+# -*- coding: utf-8 -*-
+
from sage.matrix.constructor import matrix
from sage.modules.free_module import VectorSpace
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
# TODO: make this unnecessary somehow.
from sage.misc.lazy_import import lazy_import
lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
-lazy_import('mjo.eja.eja_subalgebra',
+lazy_import('mjo.eja.eja_element_subalgebra',
'FiniteDimensionalEuclideanJordanElementSubalgebra')
from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
from mjo.eja.eja_utils import _mat2vec
Return ``self`` raised to the power ``n``.
Jordan algebras are always power-associative; see for
- example Faraut and Koranyi, Proposition II.1.2 (ii).
+ example Faraut and Korányi, Proposition II.1.2 (ii).
We have to override this because our superclass uses row
vectors instead of column vectors! We, on the other hand,
elif n == 1:
return self
else:
- return (self.operator()**(n-1))(self)
+ return (self**(n-1))*self
def apply_univariate_polynomial(self, p):
SETUP::
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
EXAMPLES::
sage: R = PolynomialRing(QQ, 't')
sage: t = R.gen(0)
sage: p = t^4 - t^3 + 5*t - 2
- sage: J = RealCartesianProductEJA(5)
+ sage: J = HadamardEJA(5)
sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
True
We should always get back an element of the algebra::
sage: set_random_seed()
- sage: p = PolynomialRing(QQ, 't').random_element()
+ sage: p = PolynomialRing(AA, 't').random_element()
sage: J = random_eja()
sage: x = J.random_element()
sage: x.apply_univariate_polynomial(p) in J
SETUP::
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+ sage: from mjo.eja.eja_algebra import HadamardEJA
EXAMPLES:
the identity element is `(t-1)` from which it follows that
the characteristic polynomial should be `(t-1)^3`::
- sage: J = RealCartesianProductEJA(3)
+ sage: J = HadamardEJA(3)
sage: J.one().characteristic_polynomial()
t^3 - 3*t^2 + 3*t - 1
Likewise, the characteristic of the zero element in the
rank-three algebra `R^{n}` should be `t^{3}`::
- sage: J = RealCartesianProductEJA(3)
+ sage: J = HadamardEJA(3)
sage: J.zero().characteristic_polynomial()
t^3
to zero on that element::
sage: set_random_seed()
- sage: x = RealCartesianProductEJA(3).random_element()
+ sage: x = HadamardEJA(3).random_element()
sage: p = x.characteristic_polynomial()
sage: x.apply_univariate_polynomial(p)
0
+ The characteristic polynomials of the zero and unit elements
+ should be what we think they are in a subalgebra, too::
+
+ sage: J = HadamardEJA(3)
+ sage: p1 = J.one().characteristic_polynomial()
+ sage: q1 = J.zero().characteristic_polynomial()
+ sage: e0,e1,e2 = J.gens()
+ sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
+ sage: p2 = A.one().characteristic_polynomial()
+ sage: q2 = A.zero().characteristic_polynomial()
+ sage: p1 == p2
+ True
+ sage: q1 == q2
+ True
+
"""
p = self.parent().characteristic_polynomial()
return p(*self.to_vector())
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: x.inner_product(y) in RR
+ sage: x,y = J.random_elements(2)
+ sage: x.inner_product(y) in RLF
True
"""
Test Lemma 1 from Chapter III of Koecher::
sage: set_random_seed()
- sage: J = random_eja()
- sage: u = J.random_element()
- sage: v = J.random_element()
+ sage: u,v = random_eja().random_elements(2)
sage: lhs = u.operator_commutes_with(u*v)
sage: rhs = v.operator_commutes_with(u^2)
sage: lhs == rhs
Chapter III, or from Baes (2.3)::
sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = random_eja().random_elements(2)
sage: Lx = x.operator()
sage: Ly = y.operator()
sage: Lxx = (x*x).operator()
Baes (2.4)::
sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: x,y,z = random_eja().random_elements(3)
sage: Lx = x.operator()
sage: Ly = y.operator()
sage: Lz = z.operator()
Baes (2.5)::
sage: set_random_seed()
- sage: J = random_eja()
- sage: u = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: u,y,z = random_eja().random_elements(3)
sage: Lu = u.operator()
sage: Ly = y.operator()
sage: Lz = z.operator()
sage: x.is_invertible() == (x.det() != 0)
True
+ Ensure that the determinant is multiplicative on an associative
+ subalgebra as in Faraut and Korányi's Proposition II.2.2::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: x,y = J.random_elements(2)
+ sage: (x*y).det() == x.det()*y.det()
+ True
+
"""
P = self.parent()
r = P.rank()
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
....: random_eja)
EXAMPLES:
Example 11.11::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinEJA(n)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: while not x.is_invertible():
....: x = J.random_element()
sage: x.inverse() == J.from_vector(x_inverse)
True
+ Trying to invert a non-invertible element throws an error:
+
+ sage: JordanSpinEJA(3).zero().inverse()
+ Traceback (most recent call last):
+ ...
