eja: add "normalize" argument to matrix algebra constructors.

This is useful for two reasons:

  1. It's nice to be able to test that some things are invariant
     under changes of basis.

  2. The min/charpoly computations will be a lot faster if we
     can use the basis over QQ (i.e. if the properties that we're
     testing in the first item hold).

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 05dea56e6c389a25713b0f99d95663c8d2998a0d..adf9581c541bf81e02c612d281a4d671c3b9df74 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -802,7 +802,7 @@ def random_eja():
 
 
 
-def _real_symmetric_basis(n, field):
+def _real_symmetric_basis(n, field, normalize):
     """
     Return a basis for the space of real symmetric n-by-n matrices.
 
@@ -814,7 +814,7 @@ def _real_symmetric_basis(n, field):
 
         sage: set_random_seed()
         sage: n = ZZ.random_element(1,5)
-        sage: B = _real_symmetric_basis(n, QQbar)
+        sage: B = _real_symmetric_basis(n, QQbar, False)
         sage: all( M.is_symmetric() for M in  B)
         True
 
@@ -829,13 +829,13 @@ def _real_symmetric_basis(n, field):
                 Sij = Eij
             else:
                 Sij = Eij + Eij.transpose()
-            # Now normalize it.
-            Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt()
+            if normalize:
+                Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt()
             S.append(Sij)
     return tuple(S)
 
 
-def _complex_hermitian_basis(n, field):
+def _complex_hermitian_basis(n, field, normalize):
     """
     Returns a basis for the space of complex Hermitian n-by-n matrices.
 
@@ -854,7 +854,7 @@ def _complex_hermitian_basis(n, field):
         sage: set_random_seed()
         sage: n = ZZ.random_element(1,5)
         sage: field = QuadraticField(2, 'sqrt2')
-        sage: B = _complex_hermitian_basis(n, field)
+        sage: B = _complex_hermitian_basis(n, field, False)
         sage: all( M.is_symmetric() for M in  B)
         True
 
@@ -884,15 +884,17 @@ def _complex_hermitian_basis(n, field):
                 Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
                 S.append(Sij_imag)
 
-    # Normalize these with our inner product before handing them back.
-    # And since we embedded them, we can drop back to the "field" that
-    # we started with instead of the complex extension "F".
-    return tuple( (s / _complex_hermitian_matrix_ip(s,s).sqrt()).change_ring(field)
-                  for s in S )
+    # Since we embedded these, we can drop back to the "field" that we
+    # started with instead of the complex extension "F".
+    S = [ s.change_ring(field) for s in S ]
+    if normalize:
+        S = [ s / _complex_hermitian_matrix_ip(s,s).sqrt() for s in S ]
+
+    return tuple(S)
 
 
 
-def _quaternion_hermitian_basis(n, field):
+def _quaternion_hermitian_basis(n, field, normalize):
     """
     Returns a basis for the space of quaternion Hermitian n-by-n matrices.
 
@@ -910,7 +912,7 @@ def _quaternion_hermitian_basis(n, field):
 
         sage: set_random_seed()
         sage: n = ZZ.random_element(1,5)
-        sage: B = _quaternion_hermitian_basis(n, QQ)
+        sage: B = _quaternion_hermitian_basis(n, QQ, False)
         sage: all( M.is_symmetric() for M in B )
         True
 
@@ -1307,7 +1309,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
         True
 
     """
-    def __init__(self, n, field=QQ, **kwargs):
+    def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
         if n > 1:
             # We'll need sqrt(2) to normalize the basis, and this
             # winds up in the multiplication table, so the whole
@@ -1318,7 +1320,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
             if p.is_irreducible():
                 field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
 
-        S = _real_symmetric_basis(n, field)
+        S = _real_symmetric_basis(n, field, normalize_basis)
         Qs = _multiplication_table_from_matrix_basis(S)
 
         fdeja = super(RealSymmetricEJA, self)
@@ -1405,7 +1407,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
         True
 
     """
-    def __init__(self, n, field=QQ, **kwargs):
+    def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
         if n > 1:
             # We'll need sqrt(2) to normalize the basis, and this
             # winds up in the multiplication table, so the whole
@@ -1416,7 +1418,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
             if p.is_irreducible():
                 field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
 
-        S = _complex_hermitian_basis(n, field)
+        S = _complex_hermitian_basis(n, field, normalize_basis)
         Qs = _multiplication_table_from_matrix_basis(S)
 
         fdeja = super(ComplexHermitianEJA, self)
@@ -1487,8 +1489,8 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
         True
 
     """
-    def __init__(self, n, field=QQ, **kwargs):
-        S = _quaternion_hermitian_basis(n, field)
+    def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
+        S = _quaternion_hermitian_basis(n, field, normalize_basis)
         Qs = _multiplication_table_from_matrix_basis(S)
 
         fdeja = super(QuaternionHermitianEJA, self)