"""
-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.categories.sets_cat import cartesian_product
-from sage.combinat.free_module import (CombinatorialFreeModule,
- CombinatorialFreeModule_CartesianProduct)
+from sage.combinat.free_module import CombinatorialFreeModule
from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEJAElement
from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
-from mjo.eja.eja_utils import _mat2vec
+from mjo.eja.eja_utils import _all2list, _mat2vec
class FiniteDimensionalEJA(CombinatorialFreeModule):
r"""
INPUT:
- - basis -- a tuple of basis elements in their matrix form.
+ - ``basis`` -- a tuple; 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`` -- a function; afunction 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`` -- a function; a 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.
+
+ - ``field`` -- a subfield of the reals (default: ``AA``); the scalar
+ field for the algebra.
+
+ - ``orthonormalize`` -- boolean (default: ``True``); whether or
+ not to orthonormalize the basis. Doing so is expensive and
+ generally rules out using the rationals as your ``field``, but
+ is required for spectral decompositions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
- - 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.
+ TESTS:
+
+ We should compute that an element subalgebra is associative even
+ if we circumvent the element method::
- - 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.
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ,orthonormalize=False)
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: basis = tuple(b.superalgebra_element() for b in A.basis())
+ sage: J.subalgebra(basis, orthonormalize=False).is_associative()
+ True
"""
Element = FiniteDimensionalEJAElement
inner_product,
field=AA,
orthonormalize=True,
- associative=False,
+ associative=None,
+ cartesian_product=False,
check_field=True,
check_axioms=True,
- prefix='e',
- category=None):
+ prefix="b"):
+
+ n = len(basis)
if check_field:
if not field.is_subring(RR):
# 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
raise ValueError("inner-product is not commutative")
- if category is None:
- category = MagmaticAlgebras(field).FiniteDimensional()
- category = category.WithBasis().Unital()
- if associative:
- # Element subalgebras can take advantage of this.
- category = category.Associative()
+ category = MagmaticAlgebras(field).FiniteDimensional()
+ category = category.WithBasis().Unital().Commutative()
+
+ if associative is None:
+ # We should figure it out. As with check_axioms, we have to do
+ # this without the help of the _jordan_product_is_associative()
+ # method because we need to know the category before we
+ # initialize the algebra.
+ associative = all( jordan_product(jordan_product(bi,bj),bk)
+ ==
+ jordan_product(bi,jordan_product(bj,bk))
+ for bi in basis
+ for bj in basis
+ for bk in basis)
+
+ if associative:
+ # Element subalgebras can take advantage of this.
+ category = category.Associative()
+ if cartesian_product:
+ # Use join() here because otherwise we only get the
+ # "Cartesian product of..." and not the things themselves.
+ category = category.join([category,
+ category.CartesianProducts()])
# 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)
+ 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.
+ # we see in things like x = 1*b1 + 2*b2.
vector_basis = basis
degree = 0
if n > 0:
- # Works on both column and square matrices...
- degree = len(basis[0].list())
+ degree = len(_all2list(basis[0]))
# Build an ambient space that fits our matrix basis when
# written out as "long vectors."
# 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 )
+ 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))
# 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 )
+ vector_basis = tuple( V(_all2list(b)) for b in basis )
W = V.span_of_basis( vector_basis, check=check_axioms)
if orthonormalize:
# 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(elt.list()))
+ elt = W.coordinate_vector(V(_all2list(elt)))
self._multiplication_table[i][j] = self.from_vector(elt)
if not orthonormalize:
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: bi = J.zero()
+ sage: bj = J.zero()
+ sage: bi_bj = J.zero()*J.zero()
+ sage: if n > 0:
+ ....: i = ZZ.random_element(n)
+ ....: j = ZZ.random_element(n)
+ ....: bi = J.monomial(i)
+ ....: bj = J.monomial(j)
+ ....: bi_bj = J.product_on_basis(i,j)
+ sage: bi*bj == bi_bj
+ True
+
+ """
# We only stored the lower-triangular portion of the
# multiplication table.
if j <= i:
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())
+ def is_associative(self):
+ r"""
+ Return whether or not this algebra's Jordan product is associative.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+ EXAMPLES::
+
+ 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
+
+ """
+ return "Associative" in self.category().axioms()
+
def _is_commutative(self):
r"""
Whether or not this algebra's multiplication table is commutative.
