F = domain_eja.base_ring()
if not (F == codomain_eja.base_ring()):
raise ValueError("domain and codomain must have the same base ring")
+ if not (F == mat.base_ring()):
+ raise ValueError("domain and matrix must have the same base ring")
# We need to supply something here to avoid getting the
# default Homset of the parent FiniteDimensionalAlgebra class,
[2 0 0]
[0 2 0]
[0 0 2]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
If you try to add two identical vector space operators but on
different EJAs, that should blow up::
sage: J1 = RealSymmetricEJA(2)
+ sage: id1 = identity_matrix(J1.base_ring(), 3)
sage: J2 = JordanSpinEJA(3)
- sage: id = identity_matrix(QQ, 3)
- sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J1,id)
- sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,J2,id)
+ sage: id2 = identity_matrix(J2.base_ring(), 3)
+ sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J1,id1)
+ sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,J2,id2)
sage: f + g
Traceback (most recent call last):
...
[1 0 0]
[0 1 0]
[0 0 1]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
"""
return FiniteDimensionalEuclideanJordanAlgebraOperator(
sage: x.operator()
Linear operator between finite-dimensional Euclidean Jordan algebras
represented by the matrix:
- [ 2 4 0]
+ [ 2 2 0]
[ 2 9 2]
- [ 0 4 16]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ [ 0 2 16]
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
sage: x.operator()*(1/2)
Linear operator between finite-dimensional Euclidean Jordan algebras
represented by the matrix:
- [ 1 2 0]
+ [ 1 1 0]
[ 1 9/2 1]
- [ 0 2 8]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ [ 0 1 8]
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
"""
- if other in self.codomain().base_ring():
- return FiniteDimensionalEuclideanJordanAlgebraOperator(
- self.domain(),
- self.codomain(),
- self.matrix()*other)
+ try:
+ if other in self.codomain().base_ring():
+ return FiniteDimensionalEuclideanJordanAlgebraOperator(
+ self.domain(),
+ self.codomain(),
+ self.matrix()*other)
+ except NotImplementedError:
+ # This can happen with certain arguments if the base_ring()
+ # is weird and doesn't know how to test membership.
+ pass
# This should eventually delegate to _composition_ after performing
# some sanity checks for us.
[-1 0 0]
[ 0 -1 0]
[ 0 0 -1]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
"""
return FiniteDimensionalEuclideanJordanAlgebraOperator(
[3 0 0]
[0 3 0]
[0 0 3]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
"""
if (n == 1):
[-1 0 0]
[ 0 -1 0]
[ 0 0 -1]
- Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
- Codomain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ Domain: Euclidean Jordan algebra of dimension 3 over...
+ Codomain: Euclidean Jordan algebra of dimension 3 over...
"""
return (self + (-other))
"""
# The matrix method returns a polynomial in 'x' but want one in 't'.
return self.matrix().minimal_polynomial().change_variable_name('t')
+
+
+ def spectral_decomposition(self):
+ """
+ Return the spectral decomposition of this operator as a list of
+ (eigenvalue, orthogonal projector) pairs.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(4,AA)
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: L0x = A(x).operator()
+ sage: Ps = [ P*l for (l,P) in L0x.spectral_decomposition() ]
+ sage: Ps[0] + Ps[1] == L0x
+ True
+
+ """
+ if not self.matrix().is_symmetric():
+ raise ValueError('algebra basis is not orthonormal')
+
+ D,P = self.matrix().jordan_form(subdivide=False,transformation=True)
+ eigenvalues = D.diagonal()
+ us = P.columns()
+ projectors = []
+ for i in range(len(us)):
+ # they won't be normalized, but they have to be
+ # for the spectral theorem to work.
+ us[i] = us[i]/us[i].norm()
+ mat = us[i].column()*us[i].row()
+ Pi = FiniteDimensionalEuclideanJordanAlgebraOperator(
+ self.domain(),
+ self.codomain(),
+ mat)
+ projectors.append(Pi)
+ return zip(eigenvalues, projectors)
projects / sage.d.git / blobdiff