return A( (self.operator_matrix()**(n-1))*self.vector() )
+ def apply_univariate_polynomial(self, p):
+ """
+ Apply the univariate polynomial ``p`` to this element.
+
+ A priori, SageMath won't allow us to apply a univariate
+ polynomial to an element of an EJA, because we don't know
+ that EJAs are rings (they are usually not associative). Of
+ course, we know that EJAs are power-associative, so the
+ operation is ultimately kosher. This function sidesteps
+ the CAS to get the answer we want and expect.
+
+ EXAMPLES::
+
+ sage: R = PolynomialRing(QQ, 't')
+ sage: t = R.gen(0)
+ sage: p = t^4 - t^3 + 5*t - 2
+ sage: J = RealCartesianProductEJA(5)
+ sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
+ True
+
+ TESTS:
+
+ We should always get back an element of the algebra::
+
+ sage: set_random_seed()
+ sage: p = PolynomialRing(QQ, 't').random_element()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: x.apply_univariate_polynomial(p) in J
+ True
+
+ """
+ if len(p.variables()) > 1:
+ raise ValueError("not a univariate polynomial")
+ P = self.parent()
+ R = P.base_ring()
+ # Convert the coeficcients to the parent's base ring,
+ # because a priori they might live in an (unnecessarily)
+ # larger ring for which P.sum() would fail below.
+ cs = [ R(c) for c in p.coefficients(sparse=False) ]
+ return P.sum( cs[k]*(self**k) for k in range(len(cs)) )
+
+
def characteristic_polynomial(self):
"""
Return my characteristic polynomial (if I'm a regular
def minimal_polynomial(self):
"""
+ Return the minimal polynomial of this element,
+ as a function of the variable `t`.
+
ALGORITHM:
We restrict ourselves to the associative subalgebra
polynomial of this element's operator matrix (in that
subalgebra). This works by Baes Proposition 2.3.16.
- EXAMPLES::
+ TESTS:
+
+ The minimal polynomial of the identity and zero elements are
+ always the same::
sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.degree() == x.minimal_polynomial().degree()
- True
+ sage: J = random_eja()
+ sage: J.one().minimal_polynomial()
+ t - 1
+ sage: J.zero().minimal_polynomial()
+ t
- ::
+ The degree of an element is (by one definition) the degree
+ of its minimal polynomial::
sage: set_random_seed()
sage: x = random_eja().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: t = PolynomialRing(J.base_ring(),'t').gen(0)
+ sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
sage: bool(actual == expected)
True
+ The minimal polynomial should always kill its element::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: p = x.minimal_polynomial()
+ sage: x.apply_univariate_polynomial(p)
+ 0
+
"""
V = self.span_of_powers()
assoc_subalg = self.subalgebra_generated_by()
# and subalgebra_generated_by() must be the same, and in
# the same order!
elt = assoc_subalg(V.coordinates(self.vector()))
- return elt.operator_matrix().minimal_polynomial()
+
+ # We get back a symbolic polynomial in 'x' but want a real
+ # polynomial in 't'.
+ p_of_x = elt.operator_matrix().minimal_polynomial()
+ return p_of_x.change_variable_name('t')
def natural_representation(self):
Euclidean Jordan algebra of degree...
"""
- n = ZZ.random_element(1,5)
- constructor = choice([RealCartesianProductEJA,
- JordanSpinEJA,
- RealSymmetricEJA,
- ComplexHermitianEJA,
- QuaternionHermitianEJA])
+
+ # The max_n component lets us choose different upper bounds on the
+ # value "n" that gets passed to the constructor. This is needed
+ # because e.g. R^{10} is reasonable to test, while the Hermitian
+ # 10-by-10 quaternion matrices are not.
+ (constructor, max_n) = choice([(RealCartesianProductEJA, 6),
+ (JordanSpinEJA, 6),
+ (RealSymmetricEJA, 5),
+ (ComplexHermitianEJA, 4),
+ (QuaternionHermitianEJA, 3)])
+ n = ZZ.random_element(1, max_n)
return constructor(n, field=QQ)
projects / sage.d.git / blobdiff