from sage.categories.map import Map
class FiniteDimensionalEuclideanJordanAlgebraOperator(Map):
+ r"""
+ An operator between two finite-dimensional Euclidean Jordan algebras.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import HadamardEJA
+ sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
+
+ EXAMPLES:
+
+ The domain and codomain must be finite-dimensional Euclidean
+ Jordan algebras; if either is not, then an error is raised::
+
+ sage: J = HadamardEJA(3)
+ sage: V = VectorSpace(J.base_ring(), 3)
+ sage: M = matrix.identity(J.base_ring(), 3)
+ sage: FiniteDimensionalEuclideanJordanAlgebraOperator(V,J,M)
+ Traceback (most recent call last):
+ ...
+ TypeError: domain must be a finite-dimensional Euclidean
+ Jordan algebra
+ sage: FiniteDimensionalEuclideanJordanAlgebraOperator(J,V,M)
+ Traceback (most recent call last):
+ ...
+ TypeError: codomain must be a finite-dimensional Euclidean
+ Jordan algebra
+
+ """
+
def __init__(self, domain_eja, codomain_eja, mat):
- # if not (
- # isinstance(domain_eja, FiniteDimensionalEuclideanJordanAlgebra) and
- # isinstance(codomain_eja, FiniteDimensionalEuclideanJordanAlgebra) ):
- # raise ValueError('(co)domains must be finite-dimensional Euclidean '
- # 'Jordan algebras')
+ from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
+
+ # I guess we should check this, because otherwise you could
+ # pass in pretty much anything algebraish.
+ if not isinstance(domain_eja,
+ FiniteDimensionalEuclideanJordanAlgebra):
+ raise TypeError('domain must be a finite-dimensional '
+ 'Euclidean Jordan algebra')
+ if not isinstance(codomain_eja,
+ FiniteDimensionalEuclideanJordanAlgebra):
+ raise TypeError('codomain must be a finite-dimensional '
+ 'Euclidean Jordan algebra')
F = domain_eja.base_ring()
if not (F == codomain_eja.base_ring()):
sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
sage: from mjo.eja.eja_algebra import (
....: JordanSpinEJA,
- ....: RealCartesianProductEJA,
+ ....: HadamardEJA,
....: RealSymmetricEJA)
EXAMPLES::
sage: J1 = JordanSpinEJA(3)
- sage: J2 = RealCartesianProductEJA(2)
+ sage: J2 = HadamardEJA(2)
sage: J3 = RealSymmetricEJA(1)
- sage: mat1 = matrix(QQ, [[1,2,3],
+ sage: mat1 = matrix(AA, [[1,2,3],
....: [4,5,6]])
- sage: mat2 = matrix(QQ, [[7,8]])
+ sage: mat2 = matrix(AA, [[7,8]])
sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,
....: J2,
....: mat1)
algebras represented by the matrix:
[39 54 69]
Domain: Euclidean Jordan algebra of dimension 3 over
- Rational Field
+ Algebraic Real Field
Codomain: Euclidean Jordan algebra of dimension 1 over
- Rational Field
+ Algebraic Real Field
"""
return FiniteDimensionalEuclideanJordanAlgebraOperator(
[1 0]
[0 1]
Domain: Euclidean Jordan algebra of dimension 2 over
- Rational Field
+ Algebraic Real Field
Codomain: Euclidean Jordan algebra of dimension 2 over
- Rational Field
+ Algebraic Real Field
"""
msg = ("Linear operator between finite-dimensional Euclidean Jordan "
return (self + (-other))
+ def is_self_adjoint(self):
+ r"""
+ Return whether or not this operator is self-adjoint.
