From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 28 Feb 2021 21:43:02 +0000 (-0500)
Subject: eja: massively speed up the trace inner-product in matrix algebras.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=ece9fa6ee0724db795df27797ca8db7e112176f2;p=sage.d.git

eja: massively speed up the trace inner-product in matrix algebras.
---

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 5bb4cee..a765e97 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -1837,18 +1837,13 @@ class MatrixEJA:
             True
 
         """
-        Xu = cls.real_unembed(X)
-        Yu = cls.real_unembed(Y)
-        tr = (Xu*Yu).trace()
-
-        try:
-            # Works in QQ, AA, RDF, et cetera.
-            return tr.real()
-        except AttributeError:
-            # A quaternion doesn't have a real() method, but does
-            # have coefficient_tuple() method that returns the
-            # coefficients of 1, i, j, and k -- in that order.
-            return tr.coefficient_tuple()[0]
+        # This does in fact compute the real part of the trace.
+        # If we compute the trace of e.g. a complex matrix M,
+        # then we do so by adding up its diagonal entries --
+        # call them z_1 through z_n. The real embedding of z_1
+        # will be a 2-by-2 REAL matrix [a, b; -b, a] whose trace
+        # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
+        return (X*Y).trace()/cls.dimension_over_reals()
 
 
 class RealMatrixEJA(MatrixEJA):