@cached_method
def characteristic_polynomial(self):
"""
+
+ .. WARNING::
+
+ This implementation doesn't guarantee that the polynomial
+ denominator in the coefficients is not identically zero, so
+ theoretically it could crash. The way that this is handled
+ in e.g. Faraut and Koranyi is to use a basis that guarantees
+ the denominator is non-zero. But, doing so requires knowledge
+ of at least one regular element, and we don't even know how
+ to do that. The trade-off is that, if we use the standard basis,
+ the resulting polynomial will accept the "usual" coordinates. In
+ other words, we don't have to do a change of basis before e.g.
+ computing the trace or determinant.
+
EXAMPLES:
The characteristic polynomial in the spin algebra is given in
def trace_inner_product(self, other):
"""
Return the trace inner product of myself and ``other``.
+
+ TESTS:
+
+ The trace inner product is commutative::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element(); y = J.random_element()
+ sage: x.trace_inner_product(y) == y.trace_inner_product(x)
+ True
+
+ The trace inner product is bilinear::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: a = QQ.random_element();
+ sage: actual = (a*(x+z)).trace_inner_product(y)
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*z.trace_inner_product(y) )
+ sage: actual == expected
+ True
+ sage: actual = x.trace_inner_product(a*(y+z))
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*x.trace_inner_product(z) )
+ sage: actual == expected
+ True
+
+ The trace inner product satisfies the compatibility
+ condition in the definition of a Euclidean Jordan algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
+ True
+
"""
if not other in self.parent():
raise TypeError("'other' must live in the same algebra")