eja: add a function to embed complex matrices in (bigger) real ones.

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 16ce256f244e96d7875bda4e3a27d005ae4b3c40..fdaccba58a8b99a2f5222054358969ce3e731882 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -727,6 +727,40 @@ def _multiplication_table_from_matrix_basis(basis):
     return Qs
 
 
+def _embed_complex_matrix(M):
+    """
+    Embed the n-by-n complex matrix ``M`` into the space of real
+    matrices of size 2n-by-2n via the map the sends each entry `z = a +
+    bi` to the block matrix ``[[a,b],[-b,a]]``.
+
+    EXAMPLES::
+
+        sage: F = QuadraticField(-1,'i')
+        sage: x1 = F(4 - 2*i)
+        sage: x2 = F(1 + 2*i)
+        sage: x3 = F(-i)
+        sage: x4 = F(6)
+        sage: M = matrix(F,2,[x1,x2,x3,x4])
+        sage: _embed_complex_matrix(M)
+        [ 4  2| 1 -2]
+        [-2  4| 2  1]
+        [-----+-----]
+        [ 0  1| 6  0]
+        [-1  0| 0  6]
+
+    """
+    n = M.nrows()
+    if M.ncols() != n:
+        raise ArgumentError("the matrix 'M' must be square")
+    field = M.base_ring()
+    blocks = []
+    for z in M.list():
+        a = z.real()
+        b = z.imag()
+        blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
+    return block_matrix(field, n, blocks)
+
+
 def RealSymmetricSimpleEJA(n):
     """
     The rank-n simple EJA consisting of real symmetric n-by-n