From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 19 Jul 2019 23:11:59 +0000 (-0400)
Subject: eja: add quaternion -> real matrix embedding.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=0f1a66728d639e4ebd1f3867dbc94e2232397606;p=sage.d.git

eja: add quaternion -> real matrix embedding.
---

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 1e5ada2..7c7c2d6 100644
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -1244,14 +1244,109 @@ def _unembed_complex_matrix(M):
         for j in xrange(n/2):
             submat = M[2*k:2*k+2,2*j:2*j+2]
             if submat[0,0] != submat[1,1]:
-                raise ValueError('bad real submatrix')
+                raise ValueError('bad on-diagonal submatrix')
             if submat[0,1] != -submat[1,0]:
-                raise ValueError('bad imag submatrix')
+                raise ValueError('bad off-diagonal submatrix')
             z = submat[0,0] + submat[0,1]*i
             elements.append(z)
 
     return matrix(F, n/2, elements)
 
+
+def _embed_quaternion_matrix(M):
+    """
+    Embed the n-by-n quaternion matrix ``M`` into the space of real
+    matrices of size 4n-by-4n by first sending each quaternion entry
+    `z = a + bi + cj + dk` to the block-complex matrix
+    ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
+    a real matrix.
+
+    EXAMPLES::
+
+        sage: Q = QuaternionAlgebra(QQ,-1,-1)
+        sage: i,j,k = Q.gens()
+        sage: x = 1 + 2*i + 3*j + 4*k
+        sage: M = matrix(Q, 1, [[x]])
+        sage: _embed_quaternion_matrix(M)
+        [ 1  2  3  4]
+        [-2  1 -4  3]
+        [-3  4  1 -2]
+        [-4 -3  2  1]
+
+    """
+    quaternions = M.base_ring()
+    n = M.nrows()
+    if M.ncols() != n:
+        raise ValueError("the matrix 'M' must be square")
+
+    F = QuadraticField(-1, 'i')
+    i = F.gen()
+
+    blocks = []
+    for z in M.list():
+        t = z.coefficient_tuple()
+        a = t[0]
+        b = t[1]
+        c = t[2]
+        d = t[3]
+        cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
+                                    [-c + d*i, a - b*i]])
+        blocks.append(_embed_complex_matrix(cplx_matrix))
+
+    # We should have real entries by now, so use the realest field
+    # we've got for the return value.
+    return block_matrix(quaternions.base_ring(), n, blocks)
+
+
+def _unembed_quaternion_matrix(M):
+    """
+    The inverse of _embed_quaternion_matrix().
+
+    EXAMPLES::
+
+        sage: M = matrix(QQ, [[ 1,  2,  3,  4],
+        ....:                 [-2,  1, -4,  3],
+        ....:                 [-3,  4,  1, -2],
+        ....:                 [-4, -3,  2,  1]])
+        sage: _unembed_quaternion_matrix(M)
+        [1 + 2*i + 3*j + 4*k]
+
+    TESTS::
+
+        sage: set_random_seed()
+        sage: Q = QuaternionAlgebra(QQ, -1, -1)
+        sage: M = random_matrix(Q, 3)
+        sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
+        True
+
+    """
+    n = ZZ(M.nrows())
+    if M.ncols() != n:
+        raise ValueError("the matrix 'M' must be square")
+    if not n.mod(4).is_zero():
+        raise ValueError("the matrix 'M' must be a complex embedding")
+
+    Q = QuaternionAlgebra(QQ,-1,-1)
+    i,j,k = Q.gens()
+
+    # Go top-left to bottom-right (reading order), converting every
+    # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
+    # quaternion block.
+    elements = []
+    for l in xrange(n/4):
+        for m in xrange(n/4):
+            submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
+            if submat[0,0] != submat[1,1].conjugate():
+                raise ValueError('bad on-diagonal submatrix')
+            if submat[0,1] != -submat[1,0].conjugate():
+                raise ValueError('bad off-diagonal submatrix')
+            z  = submat[0,0].real() + submat[0,0].imag()*i
+            z += submat[0,1].real()*j + submat[0,1].imag()*k
+            elements.append(z)
+
+    return matrix(Q, n/4, elements)
+
+
 # The usual inner product on R^n.
 def _usual_ip(x,y):
     return x.vector().inner_product(y.vector())