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.
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::
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
::
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()) )