+
+ def characteristic_polynomial(self):
+ """
+ EXAMPLES:
+
+ The characteristic polynomial in the spin algebra is given in
+ Alizadeh, Example 11.11::
+
+ sage: J = JordanSpinEJA(3)
+ sage: p = J.characteristic_polynomial(); p
+ X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
+ sage: xvec = J.one().vector()
+ sage: p(*xvec)
+ t^2 - 2*t + 1
+
+ """
+ if self._charpoly is not None:
+ return self._charpoly
+
+ r = self.rank()
+ n = self.dimension()
+
+ # Now switch to the polynomial rings.
+ names = ['X' + str(i) for i in range(1,n+1)]
+ R = PolynomialRing(self.base_ring(), names)
+ J = FiniteDimensionalEuclideanJordanAlgebra(R,
+ self._multiplication_table,
+ rank=r)
+
+ def e(k):
+ # The coordinates of e_k with respect to the basis B.
+ # But, the e_k are elements of B...
+ return identity_matrix(J.base_ring(), n).column(k-1).column()
+
+ # A matrix implementation 1
+ x = J(vector(R, R.gens()))
+ l1 = [column_matrix((x**k).vector()) for k in range(r)]
+ l2 = [e(k) for k in range(r+1, n+1)]
+ A_of_x = block_matrix(1, n, (l1 + l2))
+ xr = (x**r).vector()
+ a = []
+ denominator = A_of_x.det() # This is constant
+ for i in range(n):
+ A_cols = A_of_x.columns()
+ A_cols[i] = xr
+ numerator = column_matrix(A_of_x.base_ring(), A_cols).det()
+ ai = numerator/denominator
+ a.append(ai)
+
+ # We go to a bit of trouble here to reorder the
+ # indeterminates, so that it's easier to evaluate the
+ # characteristic polynomial at x's coordinates and get back
+ # something in terms of t, which is what we want.
+ S = PolynomialRing(self.base_ring(),'t')
+ t = S.gen(0)
+ S = PolynomialRing(S, R.variable_names())
+ t = S(t)
+
+ # We're relying on the theory here to ensure that each entry
+ # a[i] is indeed back in R, and the added negative signs are
+ # to make the whole expression sum to zero.
+ a = [R(-ai) for ai in a] # corresponds to powerx x^0 through x^(r-1)
+
+ # Note: all entries past the rth should be zero. The
+ # coefficient of the highest power (x^r) is 1, but it doesn't
+ # appear in the solution vector which contains coefficients
+ # for the other powers (to make them sum to x^r).
+ if (r < n):
+ a[r] = 1 # corresponds to x^r
+ else:
+ # When the rank is equal to the dimension, trying to
+ # assign a[r] goes out-of-bounds.
+ a.append(1) # corresponds to x^r
+
+ self._charpoly = sum( a[k]*(t**k) for k in range(len(a)) )
+ return self._charpoly
+
+