prefix='e',
category=None,
natural_basis=None,
- check=True):
+ check_field=True,
+ check_axioms=True):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
+ sage: from mjo.eja.eja_algebra import (
+ ....: FiniteDimensionalEuclideanJordanAlgebra,
+ ....: JordanSpinEJA,
+ ....: random_eja)
EXAMPLES:
TESTS:
- The ``field`` we're given must be real::
+ The ``field`` we're given must be real with ``check_field=True``::
sage: JordanSpinEJA(2,QQbar)
Traceback (most recent call last):
...
- ValueError: field is not real
+ ValueError: scalar field is not real
+
+ The multiplication table must be square with ``check_axioms=True``::
+
+ sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
+ Traceback (most recent call last):
+ ...
+ ValueError: multiplication table is not square
"""
- if check:
+ if check_field:
if not field.is_subring(RR):
# Note: this does return true for the real algebraic
- # field, and any quadratic field where we've specified
- # a real embedding.
- raise ValueError('field is not real')
+ # field, the rationals, and any quadratic field where
+ # we've specified a real embedding.
+ raise ValueError("scalar field is not real")
+
+ # The multiplication table had better be square
+ n = len(mult_table)
+ if check_axioms:
+ if not all( len(l) == n for l in mult_table ):
+ raise ValueError("multiplication table is not square")
self._natural_basis = natural_basis
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
- range(len(mult_table)),
+ range(n),
prefix=prefix,
category=category)
self.print_options(bracket='')
for ls in mult_table
]
+ if check_axioms:
+ if not self._is_commutative():
+ raise ValueError("algebra is not commutative")
+ 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 _element_constructor_(self, elt):
"""
return self.from_vector(coords)
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
"""
Return an integer "size" that is an upper bound on the size of
this algebra when it is used in a random test
interpreted to be far less than the dimension) should override
with a smaller number.
"""
- return 5
+ raise NotImplementedError
def _repr_(self):
"""
def product_on_basis(self, i, j):
return self._multiplication_table[i][j]
+ def _is_commutative(self):
+ r"""
+ Whether or not this algebra's multiplication table is commutative.
+
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid multiplication table.
+ """
+ return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+
+ def _is_jordanian(self):
+ r"""
+ Whether or not this algebra's multiplication table respects the
+ Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+
+ We only check one arrangement of `x` and `y`, so for a
+ ``True`` result to be truly true, you should also check
+ :meth:`_is_commutative`. This method should of course always
+ return ``True``, unless this algebra was constructed with
+ ``check_axioms=False`` and passed an invalid multiplication table.
+ """
+ return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
+ ==
+ (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+
+ def _inner_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's inner product `B` is
+ associative; that is, whether or not `B(xy,z) = B(x,yz)`.
+
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid multiplication table.
+ """
+
+ # Used to check whether or not something is zero in an inexact
+ # ring. This number is sufficient to allow the construction of
+ # QuaternionHermitianEJA(2, RDF) with check_axioms=True.
+ epsilon = 1e-16
+
+ for i in range(self.dimension()):
+ for j in range(self.dimension()):
+ for k in range(self.dimension()):
+ x = self.monomial(i)
+ y = self.monomial(j)
+ z = self.monomial(k)
+ diff = (x*y).inner_product(z) - x.inner_product(y*z)
+
+ if self.base_ring().is_exact():
+ if diff != 0:
+ return False
+ else:
+ if diff.abs() > epsilon:
+ return False
+
+ return True
+
@cached_method
def characteristic_polynomial_of(self):
"""
# appeal to the "long vectors" isometry.
oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]
- # Now we use basis linear algebra to find the coefficients,
+ # Now we use basic linear algebra to find the coefficients,
# of the matrices-as-vectors-linear-combination, which should
# work for the original algebra basis too.
- A = matrix.column(self.base_ring(), oper_vecs)
+ A = matrix(self.base_ring(), oper_vecs)
# We used the isometry on the left-hand side already, but we
# still need to do it for the right-hand side. Recall that we
# wanted something that summed to the identity matrix.
b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) )
- # Now if there's an identity element in the algebra, this should work.
- coeffs = A.solve_right(b)
- return self.linear_combination(zip(self.gens(), coeffs))
+ # Now if there's an identity element in the algebra, this
+ # should work. We solve on the left to avoid having to
+ # transpose the matrix "A".
