class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism):
"""
- A very thin wrapper around FiniteDimensionalAlgebraMorphism that
- does only three things:
+ A linear map between two finite-dimensional EJAs.
+
+ This is a very thin wrapper around FiniteDimensionalAlgebraMorphism
+ that does only a few things:
1. Avoids the ``unitary`` and ``check`` arguments to the constructor
that will always be ``False``. This is necessary because these
2. Inputs and outputs the underlying matrix with respect to COLUMN
vectors, unlike the parent class.
- 3. Allows us to add morphisms in the obvious way.
-
- 4. Allows us to invert morphisms.
+ 3. Allows us to add, subtract, negate, multiply (compose), and
+ invert morphisms in the obvious way.
If this seems a bit heavyweight, it is. I would have been happy to
use a the ring morphism that underlies the finite-dimensional
algebra morphism, but they don't seem to be callable on elements of
- our EJA, and you can't add/invert them.
+ our EJA, and you can't add/multiply/etc. them.
"""
def __add__(self, other):
check=False)
+ def __invert__(self):
+ """
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
+ sage: x.is_invertible()
+ True
+ sage: ~x.operator()
+ Morphism from Euclidean Jordan algebra of degree 3 over Rational
+ Field to Euclidean Jordan algebra of degree 3 over Rational Field
+ given by matrix
+ [-3/2 2 -1/2]
+ [ 1 0 0]
+ [-1/2 0 1/2]
+ sage: x.operator_matrix().inverse()
+ [-3/2 2 -1/2]
+ [ 1 0 0]
+ [-1/2 0 1/2]
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: not x.is_invertible() or (
+ ....: (~x.operator()).matrix() == x.operator_matrix().inverse() )
+ True
+
+ """
+ A = self.matrix()
+ if not A.is_invertible():
+ raise ValueError("morphism is not invertible")
+
+ P = self.parent()
+ return FiniteDimensionalEuclideanJordanAlgebraMorphism(self.parent(),
+ A.inverse())
+
+ def __mul__(self, other):
+ """
+ Compose two EJA morphisms using multiplicative notation.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: x = J.zero()
+ sage: y = J.one()
+ sage: x.operator() * y.operator()
+ Morphism from Euclidean Jordan algebra of degree 6 over Rational
+ Field to Euclidean Jordan algebra of degree 6 over Rational Field
+ given by matrix
+ [0 0 0 0 0 0]
+ [0 0 0 0 0 0]
+ [0 0 0 0 0 0]
+ [0 0 0 0 0 0]
+ [0 0 0 0 0 0]
+ [0 0 0 0 0 0]
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: (x.operator() * y.operator()) in J.Hom(J)
+ True
+
+ """
+ if not other.codomain() is self.domain():
+ raise ValueError("(co)domains must agree for composition")
+
+ return FiniteDimensionalEuclideanJordanAlgebraMorphism(
+ self.parent(),
+ self.matrix()*other.matrix() )
+
+
+ def __neg__(self):
+ """
+ Negate this morphism.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: x = J.one()
+ sage: -x.operator()
+ Morphism from Euclidean Jordan algebra of degree 3 over Rational
+ Field to Euclidean Jordan algebra of degree 3 over Rational Field
+ given by matrix
+ [-1 0 0]
+ [ 0 -1 0]
+ [ 0 0 -1]
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: -x.operator() in J.Hom(J)
+ True
+
+ """
+ return FiniteDimensionalEuclideanJordanAlgebraMorphism(
+ self.parent(),
+ -self.matrix() )
+
+
def _repr_(self):
"""
We override only the representation that is shown to the user,
return "Morphism from {} to {} given by matrix\n{}".format(
self.domain(), self.codomain(), self.matrix())
+
+ def __sub__(self, other):
+ """
+ Subtract one morphism from another using addition and negation.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: L1 = J.one().operator()
+ sage: L1 - L1
+ Morphism from Euclidean Jordan algebra of degree 3 over Rational
+ Field to Euclidean Jordan algebra of degree 3 over Rational
+ Field given by matrix
+ [0 0 0]
+ [0 0 0]
+ [0 0 0]
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: x.operator() - y.operator() in J.Hom(J)
+ True
+
+ """
+ return self + (-other)
+
+
def matrix(self):
"""
Return the matrix of this morphism with respect to a left-action
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