+ ValueError: element is not invertible
+
TESTS:
The identity element is its own inverse::
sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
True
- The zero element is never invertible::
+ Proposition II.2.3 in Faraut and Korányi says that the inverse
+ of an element is the inverse of its left-multiplication operator
+ applied to the algebra's identity, when that inverse exists::
sage: set_random_seed()
- sage: J = random_eja().zero().inverse()
- Traceback (most recent call last):
- ...
- ValueError: element is not invertible
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: (not x.operator().is_invertible()) or (
+ ....: x.operator().inverse()(J.one()) == x.inverse() )
+ True
+
+ Proposition II.2.4 in Faraut and Korányi gives a formula for
+ the inverse based on the characteristic polynomial and the
+ Cayley-Hamilton theorem for Euclidean Jordan algebras::
+
+ sage: set_random_seed()
+ sage: J = ComplexHermitianEJA(3)
+ sage: x = J.random_element()
+ sage: while not x.is_invertible():
+ ....: x = J.random_element()
+ sage: r = J.rank()
+ sage: a = x.characteristic_polynomial().coefficients(sparse=False)
+ sage: expected = (-1)^(r+1)/x.det()
+ sage: expected *= sum( a[i+1]*x^i for i in range(r) )
+ sage: x.inverse() == expected
+ True
"""
if not self.is_invertible():
sage: J.one().is_invertible()
True
- The zero element is never invertible::
+ The zero element is never invertible in a non-trivial algebra::
sage: set_random_seed()
sage: J = random_eja()
- sage: J.zero().is_invertible()
+ sage: (not J.is_trivial()) and J.zero().is_invertible()
False
"""
- zero = self.parent().zero()
+ if self.is_zero():
+ if self.parent().is_trivial():
+ return True
+ else:
+ return False
+
+ # In fact, we only need to know if the constant term is non-zero,
+ # so we can pass in the field's zero element instead.
+ zero = self.base_ring().zero()
p = self.minimal_polynomial()
return not (p(zero) == zero)
+ def is_primitive_idempotent(self):
+ """
+ Return whether or not this element is a primitive (or minimal)
+ idempotent.
+
+ A primitive idempotent is a non-zero idempotent that is not
+ the sum of two other non-zero idempotents. Remark 2.7.15 in
+ Baes shows that this is what he refers to as a "minimal
+ idempotent."
+
+ An element of a Euclidean Jordan algebra is a minimal idempotent
+ if it :meth:`is_idempotent` and if its Peirce subalgebra
+ corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes,
+ Proposition 2.7.17).
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: TrivialEJA,
+ ....: random_eja)
+
+ WARNING::
+
+ This method is sloooooow.
