X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fhurwitz.py;h=c213283099c2464316994a9d0fec7fef1f31c284;hb=e0e9c70961b93516c9423e370aab735aa8ab90c1;hp=57f65786ebf564e1f250d5c22c60e35dcd440c08;hpb=a87cbac5c3e1f3a6422b2152609f8209676fb4cb;p=sage.d.git

diff --git a/mjo/hurwitz.py b/mjo/hurwitz.py
index 57f6578..c213283 100644
--- a/mjo/hurwitz.py
+++ b/mjo/hurwitz.py
@@ -23,7 +23,6 @@ class Octonion(IndexedFreeModuleElement):
 
         Conjugating twice gets you the original element::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: x = O.random_element()
             sage: x.conjugate().conjugate() == x
@@ -58,7 +57,6 @@ class Octonion(IndexedFreeModuleElement):
 
         This method is idempotent::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: x = O.random_element()
             sage: x.real().real() == x.real()
@@ -91,7 +89,6 @@ class Octonion(IndexedFreeModuleElement):
 
         This method is idempotent::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: x = O.random_element()
             sage: x.imag().imag() == x.imag()
@@ -121,7 +118,6 @@ class Octonion(IndexedFreeModuleElement):
 
         The norm is nonnegative and belongs to the base field::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: n = O.random_element().norm()
             sage: n >= 0 and n in O.base_ring()
@@ -129,7 +125,6 @@ class Octonion(IndexedFreeModuleElement):
 
         The norm is homogeneous::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: x = O.random_element()
             sage: alpha = O.base_ring().random_element()
@@ -167,7 +162,6 @@ class Octonion(IndexedFreeModuleElement):
 
         TESTS::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: x = O.random_element()
             sage: x.is_zero() or ( x*x.inverse() == O.one() )
@@ -241,7 +235,6 @@ class Octonions(CombinatorialFreeModule):
 
         This gives the correct unit element::
 
-            sage: set_random_seed()
             sage: O = Octonions()
             sage: x = O.random_element()
             sage: x*O.one() == x and O.one()*x == x
@@ -306,6 +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.
@@ -366,6 +385,46 @@ class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
                     for j in range(self.ncols()) )
 
 
+    def is_skew_hermitian(self):
+        r"""
+
+        SETUP::
+
+            sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
+            ....:                          HurwitzMatrixAlgebra)
+
+        EXAMPLES::
+
+            sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+            sage: M = A([ [ 0,I],
+            ....:         [-I,1] ])
+            sage: M.is_skew_hermitian()
+            False
+
+        ::
+
+            sage: A = HurwitzMatrixAlgebra(2, AA, QQ)
+            sage: M = A([ [1, 1],
+            ....:         [1, 1] ])
+            sage: M.is_skew_hermitian()
+            False
+
+        ::
+
+            sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+            sage: M = A([ [2*I   ,  1 + I],
+            ....:         [-1 + I, -2*I] ])
+            sage: M.is_skew_hermitian()
+            True
+
+        """
+        # A tiny bit faster than checking equality with the conjugate
+        # transpose.
+        return all( self[i,j] == -self[j,i].conjugate()
+                    for i in range(self.nrows())
+                    for j in range(self.ncols()) )
+
+
 class HurwitzMatrixAlgebra(MatrixAlgebra):
     r"""
     A class of matrix algebras whose entries come from a Hurwitz
@@ -529,7 +588,6 @@ class OctonionMatrixAlgebra(HurwitzMatrixAlgebra):
 
     TESTS::
 
-        sage: set_random_seed()
         sage: A = OctonionMatrixAlgebra(ZZ.random_element(10))
         sage: x = A.random_element()
         sage: x*A.one() == x and A.one()*x == x
@@ -622,7 +680,6 @@ class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
 
     TESTS::
 
-        sage: set_random_seed()
         sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
         sage: x = A.random_element()
         sage: x*A.one() == x and A.one()*x == x
@@ -715,11 +772,11 @@ class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
         sage: (I,) = A.entry_algebra().gens()
         sage: A([ [1+I, 1],
         ....:     [-1, -I] ])
-        +-------+----+
-        | I + 1 | 1  |
-        +-------+----+
-        | -1    | -I |
-        +-------+----+
+        +---------+------+
+        | 1 + 1*I | 1    |
+        +---------+------+
+        | -1      | -1*I |
+        +---------+------+
 
     ::
 
@@ -732,7 +789,6 @@ class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
 
     TESTS::
 
-        sage: set_random_seed()
         sage: A = ComplexMatrixAlgebra(ZZ.random_element(10))
         sage: x = A.random_element()
         sage: x*A.one() == x and A.one()*x == x