"""
-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
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.
-
- - jordan_product -- function of two elements (in matrix form)
- that returns their jordan product in this algebra; this will
- be applied to ``basis`` to compute a multiplication table for
- the algebra.
-
- - inner_product -- function of two elements (in matrix form) that
- returns their inner product. This will be applied to ``basis`` to
- compute an inner-product table (basically a matrix) for this algebra.
+ - ``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.
"""
Element = FiniteDimensionalEJAElement
field=AA,
orthonormalize=True,
associative=False,
+ cartesian_product=False,
check_field=True,
check_axioms=True,
- prefix='e',
- category=None):
+ 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):
# 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 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.
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:
+ # 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.
- 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,
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: 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
+
+ """
# 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.
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))
+ return all( (self.gens()[i]**2)*(self.gens()[i]*self.gens()[j])
==
- (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+ (self.gens()[i])*((self.gens()[i]**2)*self.gens()[j])
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 (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
+
+ """
+ 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.gens()[i]
+ y = self.gens()[j]
+ z = self.gens()[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()):
for k in range(self.dimension()):
- x = self.monomial(i)
- y = self.monomial(j)
- z = self.monomial(k)
+ x = self.gens()[i]
+ y = self.gens()[j]
+ z = self.gens()[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
...
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
- 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)
# 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.gens()[i]] + [ self.product_on_basis(i,j)
+ for j in range(n) ]
for i in range(n) ]
return table(M, header_row=True, header_column=True, frame=True)
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)
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:
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)
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"""
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
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
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
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)
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ **kwargs)
self.rank.set_cache(n)
if n == 0:
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
+
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, 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")
- M = cartesian_product( [J.matrix_space() for J in modules] )
+ 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()
+ )
- m = len(modules)
- W = VectorSpace(field,m)
- self._matrix_basis = []
- for k in range(m):
- for a in modules[k].matrix_basis():
- v = W.zero().list()
- v[k] = a
- self._matrix_basis.append(M(v))
+ # 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)
+ ))
- self._matrix_basis = tuple(self._matrix_basis)
+ def inner_product(x, y):
+ return sum(
+ Js[i](x[i]).inner_product(Js[i](y[i])) for i in range(m)
+ )
- n = len(self._matrix_basis)
- # TODO:
- #
- # Initialize the FDEJA class, too. Does this override the
- # initialization that we did for the
- # CombinatorialFreeModule_CartesianProduct class? If not, we
- # will probably have to duplicate some of the work (i.e. one
- # of the constructors). Since the CartesianProduct one is
- # smaller, that makes the most sense to copy/paste if it comes
- # down to that.
+ # 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))
- 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):
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
- ....: RealSymmetricEJA)
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The projection morphisms are Euclidean Jordan algebra
+ operators::
sage: J1 = HadamardEJA(2)
sage: J2 = RealSymmetricEJA(2)
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 = 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::
"""
Ji = self.cartesian_factors()[i]
- # We reimplement the CombinatorialFreeModule superclass method
- # because if we don't, something gets messed up with the caching
- # and the answer changes the second time you run it. See the TESTS.
- Pi = self._module_morphism(lambda j_t: Ji.monomial(j_t[1])
- if i == j_t[0] else Ji.zero(),
- codomain=Ji)
+ # Requires the fix on Trac 31421/31422 to work!
+ Pi = super().cartesian_projection(i)
return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix())
@cached_method
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
....: RealSymmetricEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The embedding morphisms are Euclidean Jordan algebra
+ operators::
sage: J1 = HadamardEJA(2)
sage: J2 = RealSymmetricEJA(2)
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 = 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()
+ True
+ sage: J1.one().to_vector()
+ (1, 0, 0)
+ sage: iota_left(J1.one()).to_vector()
+ (1, 0, 0, 0, 0, 0)
+ sage: J2.one().to_vector()
+ (1, 0, 1)
+ sage: iota_right(J2.one()).to_vector()
+ (0, 0, 0, 1, 0, 1)
+ sage: J.one().to_vector()
+ (1, 0, 0, 1, 0, 1)
+
TESTS:
The answer never changes::
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 = 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
+ sage: (pi_left*iota_right).is_zero()
+ True
+ sage: (pi_right*iota_left).is_zero()
+ True
+
"""
Ji = self.cartesian_factors()[i]
- # We reimplement the CombinatorialFreeModule superclass method
- # because if we don't, something gets messed up with the caching
- # and the answer changes the second time you run it. See the TESTS.