+
+ EXAMPLES:
+
+ The spectral decomposition of a non-regular element should always
+ contain at least one non-minimal idempotent::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: x = sum(J.gens())
+ sage: x.is_regular()
+ False
+ sage: [ c.is_primitive_idempotent()
+ ....: for (l,c) in x.spectral_decomposition() ]
+ [False, True]
+
+ On the other hand, the spectral decomposition of a regular
+ element should always be in terms of minimal idempotents::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
+ sage: x.is_regular()
+ True
+ sage: [ c.is_primitive_idempotent()
+ ....: for (l,c) in x.spectral_decomposition() ]
+ [True, True]
+
+ TESTS:
+
+ The identity element is minimal only in an EJA of rank one::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.rank() == 1 or not J.one().is_primitive_idempotent()
+ True
+
+ A non-idempotent cannot be a minimal idempotent::
+
+ sage: set_random_seed()
+ sage: J = JordanSpinEJA(4)
+ sage: x = J.random_element()
+ sage: (not x.is_idempotent()) and x.is_primitive_idempotent()
+ False
+
+ Proposition 2.7.19 in Baes says that an element is a minimal
+ idempotent if and only if it's idempotent with trace equal to
+ unity::
+
+ sage: set_random_seed()
+ sage: J = JordanSpinEJA(4)
+ sage: x = J.random_element()
+ sage: expected = (x.is_idempotent() and x.trace() == 1)
+ sage: actual = x.is_primitive_idempotent()
+ sage: actual == expected
+ True
+
+ Primitive idempotents must be non-zero::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.zero().is_idempotent()
+ True
+ sage: J.zero().is_primitive_idempotent()
+ False
+
+ As a consequence of the fact that primitive idempotents must
+ be non-zero, there are no primitive idempotents in a trivial
+ Euclidean Jordan algebra::
+
+ sage: J = TrivialEJA()
+ sage: J.one().is_idempotent()
+ True
+ sage: J.one().is_primitive_idempotent()
+ False
+
+ """
+ if not self.is_idempotent():
+ return False
+
+ if self.is_zero():
+ return False
+
+ (_,_,J1) = self.parent().peirce_decomposition(self)
+ return (J1.dimension() == 1)
+
+
def is_nilpotent(self):
"""
Return whether or not some power of this element is zero.
TESTS:
- The identity element is never nilpotent::
+ The identity element is never nilpotent, except in a trivial EJA::
sage: set_random_seed()
- sage: random_eja().one().is_nilpotent()
+ sage: J = random_eja()
+ sage: J.one().is_nilpotent() and not J.is_trivial()
False
The additive identity is always nilpotent::
TESTS:
The zero element should never be regular, unless the parent
- algebra has dimension one::
+ algebra has dimension less than or equal to one::
sage: set_random_seed()
sage: J = random_eja()
- sage: J.dimension() == 1 or not J.zero().is_regular()
+ sage: J.dimension() <= 1 or not J.zero().is_regular()
True
The unit element isn't regular unless the algebra happens to
sage: set_random_seed()
sage: J = random_eja()
- sage: J.dimension() == 1 or not J.one().is_regular()
+ sage: J.dimension() <= 1 or not J.one().is_regular()
True
"""
aren't multiples of the identity are regular::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinEJA(n)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: x == x.coefficient(0)*J.one() or x.degree() == 2
True
TESTS:
- The zero and unit elements are both of degree one::
+ The zero and unit elements are both of degree one in nontrivial
+ algebras::
sage: set_random_seed()
sage: J = random_eja()
- sage: J.zero().degree()
- 1
- sage: J.one().degree()
- 1
+ sage: d = J.zero().degree()
+ sage: (J.is_trivial() and d == 0) or d == 1
+ True
+ sage: d = J.one().degree()
+ sage: (J.is_trivial() and d == 0) or d == 1
+ True
Our implementation agrees with the definition::
True
"""
+ if self.is_zero() and not self.parent().is_trivial():
+ # The minimal polynomial of zero in a nontrivial algebra
+ # is "t"; in a trivial algebra it's "1" by convention
+ # (it's an empty product).
