- def natural_representation(self):
- """
- Return a more-natural representation of this element.
-
- Every finite-dimensional Euclidean Jordan Algebra is a
- direct sum of five simple algebras, four of which comprise
- Hermitian matrices. This method returns the original
- "natural" representation of this element as a Hermitian
- matrix, if it has one. If not, you get the usual representation.
-
- EXAMPLES::
-
- sage: J = ComplexHermitianSimpleEJA(3)
- sage: J.one()
- e0 + e5 + e8
- sage: J.one().natural_representation()
- [1 0 0 0 0 0]
- [0 1 0 0 0 0]
- [0 0 1 0 0 0]
- [0 0 0 1 0 0]
- [0 0 0 0 1 0]
- [0 0 0 0 0 1]
-
- """
- B = self.parent().natural_basis()
- W = B[0].matrix_space()
- return W.linear_combination(zip(self.vector(), B))
-
-
- def minimal_polynomial(self):
- """
- EXAMPLES::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.degree() == x.minimal_polynomial().degree()
- True
-
- ::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.degree() == x.minimal_polynomial().degree()
- True
-
- The minimal polynomial and the characteristic polynomial coincide
- and are known (see Alizadeh, Example 11.11) for all elements of
- the spin factor algebra that aren't scalar multiples of the
- identity::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(2,10)
- sage: J = JordanSpinSimpleEJA(n)
- sage: y = J.random_element()
- sage: while y == y.coefficient(0)*J.one():
- ....: y = J.random_element()
- sage: y0 = y.vector()[0]
- sage: y_bar = y.vector()[1:]
- sage: actual = y.minimal_polynomial()
- sage: x = SR.symbol('x', domain='real')
- sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
- sage: bool(actual == expected)
- True
-
- """
- # The element we're going to call "minimal_polynomial()" on.
- # Either myself, interpreted as an element of a finite-
- # dimensional algebra, or an element of an associative
- # subalgebra.
- elt = None
-
- if self.parent().is_associative():
- elt = FiniteDimensionalAlgebraElement(self.parent(), self)
- else:
- V = self.span_of_powers()
- assoc_subalg = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- elt = assoc_subalg(V.coordinates(self.vector()))
-
- # Recursive call, but should work since elt lives in an
- # associative algebra.
- return elt.minimal_polynomial()
-
-