+
+ At least in Sage, the fact that the base field is real means
+ that the algebra elements have to be real as well (this is why
+ we embed the complex numbers and quaternions). As a result, the
+ matrix of this operator will contain only real entries, and it
+ suffices to check only symmetry, not conjugate symmetry.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA)
+
+ EXAMPLES::
+
+ sage: J = JordanSpinEJA(4)
+ sage: J.one().operator().is_self_adjoint()
+ True
+
+ """
+ return self.matrix().is_symmetric()
+
+
+ def is_zero(self):
+ r"""
+ Return whether or not this map is the zero operator.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES::
+
+ sage: J1 = JordanSpinEJA(2)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: R = J1.base_ring()
+ sage: M = matrix(R, [ [0, 0],
+ ....: [0, 0],
+ ....: [0, 0] ])
+ sage: L = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J2,M)
+ sage: L.is_zero()
+ True
+ sage: M = matrix(R, [ [0, 0],
+ ....: [0, 1],
+ ....: [0, 0] ])
+ sage: L = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J2,M)
+ sage: L.is_zero()
+ False
+
+ TESTS:
+
+ The left-multiplication-by-zero operation on a given algebra
+ is its zero map::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.zero().operator().is_zero()
+ True
+
+ """
+ return self.matrix().is_zero()
+
+
+ def inverse(self):
+ """
+ Return the inverse of this operator, if it exists.
+
+ The reason this method is not simply an alias for the built-in
+ :meth:`__invert__` is that the built-in inversion is a bit magic
+ since it's intended to be a unary operator. If we alias ``inverse``
+ to ``__invert__``, then we wind up having to call e.g. ``A.inverse``
+ without parentheses.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA, random_eja
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.operator().inverse().matrix()
+ [3/2 -1 1/2]
+ [ -1 2 -1]
+ [1/2 -1 3/2]
+ sage: x.operator().matrix().inverse()
+ [3/2 -1 1/2]
+ [ -1 2 -1]
+ [1/2 -1 3/2]
+
+ TESTS:
+
+ The identity operator is its own inverse::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: idJ = J.one().operator()
+ sage: idJ.inverse() == idJ
+ True
+
+ The inverse of the inverse is the operator we started with::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: L = x.operator()
+ sage: not L.is_invertible() or (L.inverse().inverse() == L)
+ True
+
+ """
+ return ~self
+
+
+ def is_invertible(self):
+ """
+ Return whether or not this operator is invertible.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: TrivialEJA,
+ ....: random_eja)
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.operator().matrix()
+ [ 1 1/2 0]
+ [1/2 1 1/2]
+ [ 0 1/2 1]
+ sage: x.operator().matrix().is_invertible()
+ True
+ sage: x.operator().is_invertible()
+ True
+
+ The zero operator is invertible in a trivial algebra::
+
+ sage: J = TrivialEJA()
+ sage: J.zero().operator().is_invertible()
+ True
+
+ TESTS:
+
+ The identity operator is always invertible::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.one().operator().is_invertible()
+ True
+
+ The zero operator is never invertible in a nontrivial algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: not J.is_trivial() and J.zero().operator().is_invertible()
+ False
+
+ """
+ return self.matrix().is_invertible()
+
+
def matrix(self):
"""
Return the matrix representation of this operator with respect
"""
# 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.
+
+ This is the unique spectral decomposition, up to the order of
+ the projection operators, with distinct eigenvalues. So, the
+ projections are generally onto subspaces of dimension greater
+ than one.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(4)
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: L0x = A(x).operator()
+ sage: sd = L0x.spectral_decomposition()
+ sage: l0 = sd[0][0]
+ sage: l1 = sd[1][0]
+ sage: P0 = sd[0][1]
+ sage: P1 = sd[1][1]
+ sage: P0*l0 + P1*l1 == L0x
+ True
+ sage: P0 + P1 == P0^0 # the identity
+ True
+ sage: P0^2 == P0
+ True
+ sage: P1^2 == P1
+ True
+ sage: P0*P1 == A.zero().operator()
+ True
+ sage: P1*P0 == A.zero().operator()
+ 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 list(zip(eigenvalues, projectors))
projects / sage.d.git / blobdiff