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()) )
+ return all( x*y == y*x for x in self.gens() for y in self.gens() )
def _is_jordanian(self):
r"""
for i in range(self.dimension())
for j in range(self.dimension()) )
+ def _jordan_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's Jordan product is
+ associative; that is, whether or not `x*(y*z) = (x*y)*z`
+ for all `x,y,x`.
+
+ This method should agree with :meth:`is_associative` unless
+ you lied about the value of the ``associative`` parameter
+ when you constructed the algebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(4, orthonormalize=False)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A._jordan_product_is_associative()
+ True
+
+ ::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A._jordan_product_is_associative()
+ True
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A._jordan_product_is_associative()
+ True
+
+ TESTS:
+
+ The values we've presupplied to the constructors agree with
+ the computation::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.is_associative() == J._jordan_product_is_associative()
+ True
+
+ """
+ R = self.base_ring()
+
+ # Used to check whether or not something is zero.
+ epsilon = R.zero()
+ if not R.is_exact():
+ # I don't know of any examples that make this magnitude
+ # necessary because I don't know how to make an
+ # associative algebra when the element subalgebra
+ # construction is unreliable (as it is over RDF; we can't
+ # find the degree of an element because we can't compute
+ # the rank of a matrix). But even multiplication of floats
+ # is non-associative, so *some* epsilon is needed... let's
+ # just take the one from _inner_product_is_associative?
+ epsilon = 1e-15
+
+ 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)*z - x*(y*z)
+
+ if diff.norm() > epsilon:
+ return False
+
+ return True
+
def _inner_product_is_associative(self):
r"""
Return whether or not this algebra's inner product `B` is
this algebra was constructed with ``check_axioms=False`` and
passed an invalid Jordan or inner-product.
"""
+ R = self.base_ring()
- # 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
+ # Used to check whether or not something is zero.
+ epsilon = R.zero()
+ if not R.is_exact():
+ # This choice is sufficient to allow the construction of
+ # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
+ epsilon = 1e-15
for i in range(self.dimension()):
for j in range(self.dimension()):
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
+ if diff.abs() > epsilon:
+ return False
return True
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
....: RealSymmetricEJA)
...
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]) )
+ b1 + b5
+
TESTS:
- Ensure that we can convert any element of the two non-matrix
- simple algebras (whose matrix representations are columns)
- back and forth faithfully::
+ Ensure that we can convert any element back and forth
+ faithfully between its matrix and algebra representations::
sage: set_random_seed()
- sage: J = HadamardEJA.random_instance()
- sage: x = J.random_element()
- sage: J(x.to_vector().column()) == x
- True
- sage: J = JordanSpinEJA.random_instance()
+ sage: J = random_eja()
sage: x = J.random_element()
- sage: J(x.to_vector().column()) == x
+ sage: J(x.to_matrix()) == 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
raise ValueError(msg)
try:
+ # Try to convert a vector into a column-matrix...
elt = elt.column()
except (AttributeError, TypeError):
- # Try to convert a vector into a column-matrix
+ # and ignore failure, because we weren't really expecting
+ # a vector as an argument anyway.