+ return 1
return self.subalgebra_generated_by().dimension()
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: TrivialEJA,
....: random_eja)
+ EXAMPLES:
+
+ Keeping in mind that the polynomial ``1`` evaluates the identity
+ element (also the zero element) of the trivial algebra, it is clear
+ that the polynomial ``1`` is the minimal polynomial of the only
+ element in a trivial algebra::
+
+ sage: J = TrivialEJA()
+ sage: J.one().minimal_polynomial()
+ 1
+ sage: J.zero().minimal_polynomial()
+ 1
+
TESTS:
The minimal polynomial of the identity and zero elements are
always the same::
sage: set_random_seed()
- sage: J = random_eja()
+ sage: J = random_eja(nontrivial=True)
sage: J.one().minimal_polynomial()
t - 1
sage: J.zero().minimal_polynomial()
The minimal polynomial and the characteristic polynomial coincide
and are known (see Alizadeh, Example 11.11) for all elements of
the spin factor algebra that aren't scalar multiples of the
- identity::
+ identity. We require the dimension of the algebra to be at least
+ two here so that said elements actually exist::
sage: set_random_seed()
- sage: n = ZZ.random_element(2,10)
+ sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
+ sage: n = ZZ.random_element(2, n_max)
sage: J = JordanSpinEJA(n)
sage: y = J.random_element()
sage: while y == y.coefficient(0)*J.one():
sage: x.apply_univariate_polynomial(p)
0
+ The minimal polynomial is invariant under a change of basis,
+ and in particular, a re-scaling of the basis::
+
+ sage: set_random_seed()
+ sage: n_max = RealSymmetricEJA._max_test_case_size()
+ sage: n = ZZ.random_element(1, n_max)
+ sage: J1 = RealSymmetricEJA(n)
+ sage: J2 = RealSymmetricEJA(n,normalize_basis=False)
+ sage: X = random_matrix(AA,n)
+ sage: X = X*X.transpose()
+ sage: x1 = J1(X)
+ sage: x2 = J2(X)
+ sage: x1.minimal_polynomial() == x2.minimal_polynomial()
+ True
+
"""
+ if self.is_zero():
+ # We would generate a zero-dimensional subalgebra
+ # where the minimal polynomial would be constant.
+ # That might be correct, but only if *this* algebra
+ # is trivial too.
+ if not self.parent().is_trivial():
+ # Pretty sure we know what the minimal polynomial of
+ # the zero operator is going to be. This ensures
+ # consistency of e.g. the polynomial variable returned
+ # in the "normal" case without us having to think about it.
+ return self.operator().minimal_polynomial()
+
A = self.subalgebra_generated_by()
return A(self).operator().minimal_polynomial()
"""
B = self.parent().natural_basis()
- W = B[0].matrix_space()
+ W = self.parent().natural_basis_space()
return W.linear_combination(zip(B,self.to_vector()))
+ def norm(self):
+ """
+ The norm of this element with respect to :meth:`inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: HadamardEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.norm()
+ 1.414213562373095?
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.norm()
+ 2
+
+ """
+ return self.inner_product(self).sqrt()
+
+
def operator(self):
"""
Return the left-multiplication-by-this-element
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = J.random_elements(2)
sage: x.operator()(y) == x*y
True
sage: y.operator()(x) == x*y
"""
P = self.parent()
+ left_mult_by_self = lambda y: self*y
+ L = P.module_morphism(function=left_mult_by_self, codomain=P)
return FiniteDimensionalEuclideanJordanAlgebraOperator(
P,
P,
- self.to_matrix() )
+ L.matrix() )
def quadratic_representation(self, other=None):
Alizadeh's Example 11.12::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinEJA(n)
- sage: x = J.random_element()
+ sage: x = JordanSpinEJA.random_instance().random_element()
sage: x_vec = x.to_vector()
+ sage: n = x_vec.degree()
sage: x0 = x_vec[0]
sage: x_bar = x_vec[1:]
- sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
+ sage: A = matrix(AA, 1, [x_vec.inner_product(x_vec)])
sage: B = 2*x0*x_bar.row()
sage: C = 2*x0*x_bar.column()
- sage: D = matrix.identity(QQ, n-1)
+ sage: D = matrix.identity(AA, n-1)
sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
sage: D = D + 2*x_bar.tensor_product(x_bar)
sage: Q = matrix.block(2,2,[A,B,C,D])
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = J.random_elements(2)
sage: Lx = x.operator()
sage: Lxx = (x*x).operator()
sage: Qx = x.quadratic_representation()
Property 2 (multiply on the right for :trac:`28272`):
- sage: alpha = QQ.random_element()
+ sage: alpha = J.base_ring().random_element()
sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
True
sage: not x.is_invertible() or (
....: x.quadratic_representation(x.inverse())*Qx
....: ==
- ....: 2*x.operator()*Qex - Qx )
+ ....: 2*Lx*Qex - Qx )
True
- sage: 2*x.operator()*Qex - Qx == Lxx
+ sage: 2*Lx*Qex - Qx == Lxx
True
Property 5:
+ def spectral_decomposition(self):
+ """
+ Return the unique spectral decomposition of this element.