+ return self.from_vector(A.solve_left(b))
def peirce_decomposition(self, c):
J5 = eigspace
else:
gens = tuple( self.from_vector(b) for b in eigspace.basis() )
- subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens)
+ subalg = FiniteDimensionalEuclideanJordanSubalgebra(self,
+ gens,
+ check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
return (J0, J5, J1)
- def random_elements(self, count):
+ def random_element(self, thorough=False):
+ r"""
+ Return a random element of this algebra.
+
+ Our algebra superclass method only returns a linear
+ combination of at most two basis elements. We instead
+ want the vector space "random element" method that
+ returns a more diverse selection.
+
+ INPUT:
+
+ - ``thorough`` -- (boolean; default False) whether or not we
+ should generate irrational coefficients for the random
+ element when our base ring is irrational; this slows the
+ algebra operations to a crawl, but any truly random method
+ should include them
+
+ """
+ # For a general base ring... maybe we can trust this to do the
+ # right thing? Unlikely, but.
+ V = self.vector_space()
+ v = V.random_element()
+
+ if self.base_ring() is AA:
+ # The "random element" method of the algebraic reals is
+ # stupid at the moment, and only returns integers between
+ # -2 and 2, inclusive:
+ #
+ # https://trac.sagemath.org/ticket/30875
+ #
+ # Instead, we implement our own "random vector" method,
+ # and then coerce that into the algebra. We use the vector
+ # space degree here instead of the dimension because a
+ # subalgebra could (for example) be spanned by only two
+ # vectors, each with five coordinates. We need to
+ # generate all five coordinates.
+ if thorough:
+ v *= QQbar.random_element().real()
+ else:
+ v *= QQ.random_element()
+
+ return self.from_vector(V.coordinate_vector(v))
+
+ def random_elements(self, count, thorough=False):
"""
Return ``count`` random elements as a tuple.
+ INPUT:
+
+ - ``thorough`` -- (boolean; default False) whether or not we
+ should generate irrational coefficients for the random
+ elements when our base ring is irrational; this slows the
+ algebra operations to a crawl, but any truly random method
+ should include them
+
SETUP::
sage: from mjo.eja.eja_algebra import JordanSpinEJA
True
"""
- return tuple( self.random_element() for idx in range(count) )
+ return tuple( self.random_element(thorough)
+ for idx in range(count) )
@classmethod
def random_instance(cls, field=AA, **kwargs):
Beware, this will crash for "most instances" because the
constructor below looks wrong.
"""
- if cls is TrivialEJA:
- # The TrivialEJA class doesn't take an "n" argument because
- # there's only one.
- return cls(field)
-
- n = ZZ.random_element(cls._max_test_case_size() + 1)
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, field, **kwargs)
@cached_method
Element = FiniteDimensionalEuclideanJordanAlgebraElement
-class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra):
- """
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
-
- Note: this is nothing more than the Cartesian product of ``n``
- copies of the spin algebra. Once Cartesian product algebras
- are implemented, this can go.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import HadamardEJA
-
- EXAMPLES:
-
- 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
- 0
- sage: e0*e2
- 0
- sage: e1*e1
- e1
- sage: e1*e2
- 0
- sage: e2*e2
- e2
-
- TESTS:
-
- We can change the generator prefix::
-
- sage: HadamardEJA(3, prefix='r').gens()
- (r0, r1, r2)
-
- """
- def __init__(self, n, field=AA, **kwargs):
- V = VectorSpace(field, n)
- mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
- for i in range(n) ]
-
- fdeja = super(HadamardEJA, self)
- fdeja.__init__(field, mult_table, **kwargs)
- self.rank.set_cache(n)
-
- def inner_product(self, x, y):
- """
- Faster to reimplement than to use natural representations.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import HadamardEJA
-
- 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 = x.natural_representation()
- sage: Y = y.natural_representation()
- sage: x.inner_product(y) == J.natural_inner_product(X,Y)
- True
-
- """
- return x.to_vector().inner_product(y.to_vector())
-
def random_eja(field=AA):
"""
+class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
+ r"""
+ Algebras whose basis consists of vectors with rational
+ entries. Equivalently, algebras whose multiplication tables
+ contain only rational coefficients.