- Ei = Ji._module_morphism(lambda t: self.monomial((i, t)), codomain=self)
+ # Requires the fix on Trac 31421/31422 to work!
+ Ei = super().cartesian_embedding(i)
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.
-
- SETUP::
+FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: QuaternionHermitianEJA)
+class RationalBasisCartesianProductEJA(CartesianProductEJA,
+ RationalBasisEJA):
+ r"""
+ A separate class for products of algebras for which we know a
+ rational basis.
- EXAMPLE::
+ SETUP::
- 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
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA)
- """
- 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 )
+ EXAMPLES:
+ This gives us fast characteristic polynomial computations in
+ product algebras, too::
-FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+ 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 projections(self):
-# r"""
-# Return a pair of projections onto this algebra's factors.
-
-# SETUP::
-
-# sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
-# ....: ComplexHermitianEJA,
-# ....: DirectSumEJA)
-
-# EXAMPLES::
-
-# sage: J1 = JordanSpinEJA(2)
-# sage: J2 = ComplexHermitianEJA(2)
-# sage: J = DirectSumEJA(J1,J2)
-# sage: (pi_left, pi_right) = J.projections()
-# sage: J.one().to_vector()
-# (1, 0, 1, 0, 0, 1)
-# sage: pi_left(J.one()).to_vector()
-# (1, 0)
-# sage: pi_right(J.one()).to_vector()
-# (1, 0, 0, 1)
-
-# """
-# (J1,J2) = self.factors()
-# m = J1.dimension()
-# n = J2.dimension()
-# V_basis = self.vector_space().basis()
-# # Need to specify the dimensions explicitly so that we don't
-# # wind up with a zero-by-zero matrix when we want e.g. a
-# # zero-by-two matrix (important for composing things).
-# P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
-# P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
-# pi_left = FiniteDimensionalEJAOperator(self,J1,P1)
-# pi_right = FiniteDimensionalEJAOperator(self,J2,P2)
-# return (pi_left, pi_right)
-
-# def inclusions(self):
-# r"""
-# Return the pair of inclusion maps from our factors into us.
-
-# SETUP::
-
-# sage: from mjo.eja.eja_algebra import (random_eja,
-# ....: JordanSpinEJA,
-# ....: RealSymmetricEJA,
-# ....: DirectSumEJA)
-
-# EXAMPLES::
-
-# sage: J1 = JordanSpinEJA(3)
-# sage: J2 = RealSymmetricEJA(2)
-# sage: J = DirectSumEJA(J1,J2)
-# sage: (iota_left, iota_right) = J.inclusions()
-# sage: iota_left(J1.zero()) == J.zero()
-# True
-# sage: iota_right(J2.zero()) == J.zero()
-# True
-# sage: J1.one().to_vector()
-# (1, 0, 0)
-# sage: iota_left(J1.one()).to_vector()
-# (1, 0, 0, 0, 0, 0)
-# sage: J2.one().to_vector()
-# (1, 0, 1)
-# sage: iota_right(J2.one()).to_vector()
-# (0, 0, 0, 1, 0, 1)
-# sage: J.one().to_vector()
-# (1, 0, 0, 1, 0, 1)
-
-# TESTS:
-
-# Composing a projection with the corresponding inclusion should
-# produce the identity map, and mismatching them should produce
-# the zero map::
-
-# sage: set_random_seed()
-# sage: J1 = random_eja()
-# sage: J2 = random_eja()
-# sage: J = DirectSumEJA(J1,J2)
-# sage: (iota_left, iota_right) = J.inclusions()
-# sage: (pi_left, pi_right) = J.projections()
-# sage: pi_left*iota_left == J1.one().operator()
-# True
-# sage: pi_right*iota_right == J2.one().operator()
-# True
-# sage: (pi_left*iota_right).is_zero()
-# True
-# sage: (pi_right*iota_left).is_zero()
-# True
-
-# """
-# (J1,J2) = self.factors()
-# m = J1.dimension()
-# n = J2.dimension()
-# V_basis = self.vector_space().basis()
-# # Need to specify the dimensions explicitly so that we don't
-# # wind up with a zero-by-zero matrix when we want e.g. a
-# # two-by-zero matrix (important for composing things).
-# I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
-# I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
-# iota_left = FiniteDimensionalEJAOperator(J1,self,I1)
-# iota_right = FiniteDimensionalEJAOperator(J2,self,I2)
-# return (iota_left, iota_right)
+ """
+ def __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 = ConcreteEJA.random_instance
projects / sage.d.git / blobdiff