pass
if elt not in self.matrix_space():
# 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)
sage: J = JordanSpinEJA(4)
sage: J.multiplication_table()
+----++----+----+----+----+
- | * || e0 | e1 | e2 | e3 |
+ | * || b0 | b1 | b2 | b3 |
+====++====+====+====+====+
- | e0 || e0 | e1 | e2 | e3 |
+ | b0 || b0 | b1 | b2 | b3 |
+----++----+----+----+----+
- | e1 || e1 | e0 | 0 | 0 |
+ | b1 || b1 | b0 | 0 | 0 |
+----++----+----+----+----+
- | e2 || e2 | 0 | e0 | 0 |
+ | b2 || b2 | 0 | b0 | 0 |
+----++----+----+----+----+
- | e3 || e3 | 0 | 0 | e0 |
+ | b3 || b3 | 0 | 0 | b0 |
+----++----+----+----+----+
"""
# 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) ]
+ M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j)
+ for j in range(n) ]
for i in range(n) ]
return table(M, header_row=True, header_column=True, frame=True)
sage: J = RealSymmetricEJA(2)
sage: J.basis()
- Finite family {0: e0, 1: e1, 2: e2}
+ Finite family {0: b0, 1: b1, 2: b2}
sage: J.matrix_basis()
(
[1 0] [ 0 0.7071067811865475?] [0 0]
sage: J = JordanSpinEJA(2)
sage: J.basis()
- Finite family {0: e0, 1: e1}
+ Finite family {0: b0, 1: b1}
sage: J.matrix_basis()
(
[1] [0]
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)
sage: J = HadamardEJA(5)
sage: J.one()
- e0 + e1 + e2 + e3 + e4
+ b0 + b1 + b2 + b3 + b4
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
+ b0 + b1 + b2 + b3 + b4
sage: x = sum(J.gens())
sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: A.one()
- f0
+ c0
sage: A.one().superalgebra_element()
- e0 + e1 + e2 + e3 + e4
+ b0 + b1 + b2 + b3 + b4
TESTS:
if not c.is_idempotent():
raise ValueError("element is not idempotent: %s" % c)
- from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
-
# Default these to what they should be if they turn out to be
# trivial, because eigenspaces_left() won't return eigenvalues
# corresponding to trivial spaces (e.g. it returns only the
# eigenspace corresponding to lambda=1 if you take the
# decomposition relative to the identity element).
- trivial = FiniteDimensionalEJASubalgebra(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 = FiniteDimensionalEJASubalgebra(self,
- gens,
- check_axioms=False)
+ subalg = self.subalgebra(gens, check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
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 = FiniteDimensionalEJAElement
class RationalBasisEJA(FiniteDimensionalEJA):
r"""
if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
raise TypeError("basis not rational")
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ check_field=check_field,
+ **kwargs)
+
self._rational_algebra = None
if field is not QQ:
# There's no point in constructing the extra algebra if this
jordan_product,
inner_product,
field=QQ,
+ associative=self.is_associative(),
orthonormalize=False,
check_field=False,
check_axioms=False)
- super().__init__(basis,
- jordan_product,
- inner_product,
- field=field,
- check_field=check_field,
- **kwargs)
-
@cached_method
def _charpoly_coefficients(self):
r"""
class MatrixEJA:
+ @staticmethod
+ def jordan_product(X,Y):
+ return (X*Y + Y*X)/2
+
+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ A trace inner-product for matrices that aren't embedded in the
+ reals.
+ """
+ # We take the norm (absolute value) because Octonions() isn't
+ # smart enough yet to coerce its one() into the base field.
+ return (X*Y).trace().abs()
+
+class RealEmbeddedMatrixEJA(MatrixEJA):
@staticmethod
def dimension_over_reals():
r"""
raise ValueError("the matrix 'M' must be a real embedding")
return M
- @staticmethod
- def jordan_product(X,Y):
- return (X*Y + Y*X)/2
@classmethod
def trace_inner_product(cls,X,Y):
SETUP::
- sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
- ....: ComplexHermitianEJA,
+ sage: from mjo.eja.eja_algebra import (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)
True
"""
- Xu = cls.real_unembed(X)
- Yu = cls.real_unembed(Y)
- tr = (Xu*Yu).trace()
-
- try:
- # Works in QQ, AA, RDF, et cetera.
- return tr.real()
- except AttributeError:
- # A quaternion doesn't have a real() method, but does
- # have coefficient_tuple() method that returns the
- # coefficients of 1, i, j, and k -- in that order.
- return tr.coefficient_tuple()[0]
+ # This does in fact compute the real part of the trace.