+
+ When an EJA has a basis that can be made rational, we can speed up
+ the computation of its characteristic polynomial by doing it over
+ ``QQ``. All of the named EJA constructors that we provide fall
+ into this category.
+ """
+ @cached_method
+ def _charpoly_coefficients(self):
+ r"""
+ Override the parent method with something that tries to compute
+ over a faster (non-extension) field.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
+
+ EXAMPLES:
+
+ The base ring of the resulting polynomial coefficients is what
+ it should be, and not the rationals (unless the algebra was
+ already over the rationals)::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J._charpoly_coefficients()
+ (X1^2 - X2^2 - X3^2, -2*X1)
+ sage: a0 = J._charpoly_coefficients()[0]
+ sage: J.base_ring()
+ Algebraic Real Field
+ sage: a0.base_ring()
+ Algebraic Real Field
+
+ """
+ if self.base_ring() is QQ:
+ # There's no need to construct *another* algebra over the
+ # rationals if this one is already over the rationals.
+ superclass = super(RationalBasisEuclideanJordanAlgebra, self)
+ return superclass._charpoly_coefficients()
+
+ mult_table = tuple(
+ map(lambda x: x.to_vector(), ls)
+ for ls in self._multiplication_table
+ )
+
+ # Do the computation over the rationals. The answer will be
+ # the same, because our basis coordinates are (essentially)
+ # rational.
+ J = FiniteDimensionalEuclideanJordanAlgebra(QQ,
+ mult_table,
+ check_field=False,
+ check_axioms=False)
+ a = J._charpoly_coefficients()
+ return tuple(map(lambda x: x.change_ring(self.base_ring()), a))
+
+
class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
# Play it safe, since this will be squared and the underlying
# field can have dimension 4 (quaternions) too.
return 2
Qs = self.multiplication_table_from_matrix_basis(basis)
- fdeja = super(MatrixEuclideanJordanAlgebra, self)
- fdeja.__init__(field, Qs, natural_basis=basis, **kwargs)
- return
+ super(MatrixEuclideanJordanAlgebra, self).__init__(field,
+ Qs,
+ natural_basis=basis,
+ **kwargs)
@cached_method
Override the parent method with something that tries to compute
over a faster (non-extension) field.
"""
- if self._basis_normalizers is None:
- # We didn't normalize, so assume that the basis we started
- # with had entries in a nice field.
+ if self._basis_normalizers is None or self.base_ring() is QQ:
+ # We didn't normalize, or the basis we started with had
+ # entries in a nice field already. Just compute the thing.
return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients()
- else:
- basis = ( (b/n) for (b,n) in zip(self.natural_basis(),
- self._basis_normalizers) )
-
- # Do this over the rationals and convert back at the end.
- # Only works because we know the entries of the basis are
- # integers.
- J = MatrixEuclideanJordanAlgebra(QQ,
- basis,
- normalize_basis=False)
- a = J._charpoly_coefficients()
-
- # Unfortunately, changing the basis does change the
- # coefficients of the characteristic polynomial, but since
- # these are really the coefficients of the "characteristic
- # polynomial of" function, everything is still nice and
- # unevaluated. It's therefore "obvious" how scaling the
- # basis affects the coordinate variables X1, X2, et
- # cetera. Scaling the first basis vector up by "n" adds a
- # factor of 1/n into every "X1" term, for example. So here
- # we simply undo the basis_normalizer scaling that we
- # performed earlier.
- #
- # The a[0] access here is safe because trivial algebras
- # won't have any basis normalizers and therefore won't
- # make it to this "else" branch.
- XS = a[0].parent().gens()
- subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i]
- for i in range(len(XS)) }
- return tuple( a_i.subs(subs_dict) for a_i in a )
+
+ basis = ( (b/n) for (b,n) in zip(self.natural_basis(),
+ self._basis_normalizers) )
+
+ # Do this over the rationals and convert back at the end.
+ # Only works because we know the entries of the basis are
+ # integers. The argument ``check_axioms=False`` is required
+ # because the trace inner-product method for this
+ # class is a stub and can't actually be checked.
