eja: fix complex-unembedding with respect to 5537f4534.

[sage.d.git] / mjo / eja / euclidean_jordan_algebra.py

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index e30c34128b564499d0485c3d0f9fcb68a826e6c7..1e5ada2188c3b6b3834165f221ab20b56d5c1f98 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -94,6 +94,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         The inner product associated with this Euclidean Jordan algebra.
 
         Will default to the trace inner product if nothing else.
+
+        EXAMPLES:
+
+        The inner product must satisfy its axiom for this algebra to truly
+        be 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).inner_product(z) == y.inner_product(x*z)
+            True
+
         """
         if (not x in self) or (not y in self):
             raise TypeError("arguments must live in this algebra")
@@ -160,6 +174,51 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         An element of a Euclidean Jordan algebra.
         """
 
+        def __init__(self, A, elt=None):
+            """
+            EXAMPLES:
+
+            The identity in `S^n` is converted to the identity in the EJA::
+
+                sage: J = RealSymmetricSimpleEJA(3)
+                sage: I = identity_matrix(QQ,3)
+                sage: J(I) == J.one()
+                True
+
+            This skew-symmetric matrix can't be represented in the EJA::
+
+                sage: J = RealSymmetricSimpleEJA(3)
+                sage: A = matrix(QQ,3, lambda i,j: i-j)
+                sage: J(A)
+                Traceback (most recent call last):
+                ...
+                ArithmeticError: vector is not in free module
+
+            """
+            # Goal: if we're given a matrix, and if it lives in our
+            # parent algebra's "natural ambient space," convert it
+            # into an algebra element.
+            #
+            # The catch is, we make a recursive call after converting
+            # the given matrix into a vector that lives in the algebra.
+            # This we need to try the parent class initializer first,
+            # to avoid recursing forever if we're given something that
+            # already fits into the algebra, but also happens to live
+            # in the parent's "natural ambient space" (this happens with
+            # vectors in R^n).
+            try:
+                FiniteDimensionalAlgebraElement.__init__(self, A, elt)
+            except ValueError:
+                natural_basis = A.natural_basis()
+                if elt in natural_basis[0].matrix_space():
+                    # Thanks for nothing! Matrix spaces aren't vector
+                    # spaces in Sage, so we have to figure out its
+                    # natural-basis coordinates ourselves.
+                    V = VectorSpace(elt.base_ring(), elt.nrows()**2)
+                    W = V.span( _mat2vec(s) for s in natural_basis )
+                    coords =  W.coordinates(_mat2vec(elt))
+                    FiniteDimensionalAlgebraElement.__init__(self, A, coords)
+
         def __pow__(self, n):
             """
             Return ``self`` raised to the power ``n``.
@@ -1123,13 +1182,13 @@ def _embed_complex_matrix(M):
         sage: x2 = F(1 + 2*i)
         sage: x3 = F(-i)
         sage: x4 = F(6)
-        sage: M = matrix(F,2,[x1,x2,x3,x4])
+        sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
         sage: _embed_complex_matrix(M)
-        [ 4  2| 1 -2]
-        [-2  4| 2  1]
+        [ 4 -2| 1  2]
+        [ 2  4|-2  1]
         [-----+-----]
-        [ 0  1| 6  0]
-        [-1  0| 0  6]
+        [ 0 -1| 6  0]
+        [ 1  0| 0  6]
 
     """
     n = M.nrows()
@@ -1140,7 +1199,7 @@ def _embed_complex_matrix(M):
     for z in M.list():
         a = z.real()
         b = z.imag()
-        blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
+        blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
 
     # We can drop the imaginaries here.
     return block_matrix(field.base_ring(), n, blocks)
@@ -1157,8 +1216,17 @@ def _unembed_complex_matrix(M):
         ....:                 [ 9,  10, 11, 12],
         ....:                 [-10, 9, -12, 11] ])
         sage: _unembed_complex_matrix(A)
-        [  -2*i + 1   -4*i + 3]
-        [ -10*i + 9 -12*i + 11]
+        [  2*i + 1   4*i + 3]
+        [ 10*i + 9 12*i + 11]
+
+    TESTS::
+
+        sage: set_random_seed()
+        sage: F = QuadraticField(-1, 'i')
+        sage: M = random_matrix(F, 3)
+        sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
+        True
+
     """
     n = ZZ(M.nrows())
     if M.ncols() != n:
@@ -1179,7 +1247,7 @@ def _unembed_complex_matrix(M):
                 raise ValueError('bad real submatrix')
             if submat[0,1] != -submat[1,0]:
                 raise ValueError('bad imag submatrix')
-            z = submat[0,0] + submat[1,0]*i
+            z = submat[0,0] + submat[0,1]*i
             elements.append(z)
 
     return matrix(F, n/2, elements)