iota_right == J2.one().operator()
True
"""
- (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)
+ Ji = self.cartesian_factors()[i]
+ # Requires the fix on Trac 31421/31422 to work!
+ Ei = super().cartesian_embedding(i)
+ return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
- 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::
+FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+class RationalBasisCartesianProductEJA(CartesianProductEJA,
+ RationalBasisEJA):
+ r"""
+ A separate class for products of algebras for which we know a
+ rational basis.
- 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
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES:
+
+ This gives us fast characteristic polynomial computations in
+ product algebras, too::
- """
- (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))
+ sage: J1 = JordanSpinEJA(2)
+ sage: J2 = RealSymmetricEJA(3)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.characteristic_polynomial_of().degree()
+ 5
+ sage: J.rank()
+ 5
+
+ """
+ def __init__(self, algebras, **kwargs):
+ CartesianProductEJA.__init__(self, algebras, **kwargs)
+
+ self._rational_algebra = None
+ if self.vector_space().base_field() is not QQ:
+ self._rational_algebra = cartesian_product([
+ r._rational_algebra for r in algebras
+ ])
+RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA
-random_eja = ConcreteEuclideanJordanAlgebra.random_instance
+random_eja = ConcreteEJA.random_instance