+ J = MatrixEuclideanJordanAlgebra(QQ,
+ basis,
+ normalize_basis=False,
+ check_field=False,
+ check_axioms=False)
+ a = J._charpoly_coefficients()
+
+ # Unfortunately, changing the basis does change the
+ # coefficients of the characteristic polynomial, but since
+ # these are really the coefficients of the "characteristic
+ # polynomial of" function, everything is still nice and
+ # unevaluated. It's therefore "obvious" how scaling the
+ # basis affects the coordinate variables X1, X2, et
+ # cetera. Scaling the first basis vector up by "n" adds a
+ # factor of 1/n into every "X1" term, for example. So here
+ # we simply undo the basis_normalizer scaling that we
+ # performed earlier.
+ #
+ # The a[0] access here is safe because trivial algebras
+ # won't have any basis normalizers and therefore won't
+ # make it to this "else" branch.
+ XS = a[0].parent().gens()
+ subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i]
+ for i in range(len(XS)) }
+ return tuple( a_i.subs(subs_dict) for a_i in a )
@staticmethod
Yu = cls.real_unembed(Y)
tr = (Xu*Yu).trace()
- if tr in RLF:
- # It's real already.
- return tr
-
- # Otherwise, try the thing that works for complex numbers; and
- # if that doesn't work, the thing that works for quaternions.
try:
- return tr.vector()[0] # real part, imag part is index 1
+ # Works in QQ, AA, RDF, et cetera.
+ return tr.real()
except AttributeError:
- # A quaternions doesn't have a vector() method, but does
+ # 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]
The dimension of this algebra is `(n^2 + n) / 2`::
sage: set_random_seed()
- sage: n_max = RealSymmetricEJA._max_test_case_size()
+ sage: n_max = RealSymmetricEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = RealSymmetricEJA(n)
sage: J.dimension() == (n^2 + n)/2
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
return 4 # Dimension 10
def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n, field)
- super(RealSymmetricEJA, self).__init__(field, basis, **kwargs)
+ super(RealSymmetricEJA, self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
self.rank.set_cache(n)
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
+ sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_random_instance_size()
sage: n = ZZ.random_element(n_max)
sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
The dimension of this algebra is `n^2`::
sage: set_random_seed()
- sage: n_max = ComplexHermitianEJA._max_test_case_size()
+ sage: n_max = ComplexHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = ComplexHermitianEJA(n)
sage: J.dimension() == n^2
def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
- super(ComplexHermitianEJA,self).__init__(field, basis, **kwargs)
+ super(ComplexHermitianEJA,self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
self.rank.set_cache(n)
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
+ sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_random_instance_size()
sage: n = ZZ.random_element(n_max)
sage: Q = QuaternionAlgebra(QQ,-1,-1)
sage: X = random_matrix(Q, n)
The dimension of this algebra is `2*n^2 - n`::
sage: set_random_seed()
- sage: n_max = QuaternionHermitianEJA._max_test_case_size()
+ sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
- super(QuaternionHermitianEJA,self).__init__(field, basis, **kwargs)
+ super(QuaternionHermitianEJA,self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
+
+
+class HadamardEJA(RationalBasisEuclideanJordanAlgebra):
+ """
+ Return the Euclidean Jordan Algebra corresponding to the set
+ `R^n` under the Hadamard product.
+
+ Note: this is nothing more than the Cartesian product of ``n``
+ copies of the spin algebra. Once Cartesian product algebras
+ are implemented, this can go.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import HadamardEJA
+
+ EXAMPLES:
+
+ 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
+ 0
+ sage: e0*e2
+ 0
+ sage: e1*e1
+ e1
+ sage: e1*e2
+ 0
+ sage: e2*e2
+ e2
+
+ TESTS:
+
+ We can change the generator prefix::
+
+ sage: HadamardEJA(3, prefix='r').gens()
+ (r0, r1, r2)
+
+ """
+ def __init__(self, n, field=AA, **kwargs):
+ V = VectorSpace(field, n)
+ mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
+ for i in range(n) ]
+
+ super(HadamardEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
self.rank.set_cache(n)
+ @staticmethod
+ def _max_random_instance_size():
+ return 5
+
+ def inner_product(self, x, y):
+ """
+ Faster to reimplement than to use natural representations.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import HadamardEJA
+
+ 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 = x.natural_representation()
+ sage: Y = y.natural_representation()
+ sage: x.inner_product(y) == J.natural_inner_product(X,Y)
+ True
+
+ """
+ return x.to_vector().inner_product(y.to_vector())
+
-class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra):
+class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra):
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 =
- (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
- symmetric positive-definite "bilinear form" matrix. It has
- dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
- when ``B`` is the identity matrix of order ``n-1``.
