class Element(FiniteDimensionalAlgebraElement):
"""
An element of a Euclidean Jordan algebra.
+
+ Since EJAs are commutative, the "right multiplication" matrix is
+ also the left multiplication matrix and must be symmetric::
+
+ sage: set_random_seed()
+ sage: J = eja_ln(5)
+ sage: J.random_element().matrix().is_symmetric()
+ True
+
"""
def __pow__(self, n):
return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+ def span_of_powers(self):
+ """
+ Return the vector space spanned by successive powers of
+ this element.
+ """
+ # The dimension of the subalgebra can't be greater than
+ # the big algebra, so just put everything into a list
+ # and let span() get rid of the excess.
+ V = self.vector().parent()
+ return V.span( (self**d).vector() for d in xrange(V.dimension()) )
+
+
def degree(self):
"""
Compute the degree of this element the straightforward way
sage: (e0 - e1).degree()
2
+ In the spin factor algebra (of rank two), all elements that
+ aren't multiples of the identity are regular::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x == x.coefficient(0)*J.one() or x.degree() == 2
+ True
+
"""
- d = 0
- V = self.vector().parent()
- vectors = [(self**d).vector()]
- while V.span(vectors).dimension() > d:
- d += 1
- vectors.append((self**d).vector())
- return d
+ return self.span_of_powers().dimension()
+
+
+ def subalgebra_generated_by(self):
+ """
+ Return the subalgebra of the parent EJA generated by this element.
+ """
+ # First get the subspace spanned by the powers of myself...
+ V = self.span_of_powers()
+ F = self.base_ring()
+
+ # Now figure out the entries of the right-multiplication
+ # matrix for the successive basis elements b0, b1,... of
+ # that subspace.
+ mats = []
+ for b_right in V.basis():
+ eja_b_right = self.parent()(b_right)
+ b_right_rows = []
+ # The first row of the right-multiplication matrix by
+ # b1 is what we get if we apply that matrix to b1. The
+ # second row of the right multiplication matrix by b1
+ # is what we get when we apply that matrix to b2...
+ for b_left in V.basis():
+ eja_b_left = self.parent()(b_left)
+ # Multiply in the original EJA, but then get the
+ # coordinates from the subalgebra in terms of its
+ # basis.
+ this_row = V.coordinates((eja_b_left*eja_b_right).vector())
+ b_right_rows.append(this_row)
+ b_right_matrix = matrix(F, b_right_rows)
+ mats.append(b_right_matrix)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(F, mats)
+
def minimal_polynomial(self):
- return self.matrix().minimal_polynomial()
+ """
+ EXAMPLES::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(n)
+ sage: x = J.random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.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).abs()
+ sage: J = eja_ln(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
+
+ """
+ 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!
+ subalg_self = assoc_subalg(V.coordinates(self.vector()))
+ return subalg_self.matrix().minimal_polynomial()
+
def characteristic_polynomial(self):
return self.matrix().characteristic_polynomial()
Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
for i in xrange(dimension) ]
- # Assuming associativity is wrong here, but it works to
- # temporarily trick the Jordan algebra constructor into using the
- # multiplication table.
return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
projects / sage.d.git / blobdiff