+ # If we compute the trace of e.g. a complex matrix M,
+ # then we do so by adding up its diagonal entries --
+ # call them z_1 through z_n. The real embedding of z_1
+ # will be a 2-by-2 REAL matrix [a, b; -b, a] whose trace
+ # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
+ return (X*Y).trace()/cls.dimension_over_reals()
-
-class RealMatrixEJA(MatrixEJA):
- @staticmethod
- def dimension_over_reals():
- return 1
-
-
-class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
+class RealSymmetricEJA(ConcreteEJA, MatrixEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
EXAMPLES::
sage: J = RealSymmetricEJA(2)
- sage: e0, e1, e2 = J.gens()
- sage: e0*e0
- e0
- sage: e1*e1
- 1/2*e0 + 1/2*e2
- sage: e2*e2
- e2
+ sage: b0, b1, b2 = J.gens()
+ sage: b0*b0
+ b0
+ sage: b1*b1
+ 1/2*b0 + 1/2*b2
+ sage: b2*b2
+ b2
In theory, our "field" can be any subfield of the reals::
- sage: RealSymmetricEJA(2, field=RDF)
+ sage: RealSymmetricEJA(2, field=RDF, check_axioms=True)
Euclidean Jordan algebra of dimension 3 over Real Double Field
- sage: RealSymmetricEJA(2, field=RR)
+ sage: RealSymmetricEJA(2, field=RR, check_axioms=True)
Euclidean Jordan algebra of dimension 3 over Real Field with
53 bits of precision
"""
@classmethod
- def _denormalized_basis(cls, n):
+ def _denormalized_basis(cls, n, field):
"""
Return a basis for the space of real symmetric n-by-n matrices.
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = RealSymmetricEJA._denormalized_basis(n)
+ sage: B = RealSymmetricEJA._denormalized_basis(n,ZZ)
sage: all( M.is_symmetric() for M in B)
True
S = []
for i in range(n):
for j in range(i+1):
- Eij = matrix(ZZ, n, lambda k,l: k==i and l==j)
+ Eij = matrix(field, n, lambda k,l: k==i and l==j)
if i == j:
Sij = Eij
else:
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
- def __init__(self, n, **kwargs):
+ def __init__(self, n, field=AA, **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)
+ associative = False
+ if n <= 1:
+ associative = True
+
+ super().__init__(self._denormalized_basis(n,field),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=field,
+ associative=associative,
+ **kwargs)
# TODO: this could be factored out somehow, but is left here
# because the MatrixEJA is not presently a subclass of the
# FDEJA class that defines rank() and one().
self.rank.set_cache(n)
- idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
+ idV = self.matrix_space().one()
self.one.set_cache(self(idV))
-class ComplexMatrixEJA(MatrixEJA):
+class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
# A manual dictionary-cache for the complex_extension() method,
# since apparently @classmethods can't also be @cached_methods.
_complex_extension = {}
True
"""
- super(ComplexMatrixEJA,cls).real_embed(M)
+ super().real_embed(M)
n = M.nrows()
# We don't need any adjoined elements...
True
"""
- super(ComplexMatrixEJA,cls).real_unembed(M)
+ super().real_unembed(M)
n = ZZ(M.nrows())
d = cls.dimension_over_reals()
F = cls.complex_extension(M.base_ring())
In theory, our "field" can be any subfield of the reals::
- sage: ComplexHermitianEJA(2, field=RDF)
+ sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True)
Euclidean Jordan algebra of dimension 4 over Real Double Field
- sage: ComplexHermitianEJA(2, field=RR)
+ sage: ComplexHermitianEJA(2, field=RR, check_axioms=True)
Euclidean Jordan algebra of dimension 4 over Real Field with
53 bits of precision
"""
@classmethod
- def _denormalized_basis(cls, n):
+ def _denormalized_basis(cls, n, field):
"""
Returns a basis for the space of complex Hermitian n-by-n matrices.
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = ComplexHermitianEJA._denormalized_basis(n)
+ sage: B = ComplexHermitianEJA._denormalized_basis(n,ZZ)
sage: all( M.is_symmetric() for M in B)
True
"""
- field = ZZ
- R = PolynomialRing(field, 'z')
+ R = PolynomialRing(ZZ, 'z')
z = R.gen()
- F = field.extension(z**2 + 1, 'I')
+ F = ZZ.extension(z**2 + 1, 'I')
I = F.gen(1)
# This is like the symmetric case, but we need to be careful:
# "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".
+ # Since we embedded the entries, we can drop back to the
+ # desired real "field" instead of the extension "F".
return tuple( s.change_ring(field) for s in S )
- def __init__(self, n, **kwargs):
+ def __init__(self, n, field=AA, **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)
+ associative = False
+ if n <= 1:
+ associative = True
+
+ super().__init__(self._denormalized_basis(n,field),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=field,
+ associative=associative,
+ **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().