+ (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
+ a symmetric positive-definite "bilinear form" matrix. Its
+ dimension is the size of `B`, and it has rank two in dimensions
+ larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
+ the identity matrix of order ``n``.
+
+ We insist that the one-by-one upper-left identity block of `B` be
+ passed in as well so that we can be passed a matrix of size zero
+ to construct a trivial algebra.
SETUP::
When no bilinear form is specified, the identity matrix is used,
and the resulting algebra is the Jordan spin algebra::
- sage: J0 = BilinearFormEJA(3)
+ sage: B = matrix.identity(AA,3)
+ sage: J0 = BilinearFormEJA(B)
sage: J1 = JordanSpinEJA(3)
sage: J0.multiplication_table() == J0.multiplication_table()
True
We can create a zero-dimensional algebra::
- sage: J = BilinearFormEJA(0)
+ sage: B = matrix.identity(AA,0)
+ sage: J = BilinearFormEJA(B)
sage: J.basis()
Finite family {}
sage: set_random_seed()
sage: n = ZZ.random_element(5)
sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
- sage: B = M.transpose()*M
- sage: J = BilinearFormEJA(n, B=B)
+ sage: B11 = matrix.identity(QQ,1)
+ sage: B22 = M.transpose()*M
+ sage: B = block_matrix(2,2,[ [B11,0 ],
+ ....: [0, B22 ] ])
+ sage: J = BilinearFormEJA(B)
sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
sage: V = J.vector_space()
sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
sage: actual == expected
True
"""
- def __init__(self, n, field=AA, B=None, **kwargs):
- if B is None:
- self._B = matrix.identity(field, max(0,n-1))
- else:
- self._B = B
+ def __init__(self, B, field=AA, **kwargs):
+ self._B = B
+ n = B.nrows()
+
+ if not B.is_positive_definite():
+ raise TypeError("matrix B is not positive-definite")
V = VectorSpace(field, n)
mult_table = [[V.zero() for j in range(n)] for i in range(n)]
xbar = x[1:]
y0 = y[0]
ybar = y[1:]
- z0 = x0*y0 + (self._B*xbar).inner_product(ybar)
+ z0 = (B*x).inner_product(y)
zbar = y0*xbar + x0*ybar
z = V([z0] + zbar.list())
mult_table[i][j] = z
# The rank of this algebra is two, unless we're in a
# one-dimensional ambient space (because the rank is bounded
# by the ambient dimension).
- fdeja = super(BilinearFormEJA, self)
- fdeja.__init__(field, mult_table, **kwargs)
+ super(BilinearFormEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
self.rank.set_cache(min(n,2))
+ @staticmethod
+ def _max_random_instance_size():
+ return 5
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if n == 0:
+ # Special case needed since we use (n-1) below.
+ B = matrix.identity(field, 0)
+ return cls(B, field, **kwargs)
+
+ B11 = matrix.identity(field,1)
+ M = matrix.random(field, n-1)
+ I = matrix.identity(field, n-1)
+ alpha = field.zero()
+ while alpha.is_zero():
+ alpha = field.random_element().abs()
+ B22 = M.transpose()*M + alpha*I
+
+ from sage.matrix.special import block_matrix
+ B = block_matrix(2,2, [ [B11, ZZ(0) ],
+ [ZZ(0), B22 ] ])
+
+ return cls(B, field, **kwargs)
+
def inner_product(self, x, y):
r"""
Half of the trace inner product.
paper::
sage: set_random_seed()
- sage: n = ZZ.random_element(2,5)
- sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
- sage: B = M.transpose()*M
- sage: J = BilinearFormEJA(n, B=B)
+ sage: J = BilinearFormEJA.random_instance()
+ sage: n = J.dimension()
sage: x = J.random_element()
sage: y = J.random_element()
- sage: x.inner_product(y) == (x*y).trace()/2
+ sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
True
"""
- xvec = x.to_vector()
- xbar = xvec[1:]
- yvec = y.to_vector()
- ybar = yvec[1:]
- return x[0]*y[0] + (self._B*xbar).inner_product(ybar)
+ return (self._B*x.to_vector()).inner_product(y.to_vector())
class JordanSpinEJA(BilinearFormEJA):
def __init__(self, n, field=AA, **kwargs):
