matrix_algebra: factor out Hurwitz subclass.

diff --git a/mjo/matrix_algebra.py b/mjo/matrix_algebra.py
index 668a1c44d839e6b5704aa72acb90970136eca571..4049ef653a8a2688c2b9d4399f2b5386212aae47 100644 (file)
--- a/mjo/matrix_algebra.py
+++ b/mjo/matrix_algebra.py
@@ -50,11 +50,11 @@ class MatrixAlgebraElement(IndexedFreeModuleElement):
 
         EXAMPLES::
 
-            sage: MatrixAlgebra(ZZ,ZZ,2).one()
+            sage: MatrixAlgebra(ZZ,ZZ,2).zero()
             +---+---+
-            | 1 | 0 |
+            | 0 | 0 |
             +---+---+
-            | 0 | 1 |
+            | 0 | 0 |
             +---+---+
 
         """
@@ -71,8 +71,9 @@ class MatrixAlgebraElement(IndexedFreeModuleElement):
 
         EXAMPLES::
 
-            sage: MatrixAlgebra(ZZ,ZZ,2).one().list()
-            [1, 0, 0, 1]
+            sage: A = MatrixAlgebra(ZZ,ZZ,2)
+            sage: A([[1,2],[3,4]]).list()
+            [1, 2, 3, 4]
 
         """
         return sum( self.rows(), [] )
@@ -87,15 +88,15 @@ class MatrixAlgebraElement(IndexedFreeModuleElement):
 
         EXAMPLES::
 
-            sage: M = MatrixAlgebra(ZZ,ZZ,2).one()
+            sage: M = MatrixAlgebra(ZZ,ZZ,2)([[1,2],[3,4]])
             sage: M[0,0]
             1
             sage: M[0,1]
-            0
+            2
             sage: M[1,0]
-            0
+            3
             sage: M[1,1]
-            1
+            4
 
         """
         i,j = indices
@@ -117,7 +118,9 @@ class MatrixAlgebraElement(IndexedFreeModuleElement):
             sage: entries = MatrixSpace(ZZ,2)
             sage: scalars = ZZ
             sage: M = MatrixAlgebra(entries, scalars, 2)
-            sage: M.one().trace()
+            sage: I = entries.one()
+            sage: Z = entries.zero()
+            sage: M([[I,Z],[Z,I]]).trace()
             [2 0]
             [0 2]
 
@@ -143,25 +146,6 @@ class MatrixAlgebraElement(IndexedFreeModuleElement):
         """
         return self.parent()
 
-    # onlt valid in HurwitzMatrixAlgebra subclass
-    # def is_hermitian(self):
-    #     r"""
-
-    #     SETUP::
-
-    #         sage: from mjo.octonions import OctonionMatrixAlgebra
-
-    #     EXAMPLES::
-
-    #         sage: MS = OctonionMatrixAlgebra(3)
-    #         sage: MS.one().is_hermitian()
-    #         True
-
-    #     """
-    #     return all( self[i,j] == self[j,i].conjugate()
-    #                 for i in range(self.nrows())
-    #                 for j in range(self.ncols()) )
-
 
 class MatrixAlgebra(CombinatorialFreeModule):
     r"""
@@ -234,18 +218,6 @@ class MatrixAlgebra(CombinatorialFreeModule):
         else:
             return self.zero()
 
-    # TODO: only makes sense if I'm unital.
-    def one(self):
-        r"""
-        SETUP::
-
-            sage: from mjo.matrix_algebra import MatrixAlgebra
-
-        """
-        return sum( (self.monomial((i,i,self.entry_algebra().one()))
-                     for i in range(self.nrows()) ),
-                    self.zero() )
-
     def from_list(self, entries):
         r"""
         Construct an element of this algebra from a list of lists of
@@ -255,6 +227,16 @@ class MatrixAlgebra(CombinatorialFreeModule):
 
             sage: from mjo.matrix_algebra import MatrixAlgebra
 
+        EXAMPLES::
+
+            sage: A = MatrixAlgebra(QQbar, ZZ, 2)
+            sage: A.from_list([[0,I],[-I,0]])
+            +----+---+
+            | 0  | I |
+            +----+---+
+            | -I | 0 |
+            +----+---+
+
         """
         nrows = len(entries)
         ncols = 0
@@ -265,13 +247,68 @@ class MatrixAlgebra(CombinatorialFreeModule):
             raise ValueError("list must be square")
 
         def convert(e_ij):
-            # We have to pass through vectors to convert from the
-            # given entry algebra to ours. Otherwise we can fail
-            # to convert an element of (for example) Octonions(QQ)
-            # to Octonions(AA).
-            return self.entry_algebra().from_vector(e_ij.to_vector())
+            if e_ij in self.entry_algebra():
+                # Don't re-create an element if it already lives where
+                # it should!
+                return e_ij
+
+            try:
+                # This branch works with e.g. QQbar, where no
+                # to/from_vector() methods are available.
+                return self.entry_algebra()(e_ij)
+            except TypeError:
+                # We have to pass through vectors to convert from the
+                # given entry algebra to ours. Otherwise we can fail to
+                # convert an element of (for example) Octonions(QQ) to
+                # Octonions(AA).
+                return self.entry_algebra().from_vector(e_ij.to_vector())
 
         return sum( (self.monomial( (i,j, convert(entries[i][j])) )
                      for i in range(nrows)
                      for j in range(ncols) ),
                     self.zero() )
+
+    def _element_constructor_(self, elt):
+        if elt in self:
+            return self
+        else:
+            return self.from_list(elt)
+
+
+class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
+    def is_hermitian(self):
+        r"""
+
+        SETUP::
+
+            sage: from mjo.matrix_algebra import HurwitzMatrixAlgebra
+
+        EXAMPLES::
+
+            sage: A = HurwitzMatrixAlgebra(QQbar, ZZ, 2)
+            sage: M = A([ [ 0,I],
+            ....:         [-I,0] ])
+            sage: M.is_hermitian()
+            True
+
+        """
+        return all( self[i,j] == self[j,i].conjugate()
+                    for i in range(self.nrows())
+                    for j in range(self.ncols()) )
+
+
+class HurwitzMatrixAlgebra(MatrixAlgebra):
+    Element = HurwitzMatrixAlgebraElement
+
+    def one(self):
+        r"""
+        SETUP::
+
+            sage: from mjo.matrix_algebra import HurwitzMatrixAlgebra
+
+        """
+        return sum( (self.monomial((i,i,self.entry_algebra().one()))
+                     for i in range(self.nrows()) ),
+                    self.zero() )
+
+

diff --git a/mjo/octonions.py b/mjo/octonions.py
index bd014c22eb4c180117cf70291e4f2357c091bd20..289c46a69b075f22409b69f827d7c8efbfee375d 100644 (file)
--- a/mjo/octonions.py
+++ b/mjo/octonions.py
@@ -374,8 +374,8 @@ class OctonionMatrixAlgebra(MatrixAlgebra):
         sage: O = Octonions(QQ)
         sage: e0,e1,e2,e3,e4,e5,e6,e7 = O.gens()
         sage: MS = OctonionMatrixAlgebra(2)
-        sage: MS.from_list([ [e0+e4, e1+e5],
-        ....:                [e2-e6, e3-e7] ])
+        sage: MS([ [e0+e4, e1+e5],
+        ....:      [e2-e6, e3-e7] ])
         +---------+---------+
         | e0 + e4 | e1 + e5 |
         +---------+---------+