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class QuaternionMatrixEJA(MatrixEJA):
+class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
# A manual dictionary-cache for the quaternion_extension() method,
# since apparently @classmethods can't also be @cached_methods.
True
"""
- super(QuaternionMatrixEJA,cls).real_embed(M)
+ super().real_embed(M)
quaternions = M.base_ring()
n = M.nrows()
True
"""
- super(QuaternionMatrixEJA,cls).real_unembed(M)
+ super().real_unembed(M)
n = ZZ(M.nrows())
d = cls.dimension_over_reals()
In theory, our "field" can be any subfield of the reals::
- sage: QuaternionHermitianEJA(2, field=RDF)
+ sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True)
Euclidean Jordan algebra of dimension 6 over Real Double Field
- sage: QuaternionHermitianEJA(2, field=RR)
+ sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True)
Euclidean Jordan algebra of dimension 6 over Real Field with
53 bits of precision
"""
@classmethod
- def _denormalized_basis(cls, n):
+ def _denormalized_basis(cls, n, field):
"""
Returns a basis for the space of quaternion Hermitian n-by-n matrices.
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = QuaternionHermitianEJA._denormalized_basis(n)
+ sage: B = QuaternionHermitianEJA._denormalized_basis(n,ZZ)
sage: all( M.is_symmetric() for M in B )
True
"""
- field = ZZ
Q = QuaternionAlgebra(QQ,-1,-1)
I,J,K = Q.gens()
# "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".
+ # Since we embedded the entries, we can drop back to the
+ # desired real "field" instead of the quaternion algebra "Q".
return tuple( s.change_ring(field) for s in S )
- def __init__(self, n, **kwargs):
+ def __init__(self, n, field=AA, **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)
+ associative = False
+ if n <= 1:
+ associative = True
+
+ super().__init__(self._denormalized_basis(n,field),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=field,
+ associative=associative,
+ **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().
This multiplication table can be verified by hand::
sage: J = HadamardEJA(3)
- sage: e0,e1,e2 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
+ sage: b0,b1,b2 = J.gens()
+ sage: b0*b0
+ b0
+ sage: b0*b1
0
- sage: e0*e2
+ sage: b0*b2
0
- sage: e1*e1
- e1
- sage: e1*e2
+ sage: b1*b1
+ b1
+ sage: b1*b2
0
- sage: e2*e2
- e2
+ sage: b2*b2
+ b2
TESTS:
(r0, r1, r2)
"""
- def __init__(self, n, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
if n == 0:
jordan_product = lambda x,y: x
inner_product = lambda x,y: x
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)
+ column_basis = tuple( b.column()
+ for b in FreeModule(field, n).basis() )
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ associative=True,
+ **kwargs)
self.rank.set_cache(n)
if n == 0:
True
"""
- def __init__(self, B, **kwargs):
+ def __init__(self, B, field=AA, **kwargs):
# 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...
return P([z0] + zbar.list())
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)
+ column_basis = tuple( b.column()
+ for b in FreeModule(field, n).basis() )
+
+ # TODO: I haven't actually checked this, but it seems legit.
+ associative = False
+ if n <= 2:
+ associative = True
+
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ associative=associative,
+ **kwargs)
# The rank of this algebra is two, unless we're in a
# one-dimensional ambient space (because the rank is bounded
This multiplication table can be verified by hand::
sage: J = JordanSpinEJA(4)
- sage: e0,e1,e2,e3 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
- e1
- sage: e0*e2
- e2
- sage: e0*e3
- e3
- sage: e1*e2
+ sage: b0,b1,b2,b3 = J.gens()
+ sage: b0*b0
+ b0
+ sage: b0*b1
+ b1
+ sage: b0*b2
+ b2
+ sage: b0*b3
+ b3
+ sage: b1*b2
0
- sage: e1*e3
+ sage: b1*b3
0
- sage: e2*e3
+ sage: b2*b3
0
We can change the generator prefix::
True
"""
- def __init__(self, n, **kwargs):
+ def __init__(self, n, *args, **kwargs):
# This is a special case of the BilinearFormEJA with the
# identity matrix as its bilinear form.
B = matrix.identity(ZZ, n)