# This is a special case of the BilinearFormEJA with the identity
# matrix as its bilinear form.
- return super(JordanSpinEJA, self).__init__(n, field, **kwargs)
+ B = matrix.identity(field, n)
+ super(JordanSpinEJA, self).__init__(B, field, **kwargs)
class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
def __init__(self, field=AA, **kwargs):
mult_table = []
- fdeja = super(TrivialEJA, self)
+ super(TrivialEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
# The rank is zero using my definition, namely the dimension of the
# largest subalgebra generated by any element.
- fdeja.__init__(field, mult_table, **kwargs)
self.rank.set_cache(0)
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ # We don't take a "size" argument so the superclass method is
+ # inappropriate for us.
+ return cls(field, **kwargs)
+
+class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ r"""
+ The external (orthogonal) direct sum of two other Euclidean Jordan
+ algebras. Essentially the Cartesian product of its two factors.
+ Every Euclidean Jordan algebra decomposes into an orthogonal
+ direct sum of simple Euclidean Jordan algebras, so no generality
+ is lost by providing only this construction.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(3)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: J.dimension()
+ 8
+ sage: J.rank()
+ 5
+
+ """
+ def __init__(self, J1, J2, field=AA, **kwargs):
+ 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(n) ]
+ for i in range(n) ]
+ for i in range(n1):
+ for j in range(n1):
+ p = (J1.monomial(i)*J1.monomial(j)).to_vector()
+ mult_table[i][j] = V(p.list() + [field.zero()]*n2)
+
+ for i in range(n2):
+ for j in range(n2):
+ p = (J2.monomial(i)*J2.monomial(j)).to_vector()
+ mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
+
+ super(DirectSumEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(J1.rank() + J2.rank())
+
+
+ def factors(self):
+ r"""
+ Return the pair of this algebra's factors.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(2,QQ)
+ sage: J2 = JordanSpinEJA(3,QQ)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: J.factors()
+ (Euclidean Jordan algebra of dimension 2 over Rational Field,
+ Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+ """
+ return self._factors
+
+ def projections(self):
+ r"""
+ Return a pair of projections onto this algebra's factors.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: ComplexHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = JordanSpinEJA(2)
+ sage: J2 = ComplexHermitianEJA(2)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: (pi_left, pi_right) = J.projections()
+ sage: J.one().to_vector()
+ (1, 0, 1, 0, 0, 1)
+ sage: pi_left(J.one()).to_vector()
+ (1, 0)
+ sage: pi_right(J.one()).to_vector()
+ (1, 0, 0, 1)
+
+ """
+ (J1,J2) = self.factors()
+ n = J1.dimension()
+ pi_left = lambda x: J1.from_vector(x.to_vector()[:n])
+ pi_right = lambda x: J2.from_vector(x.to_vector()[n:])
+ 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 (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)
+
+ """
+ (J1,J2) = self.factors()
+ n = J1.dimension()
+ V_basis = self.vector_space().basis()
+ I1 = matrix.column(self.base_ring(), V_basis[:n])
+ I2 = matrix.column(self.base_ring(), V_basis[n:])
+ iota_left = lambda x: self.from_vector(I1*x.to_vector())
+ iota_right = lambda x: self.from_vector(I2*+x.to_vector())
+ return (iota_left, iota_right)
+
+ def inner_product(self, x, y):
+ r"""
+ The standard Cartesian inner-product.
+
+ We project ``x`` and ``y`` onto our factors, and add up the
+ inner-products from the subalgebras.
+
+ SETUP::
+
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: QuaternionHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLE::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = QuaternionHermitianEJA(2,QQ,normalize_basis=False)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: x1 = J1.one()
+ sage: x2 = x1
+ sage: y1 = J2.one()
+ sage: y2 = y1
+ sage: x1.inner_product(x2)
+ 3
+ sage: y1.inner_product(y2)
+ 2
+ sage: J.one().inner_product(J.one())
+ 5
+
+ """
+ (pi_left, pi_right) = self.projections()
+ x1 = pi_left(x)
+ x2 = pi_right(x)
+ y1 = pi_left(y)
+ y2 = pi_right(y)
+
+ return (x1.inner_product(y1) + x2.inner_product(y2))
projects / sage.d.git / blobdiff