+ return (self*other).operator().matrix().trace()
+
+
+ def operator_trace_norm(self):
+ """
+ The norm of this element with respect to
+ :meth:`operator_trace_inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: HadamardEJA)
+
+ EXAMPLES:
+
+ On a simple algebra, this will differ from :meth:`trace_norm`
+ by the scalar factor ``(n/r).sqrt()``, where `n` is the
+ dimension of the algebra and `r` its rank. This follows from
+ the corresponding result (Proposition III.4.2 of Faraut and
+ Korányi) for the trace inner product::
+
+ sage: J = HadamardEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.operator_trace_norm()
+ 1.414213562373095?
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.operator_trace_norm()
+ 4
+
+ """
+ return self.operator_trace_inner_product(self).sqrt()
+
+
+class CartesianProductParentEJAElement(EJAElement):
+ r"""
+ An intermediate class for elements that have a Cartesian
+ product as their parent algebra.
+
+ This is needed because the ``to_matrix`` method (which gives you a
+ representation from the superalgebra) needs to do special stuff
+ for Cartesian products. Specifically, an EJA subalgebra of a
+ Cartesian product EJA will not itself be a Cartesian product (it
+ has its own basis) -- but we want ``to_matrix()`` to be able to
+ give us a Cartesian product representation.
+ """
+ def to_matrix(self):
+ # An override is necessary to call our custom _scale().
+ B = self.parent().matrix_basis()
+ W = self.parent().matrix_space()
+
+ # Aaaaand linear combinations don't work in Cartesian
+ # product spaces, even though they provide a method with
+ # that name. This is hidden in a subclass because the
+ # _scale() function is slow.
+ pairs = zip(B, self.to_vector())
+ return W.sum( _scale(b, alpha) for (b,alpha) in pairs )
+
+class CartesianProductEJAElement(CartesianProductParentEJAElement):
+ def det(self):
+ r"""
+ Compute the determinant of this product-element using the
+ determianants of its factors.
+
+ This result Follows from the spectral decomposition of (say)
+ the pair `(x,y)` in terms of the Jordan frame `\left\{ (c_1,
+ 0),(c_2, 0),...,(0,d_1),(0,d_2),... \right\}.
+ """
+ from sage.misc.misc_c import prod
+ return prod( f.det() for f in self.cartesian_factors() )