# But also don't pass check_field=False here, because the user
# can pass in a field!
- super(JordanSpinEJA, self).__init__(B, **kwargs)
+ super().__init__(B, *args, **kwargs)
@staticmethod
def _max_random_instance_size():
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)
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ **kwargs)
+
# The rank is zero using my definition, namely the dimension of the
# largest subalgebra generated by any element.
self.rank.set_cache(0)
return cls(**kwargs)
-class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct,
- FiniteDimensionalEJA):
+class CartesianProductEJA(FiniteDimensionalEJA):
r"""
The external (orthogonal) direct sum of two or more Euclidean
Jordan algebras. Every Euclidean Jordan algebra decomposes into an
SETUP::
- sage: from mjo.eja.eja_algebra import (CartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: CartesianProductEJA,
....: HadamardEJA,
....: JordanSpinEJA,
....: RealSymmetricEJA)
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()
+ 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
+
+ Cartesian products of Cartesian products work::
+
+ sage: J1 = JordanSpinEJA(1)
+ sage: J2 = JordanSpinEJA(1)
+ sage: J3 = JordanSpinEJA(1)
+ sage: J = cartesian_product([J1,cartesian_product([J2,J3])])
+ sage: J.multiplication_table()
+ +----++----+----+----+
+ | * || b0 | b1 | b2 |
+ +====++====+====+====+
+ | b0 || b0 | 0 | 0 |
+ +----++----+----+----+
+ | b1 || 0 | b1 | 0 |
+ +----++----+----+----+
+ | b2 || 0 | 0 | b2 |
+ +----++----+----+----+
+ sage: HadamardEJA(3).multiplication_table()
+ +----++----+----+----+
+ | * || b0 | b1 | b2 |
+ +====++====+====+====+
+ | b0 || b0 | 0 | 0 |
+ +----++----+----+----+
+ | b1 || 0 | b1 | 0 |
+ +----++----+----+----+
+ | b2 || 0 | 0 | b2 |
+ +----++----+----+----+
+
TESTS:
All factors 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, modules, **kwargs):
- CombinatorialFreeModule_CartesianProduct.__init__(self,
- modules,
- **kwargs)
- field = modules[0].base_ring()
- if not all( J.base_ring() == field for J in modules ):
+ Element = FiniteDimensionalEJAElement
+
+
+ def __init__(self, factors, **kwargs):
+ m = len(factors)
+ if m == 0:
+ return TrivialEJA()
+
+ self._sets = factors
+
+ field = factors[0].base_ring()
+ if not all( J.base_ring() == field for J in factors ):
raise ValueError("all factors must share the same base field")
- basis = tuple( b.to_vector().column() for b in self.basis() )
+ associative = all( f.is_associative() for f in factors )
- # Define jordan/inner products that operate on the basis.
- def jordan_product(x_mat,y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_jordan_product(x,y).to_vector().column()
+ MS = self.matrix_space()
+ basis = []
+ zero = MS.zero()
+ for i in range(m):
+ for b in factors[i].matrix_basis():
+ z = list(zero)
+ z[i] = b
+ basis.append(z)
- def inner_product(x_mat, y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_inner_product(x,y)
+ basis = tuple( MS(b) for b in basis )
- # Use whatever category the superclass came up with. Usually
- # some join of the EJA and Cartesian product
- # categories. There's no need to check the field since it
- # already came from an EJA.
+ # Define jordan/inner products that operate on that matrix_basis.
+ def jordan_product(x,y):
+ return MS(tuple(
+ (factors[i](x[i])*factors[i](y[i])).to_matrix()
+ for i in range(m)
+ ))
+
+ def inner_product(x, y):
+ return sum(
+ factors[i](x[i]).inner_product(factors[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.
inner_product,
field=field,
orthonormalize=False,
+ associative=associative,
+ cartesian_product=True,
check_field=False,
- check_axioms=False,
- category=self.category())
+ check_axioms=False)
+
+ ones = tuple(J.one().to_matrix() for J in factors)
+ self.one.set_cache(self(ones))
+ self.rank.set_cache(sum(J.rank() for J in factors))
+
+ def cartesian_factors(self):
+ # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
+ return self._sets
+
+ def cartesian_factor(self, i):
+ r"""
+ Return the ``i``th factor of this algebra.
