+ The zero element is never invertible in a non-trivial algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: (not J.is_trivial()) and J.zero().is_invertible()
+ False
+
+ Test that the fast (cached) and slow algorithms give the same
+ answer::
+
+ sage: set_random_seed() # long time
+ sage: J = random_eja(field=QQ, orthonormalize=False) # long time
+ sage: x = J.random_element() # long time
+ sage: slow = x.is_invertible() # long time
+ sage: _ = J._charpoly_coefficients() # long time
+ sage: fast = x.is_invertible() # long time
+ sage: slow == fast # long time
+ True
+ """
+ if self.is_zero():
+ if self.parent().is_trivial():
+ return True
+ else:
+ return False
+
+ if self.parent()._charpoly_coefficients.is_in_cache():
+ # The determinant will be quicker than inverting the
+ # quadratic representation, most likely.
+ return (not self.det().is_zero())
+
+ # The easiest way to determine if I'm invertible is to try.
+ try:
+ inv = (~self.quadratic_representation())(self)
+ self.inverse.set_cache(inv)
+ return True
+ except ZeroDivisionError:
+ return False
+
+
+ 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::