+
+ ALGORITHM:
+
+ Following Faraut and Korányi's Theorem III.1.1, we restrict this
+ element's left-multiplication-by operator to the subalgebra it
+ generates. We then compute the spectral decomposition of that
+ operator, and the spectral projectors we get back must be the
+ left-multiplication-by operators for the idempotents we
+ seek. Thus applying them to the identity element gives us those
+ idempotents.
+
+ Since the eigenvalues are required to be distinct, we take
+ the spectral decomposition of the zero element to be zero
+ times the identity element of the algebra (which is idempotent,
+ obviously).
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+
+ EXAMPLES:
+
+ The spectral decomposition of the identity is ``1`` times itself,
+ and the spectral decomposition of zero is ``0`` times the identity::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: J.one()
+ e0 + e2 + e5
+ sage: J.one().spectral_decomposition()
+ [(1, e0 + e2 + e5)]
+ sage: J.zero().spectral_decomposition()
+ [(0, e0 + e2 + e5)]
+
+ TESTS::
+
+ sage: J = RealSymmetricEJA(4)
+ sage: x = sum(J.gens())
+ sage: sd = x.spectral_decomposition()
+ sage: l0 = sd[0][0]
+ sage: l1 = sd[1][0]
+ sage: c0 = sd[0][1]
+ sage: c1 = sd[1][1]
+ sage: c0.inner_product(c1) == 0
+ True
+ sage: c0.is_idempotent()
+ True
+ sage: c1.is_idempotent()
+ True
+ sage: c0 + c1 == J.one()
+ True
+ sage: l0*c0 + l1*c1 == x
+ True
+
+ The spectral decomposition should work in subalgebras, too::
+
+ sage: J = RealSymmetricEJA(4)
+ sage: (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9) = J.gens()
+ sage: A = 2*e5 - 2*e8
+ sage: (lambda1, c1) = A.spectral_decomposition()[1]
+ sage: (J0, J5, J1) = J.peirce_decomposition(c1)
+ sage: (f0, f1, f2) = J1.gens()
+ sage: f0.spectral_decomposition()
+ [(0, 1.000000000000000?*f2), (1, 1.000000000000000?*f0)]
+
+ """
+ A = self.subalgebra_generated_by(orthonormalize_basis=True)
+ result = []
+ for (evalue, proj) in A(self).operator().spectral_decomposition():
+ result.append( (evalue, proj(A.one()).superalgebra_element()) )
+ return result
- def subalgebra_generated_by(self):
+ def subalgebra_generated_by(self, orthonormalize_basis=False):
"""
Return the associative subalgebra of the parent EJA generated
by this element.
+ Since our parent algebra is unital, we want "subalgebra" to mean
+ "unital subalgebra" as well; thus the subalgebra that an element
+ generates will itself be a Euclidean Jordan algebra after
+ restricting the algebra operations appropriately. This is the
+ subalgebra that Faraut and Korányi work with in section II.2, for
+ example.
