eja: add example of Gram-Schmidt with matrices.

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 4d70e062c44e98ba90584fb0f97d3bbfef91e491..b6e0c7d38eaf1d9114973bba4f5ad0674cb8a8a2 100644 (file)
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -1,9 +1,6 @@
 from sage.functions.other import sqrt
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
-from sage.rings.number_field.number_field import NumberField
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.real_lazy import RLF
 
 def _mat2vec(m):
         return vector(m.base_ring(), m.list())
@@ -64,6 +61,23 @@ def gram_schmidt(v, inner_product=None):
         sage: ip(u[1],u[2]).is_zero()
         True
 
+    This Gram-Schmidt routine can be used on matrices as well, so long
+    as an appropriate inner-product is provided::
+
+        sage: E11 = matrix(QQ, [ [1,0],
+        ....:                    [0,0] ])
+        sage: E12 = matrix(QQ, [ [0,1],
+        ....:                    [1,0] ])
+        sage: E22 = matrix(QQ, [ [0,0],
+        ....:                    [0,1] ])
+        sage: I = matrix.identity(QQ,2)
+        sage: trace_ip = lambda X,Y: (X*Y).trace()
+        sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
+        [
+        [1 0]  [          0 1/2*sqrt(2)]  [0 0]
+        [0 0], [1/2*sqrt(2)           0], [0 1]
+        ]
+
     TESTS:
 
     Ensure that zero vectors don't get in the way::