+ """
+ return self._sets[i]
+
+ def _repr_(self):
+ # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
+ from sage.categories.cartesian_product import cartesian_product
+ return cartesian_product.symbol.join("%s" % factor
+ for factor in self._sets)
- self.rank.set_cache(sum(J.rank() for J in modules))
+ def matrix_space(self):
+ r"""
+ Return the space that our matrix basis lives in as a Cartesian
+ product.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES::
+
+ 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)
+
+ """
+ from sage.categories.cartesian_product import cartesian_product
+ return cartesian_product( [J.matrix_space()
+ for J in self.cartesian_factors()] )
@cached_method
def cartesian_projection(self, i):
True
"""
- Ji = self.cartesian_factors()[i]
- # Requires the fix on Trac 31421/31422 to work!
- Pi = super().cartesian_projection(i)
+ offset = sum( self.cartesian_factor(k).dimension()
+ for k in range(i) )
+ Ji = self.cartesian_factor(i)
+ Pi = self._module_morphism(lambda j: Ji.monomial(j - offset),
+ codomain=Ji)
+
return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix())
@cached_method
True
"""
- Ji = self.cartesian_factors()[i]
- # Requires the fix on Trac 31421/31422 to work!
- Ei = super().cartesian_embedding(i)
+ offset = sum( self.cartesian_factor(k).dimension()
+ for k in range(i) )
+ Ji = self.cartesian_factor(i)
+ Ei = Ji._module_morphism(lambda j: self.monomial(j + offset),
+ codomain=self)
return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
- def cartesian_jordan_product(self, x, y):
- r"""
- The componentwise Jordan product.
-
- We project ``x`` and ``y`` onto our factors, and add up the
- Jordan products from the subalgebras. This may still be useful
- after (if) the default Jordan product in the Cartesian product
- algebra is overridden.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: JordanSpinEJA)
-
- EXAMPLE::
-
- sage: J1 = HadamardEJA(3)
- sage: J2 = JordanSpinEJA(3)
- sage: J = cartesian_product([J1,J2])
- sage: x1 = J1.from_vector(vector(QQ,(1,2,1)))
- sage: y1 = J1.from_vector(vector(QQ,(1,0,2)))
- sage: x2 = J2.from_vector(vector(QQ,(1,2,3)))
- sage: y2 = J2.from_vector(vector(QQ,(1,1,1)))
- sage: z1 = J.from_vector(vector(QQ,(1,2,1,1,2,3)))
- sage: z2 = J.from_vector(vector(QQ,(1,0,2,1,1,1)))
- sage: (x1*y1).to_vector()
- (1, 0, 2)
- sage: (x2*y2).to_vector()
- (6, 3, 4)
- sage: J.cartesian_jordan_product(z1,z2).to_vector()
- (1, 0, 2, 6, 3, 4)
-
- """
- m = len(self.cartesian_factors())
- projections = ( self.cartesian_projection(i) for i in range(m) )
- products = ( P(x)*P(y) for P in projections )
- return self._cartesian_product_of_elements(tuple(products))
-
- def cartesian_inner_product(self, x, y):
- r"""
- The standard componentwise Cartesian inner-product.
- We project ``x`` and ``y`` onto our factors, and add up the
- inner-products from the subalgebras. This may still be useful
- after (if) the default inner product in the Cartesian product
- algebra is overridden.
+FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
- SETUP::
+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)
+ SETUP::
- EXAMPLE::
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA)
- sage: J1 = HadamardEJA(3,field=QQ)
- sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: J = cartesian_product([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: z1 = J._cartesian_product_of_elements((x1,y1))
- sage: z2 = J._cartesian_product_of_elements((x2,y2))
- sage: J.cartesian_inner_product(z1,z2)
- 5
+ EXAMPLES:
- """
- m = len(self.cartesian_factors())
- projections = ( self.cartesian_projection(i) for i in range(m) )
- return sum( P(x).inner_product(P(y)) for P in projections )
+ This gives us fast characteristic polynomial computations in
+ product algebras, too::
- Element = FiniteDimensionalEJAElement
+ 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)
-FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
-random_eja = ConcreteEJA.random_instance
+ 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
+
+def random_eja(*args, **kwargs):
+ J1 = ConcreteEJA.random_instance(*args, **kwargs)
+
+ # This might make Cartesian products appear roughly as often as
+ # any other ConcreteEJA.
+ if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0:
+ # Use random_eja() again so we can get more than two factors.
+ J2 = random_eja(*args, **kwargs)
+ J = cartesian_product([J1,J2])
+ return J
+ else:
+ return J1
projects / sage.d.git / blobdiff