+
SETUP::
sage: from mjo.eja.eja_algebra import random_eja
sage: set_random_seed()
sage: x0 = random_eja().random_element()
sage: A = x0.subalgebra_generated_by()
- sage: x = A.random_element()
- sage: y = A.random_element()
- sage: z = A.random_element()
+ sage: x,y,z = A.random_elements(3)
sage: (x*y)*z == x*(y*z)
True
sage: A(x^2) == A(x)*A(x)
True
+ By definition, the subalgebra generated by the zero element is
+ the one-dimensional algebra generated by the identity
+ element... unless the original algebra was trivial, in which
+ case the subalgebra is trivial too::
+
+ sage: set_random_seed()
+ sage: A = random_eja().zero().subalgebra_generated_by()
+ sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
+ True
+
"""
- return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
+ return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
def subalgebra_idempotent(self):
sage: from mjo.eja.eja_algebra import random_eja
- TESTS::
+ TESTS:
+
+ Ensure that we can find an idempotent in a non-trivial algebra
+ where there are non-nilpotent elements::
sage: set_random_seed()
- sage: J = random_eja()
+ sage: J = random_eja(nontrivial=True)
sage: x = J.random_element()
sage: while x.is_nilpotent():
....: x = J.random_element()
True
"""
+ if self.parent().is_trivial():
+ return self
+
if self.is_nilpotent():
raise ValueError("this only works with non-nilpotent elements!")
# will be minimal for some natural number s...
s = 0
minimal_dim = J.dimension()
- for i in xrange(1, minimal_dim):
+ for i in range(1, minimal_dim):
this_dim = (u**i).operator().matrix().image().dimension()
if this_dim < minimal_dim:
minimal_dim = this_dim
"""
Return my trace, the sum of my eigenvalues.
+ In a trivial algebra, however you want to look at it, the trace is
+ an empty sum for which we declare the result to be zero.
+
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: RealCartesianProductEJA,
+ ....: HadamardEJA,
+ ....: TrivialEJA,
....: random_eja)
EXAMPLES::
+ sage: J = TrivialEJA()
+ sage: J.zero().trace()
+ 0
+
+ ::
sage: J = JordanSpinEJA(3)
sage: x = sum(J.gens())
sage: x.trace()
::
- sage: J = RealCartesianProductEJA(5)
+ sage: J = HadamardEJA(5)
sage: J.one().trace()
5
sage: set_random_seed()
sage: J = random_eja()
- sage: J.random_element().trace() in J.base_ring()
+ sage: J.random_element().trace() in RLF
True
"""
P = self.parent()
r = P.rank()
+
+ if r == 0:
+ # Special case for the trivial algebra where
+ # the trace is an empty sum.
+ return P.base_ring().zero()
+
p = P._charpoly_coeff(r-1)
# The _charpoly_coeff function already adds the factor of
# -1 to ensure that _charpoly_coeff(r-1) is really what
TESTS:
- The trace inner product is commutative::
+ The trace inner product is commutative, bilinear, and associative::
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element(); y = J.random_element()
+ sage: x,y,z = J.random_elements(3)
+ sage: # commutative
sage: x.trace_inner_product(y) == y.trace_inner_product(x)
True
-
- The trace inner product is bilinear::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
- sage: a = QQ.random_element();
+ sage: # bilinear
+ sage: a = J.base_ring().random_element();
sage: actual = (a*(x+z)).trace_inner_product(y)
sage: expected = ( a*x.trace_inner_product(y) +
....: a*z.trace_inner_product(y) )
....: a*x.trace_inner_product(z) )
sage: actual == expected
True
-
- The trace inner product satisfies the compatibility
- condition in the definition of a Euclidean Jordan algebra::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: # associative
sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
True
raise TypeError("'other' must live in the same algebra")
return (self*other).trace()
+
+
+ def trace_norm(self):
+ """
+ The norm of this element with respect to :meth:`trace_inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: HadamardEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.trace_norm()
+ 1.414213562373095?
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.trace_norm()
+ 2.828427124746190?
+
+ """
+ return self.trace_inner_product(self).sqrt()