eja: test the nonnegativity of trace() for squares.

[sage.d.git] / mjo / hurwitz.py

diff --git a/mjo/hurwitz.py b/mjo/hurwitz.py
index 43ed8f20782c074022a3a0f46f68c7dad00dae5f..6767fe12fcd2d433c3b3a12324d6aab302d1b57b 100644 (file)
--- a/mjo/hurwitz.py
+++ b/mjo/hurwitz.py
@@ -299,6 +299,32 @@ class Octonions(CombinatorialFreeModule):
 
 
 class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
+    def conjugate(self):
+        r"""
+        Return the entrywise conjugate of this matrix.
+
+        SETUP::
+
+            sage: from mjo.hurwitz import ComplexMatrixAlgebra
+
+        EXAMPLES::
+
+            sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+            sage: M = A([ [ I,   1 + 2*I],
+            ....:         [ 3*I,     4*I] ])
+            sage: M.conjugate()
+            +------+----------+
+            | -I   | -2*I + 1 |
+            +------+----------+
+            | -3*I | -4*I     |
+            +------+----------+
+
+        """
+        entries = [ [ self[i,j].conjugate()
+                      for j in range(self.ncols())]
+                    for i in range(self.nrows()) ]
+        return self.parent()._element_constructor_(entries)
+
     def conjugate_transpose(self):
         r"""
         Return the conjugate-transpose of this matrix.
@@ -359,8 +385,9 @@ class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
                     for j in range(self.ncols()) )
 
 
-    def is_skew_hermitian(self):
+    def is_skew_symmetric(self):
         r"""
+        Return whether or not this matrix is skew-symmetric.
 
         SETUP::
 
@@ -372,15 +399,23 @@ class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
             sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
             sage: M = A([ [ 0,I],
             ....:         [-I,1] ])
-            sage: M.is_skew_hermitian()
+            sage: M.is_skew_symmetric()
             False
 
+        ::
+
+            sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+            sage: M = A([ [   0, 1+I],
+            ....:         [-1-I,   0] ])
+            sage: M.is_skew_symmetric()
+            True
+
         ::
 
             sage: A = HurwitzMatrixAlgebra(2, AA, QQ)
             sage: M = A([ [1, 1],
             ....:         [1, 1] ])
-            sage: M.is_skew_hermitian()
+            sage: M.is_skew_symmetric()
             False
 
         ::
@@ -388,13 +423,13 @@ class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
             sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
             sage: M = A([ [2*I   ,  1 + I],
             ....:         [-1 + I, -2*I] ])
-            sage: M.is_skew_hermitian()
-            True
+            sage: M.is_skew_symmetric()
+            False
 
         """
-        # A tiny bit faster than checking equality with the conjugate
-        # transpose.
-        return all( self[i,j] == -self[j,i].conjugate()
+        # A tiny bit faster than checking equality with the negation
+        # of the transpose.
+        return all( self[i,j] == -self[j,i]
                     for i in range(self.nrows())
                     for j in range(self.ncols()) )