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 "
sage: idJ.inverse() == idJ
True
- The zero operator is never invertible::
+ The inverse of the inverse is the operator we started with::
sage: set_random_seed()
- sage: J = random_eja()
- sage: J.zero().operator().inverse()
- Traceback (most recent call last):
- ...
- ZeroDivisionError: input matrix must be nonsingular
+ sage: x = random_eja().random_element()
+ sage: L = x.operator()
+ sage: not L.is_invertible() or (L.inverse().inverse() == L)
+ True
"""
return ~self
SETUP::
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA, random_eja
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: TrivialEJA,
+ ....: random_eja)
EXAMPLES::
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: J.one().operator().is_invertible()
True
- The zero operator is never invertible::
+ The zero operator is never invertible in a nontrivial algebra::
sage: set_random_seed()
sage: J = random_eja()
- sage: J.zero().operator().is_invertible()
+ sage: not J.is_trivial() and J.zero().operator().is_invertible()
False
"""
EXAMPLES::
- sage: J = RealSymmetricEJA(4,AA)
+ sage: J = RealSymmetricEJA(4)
sage: x = sum(J.gens())
sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
sage: L0x = A(x).operator()
self.codomain(),
mat)
projectors.append(Pi)
- return zip(eigenvalues, projectors)
+ return list(zip(eigenvalues, projectors))